Quantum Physics in the Sinhala Buddhist culture
Nalin de Silva
Department of Mathematics, University of Kelaniya, Kelaniya
e-mail: nalink@kln.ac.lk
Department of Mathematics, University of Kelaniya, Kelaniya
e-mail: nalink@kln.ac.lk
Abstract
It is shown that neither the wave picture nor the ordinary particle picture
offers a satisfactory explanation of the double–slit experiment. The western
Physicists who have been successful in formulating theories in the
Newtonian Paradigm based in their culture find it difficult to interpret
Quantum Physics which deals with particles that are not sensory perceptible.
A different interpretation of Quantum Physics based in the Sinhala Buddhist
culture is presented in what follows. According to the new interpretation
Quantum particles have different properties from those of Classical
Newtonian particles. The interference patterns are explained in terms of
particles each of which passes through both slits.
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Quantum Physics in the Sinhala Buddhist culture
INTRODUCTION
Planck introduced his ideas on quanta or packets of energy towards the end of the
nineteenth century. In that sense Quantum Physics is more than one hundred years old.
From the very beginning Quantum Physics came up with strange phenomena that made
the western Physicists to disbelieve what they themselves were proposing to understand
the new features that were being observed.
The so-called double-slit experiment1 continues to baffle the western Physicists who are
glued to twofold two valued logic that is behind western thinking. As it was one of the
most fundamental experiments that they could not understand in Quantum Physics the
Nobel Prize winning Physicist Richard Feynmann once declared that no body understood
Quantum Physics! This statement by Feynmann makes one to delve into the meaning of
understanding. In other words one has to understand what is meant by understanding.
However, it is clear that if one is confined to twofold formal logic, and linear thinking
one would be confused by a statement such as understanding what is meant by
understanding. A decade ago the western intellectuals who are only familiar with linear
thinking and not with cyclic thinking would have left deliberations into such statements
to whom they call mystics, as such statements did not come within their “rational" way of
thinking.
The principle of superposition which was familiar to Classical Physicists as well, has
taken an entirely different meaning with respect to Quantum Physics. The essence of the
principle can be explained as follows. If x and y are two solutions of what is called a
linear differential equation then x+y is also a solution of the same differential equation.
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This is a simpler version of what is generally known as the principle of superposition. In
Classical Physics two magnets giving rise to two different magnetic fields would
combine to give one magnetic field, and a compass that is brought to the resulting
magnetic field would respond to the resulting field, and not to the field of any one of the
magnets. It has to be emphasised that a magnet is in only one state, corresponding to the
respective magnetic field and it is the two fields of the two magnets that combine to give
one field though one would not find a single magnet that gives rise to the resultant field.
We could describe this phenomenon as that of two or more becoming one. However, in
the Quantum world things are different, and the principle of superposition has an unusual
interpretation.
THE WAVE NATURE OF PARTICLES
In order to discuss the new interpretation of the principle of superposition we first
consider the so called double-slit experiment where a stream of electrons (in general
particles or photons) is made to pass through two slits and then to strike a screen. If both
slits are open an interference pattern is observed on the screen. Now in Quantum Physics
it is said that particles such as electrons posses wave properties and photons (light)
exhibit particle properties in addition to their normal properties. Interference patterns are
supposed to result from wave properties and according to the western Physicists the wave
theory successfully explains the formation of such patterns in the case of a stream of
particles fired from a source to strike the screen after passing through the slits. The
western Physicists would claim that the double-slit experiment demonstrates that particles
such as electrons do exhibit wave properties.
The double-slit experiment has been carried out with only one electron passing through
the slits one at a time2 (electrons at very low intensities) instead of a stream of particles
released almost simultaneously to pass through the two slits. Even at very low intensities
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interference patterns have been observed after sufficiently large number of electrons had
been fired from the source (Figure 1). The western Physicists have been puzzled by this
phenomenon. In the case of several electrons passing through the slits simultaneously it
could be explained using the wave properties of the particles, in other words resorting to
the wave picture. Unfortunately in the case of electrons being shot one at a time this
explanation was not possible as what was observed on the screen was not a faint
interference pattern corresponding to one electron but an electron striking the screen at a
single point on the screen. These points in sufficiently large numbers, corresponding to a
large number of electrons, finally gave rise to an interference pattern. The wave nature is
only a way of speaking, as even in the case of large number of particles what is observed
is a collection of points and not waves interfering with each other.
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Figure 1: The interference pattern produced by a single-photon source with (a) 30, (b) 300, and
(c) 3000 photons registered. In contrast, the decoherent distribution of (d) 30, (e)
300, and (f) 3000 photons lacks the dark fringes. (Courtesy Afshra)
The western Physicists also believe that an electron as a particle could pass through only
one of the slits and a related question that has been asked is whether it was possible to
find out the slit through which an electron passes on its way to the screen. Various
mechanisms, including “capturing” the electron using Geiger counters, have been tried to
“detect the path” of the electron, and it has been found that if the particular slit through
which the electron passed was detected then the interference patterns were washed out. In
other words determining the particle properties of the electron erased its wave properties.
Bohr, who was instrumental in formulating the Copenhagen interpretation3, was of the
view that one could observe either the particle properties or the wave properties but not
both, and the inability to observe both wave and particle properties simultaneously came
to be referred to as complementarity. The experiments that attempted to determine the slit
through which the electron passed were known as which-way (welcherweg) experiments
as they attempted to find the way or the path of the particle from the source to the screen.
The outcome of these experiments made it clear that the which-way experiments washed
out the interference patterns. It was believed that at any given time the electrons exhibited
either the particle properties or wave properties but not both.
However, what the western Physicists failed to recognize was that in the case of one
electron shot at a time there was no weak interference pattern observed on the screen for
each electron thus illustrating that a single electron did not exhibit any wave properties.
The electron strikes the screen at one point, and it is the collection of a large number of
such points or images on the screen that gave the interference pattern. In the case of a
stream of electrons fired to strike the screen each electron would have met the screen at
one point and the collection of such points or images would have given rise to an
interference pattern. Thus we could say that the interference patterns are obtained not as a
result of the “wave nature” of electrons but due to the collectiveness of a large number of
electrons that strike the screen. The “wave nature” arises out of “particle” properties and
not due to “wave properties”. Afshar4 comes closer to this view when he states “in other
words, evidence for coherent wave-like behavior is not a single particle property, but an
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ensemble or multi-particle property”. We are of the opinion that in the double-slit
experiments no wave properties are observed contrary to what is generally believed. It is
the particle properties that are observed, though not necessarily those of ordinary
classical particles.
As a case in point this does not mean that a particle in Quantum Physics has a definite
path from the source to the screen through one of the slits, as could be expected in the
case of classical particles. For a particle to have a path it should posses both position and
momentum simultaneously. A path at any point (assuming that it is a continuous path)
should have a well defined tangent. In the case of a particle moving, the direction of the
velocity (and the momentum) of the particle at any given point defines the unit tangent
vector to its path. Conversely the tangent to the path at any point defines the direction of
the velocity and the momentum of the particle at that point. However, according to the
Uncertainty Principle, both the momentum and the position of a particle cannot be
determined simultaneously, and if the position is known then the momentum cannot be
determined. Without the momentum the direction of the velocity of the particle and hence
the tangent vector cannot be known implying that a continuous curve is not traced by a
particle in space. On the other hand if the momentum of the particle is known then only
the direction and magnitude of the velocity (momentum) and properties of other non
conjugate observables such as spin of the particle are known, without the position being
known. Thus the particle can be everywhere, with variable probabilities of finding the
particle at different points, but at each point the particle being moving in parallel
directions with the same speed. However, as will be explained later, this does not mean
that we could observe the particle everywhere.
In the light of the uncertainty principle it is futile to design experiments to find out the
path of a particle. The so-called which-way experiments have been designed to detect the
slit through which the particle moves, on the assumption that the particle moves through
one slit only. The which-way experiment actually stops the particle from reaching the
screen and hence there is no possibility of obtaining any interference pattern. It is not a
case of observing particle properties destroying the wave properties of matter, but an
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instance of creating a situation where the particle is either not allowed to strike the screen
or pass through only one slit deliberately. In effect it is the particle properties exhibited
at the screen that are cut off.
What is important is to note that interference patterns are observed only if both slits are
kept open, and if the particles are free to reach the screen. If one slit is closed or obstacles
are set up in the guise of which-way experiments or otherwise, so as not to allow the
particles to reach the screen then no interference patterns are observed. The most
important factor is the opening of the two slits. In the case of which-way experiments as
well, what is effectively done is to close one of the slits as particles through that slit are
not allowed to reach the screen. With only one slit open while the other slit is effectively
closed with the which-way experiment apparatus, no interference patterns are observed.
The probability of an electron striking the screen at a given point with only one slit open,
is not the same as that when both slits are open. Thus when a large number of particles
strike the screen, the different probabilities give rise to different “patterns” which are
essentially collection of points where the particles meet the screen. The “interference
patterns” observed when both slits are open is replaced by “chaotic patterns” when one of
the slits is closed (Figure 1 – interference and decoherence). The “interference patterns”
as well as the “chaotic patterns” are the results of particle properties, the difference being
due to the number of slits that are open. If both slits are closed there is no pattern at all as
no particle would reach the screen under such conditions. When one of the slits is open
the particle can be only at that position where the slit is whereas when both slits are open
there is a probability that the particle could be at both the slits “before” reaching the
screen.
The western Physicists are obsessed with the idea that a particle can be only at one
position at a given time. While this may be the experience with our sensory perceptible
particles (objects) or what we may call ordinary Newtonian classical objects such as
billiard balls, it need not be the case with Quantum particles. However, from the
beginning of Quantum Physics, it appears that the western Physicists have been of the
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view that a particle can be at one position at a given time whether it is being observed or
not. Hence they seem to have assumed that on its “journey to the screen from the source”
a particle could pass through only one of the slits. They have worked on the assumption
that even if both slits are open the particle passes through only one of the slits but
behaves differently to create interference patterns as if the particle is “aware” that both
slits are open. According to the view of the western Physicists if only one slit is open the
particle having “known” that the other slit is closed passes through the open slit and
“decides” not to form any interference patterns. It is clear that the explanation given by
the western Physicists for the formation of interference patterns on the basis of the
particle picture is not satisfactory. We saw earlier that the explanation given in the wave
picture is also not satisfactory as a single electron fired from the source does not form a
faint interference pattern on the screen. If the particles behave like waves then even a
single particle should behave like a wave and produce a faint interference pattern, having
interfered with itself. What is emphasised here is that the final interference pattern is not
the sum of faint interference patterns due to single particles, but an apparent pattern
formed by a collection of images on the screen due to the particles. There is no
interference pattern as such but only a collection of the points where the particles strike
the screen, or of the images formed by the particles that were able to reach the screen.
The images finally depend on the probability that a particle would be at a given position.
Before we proceed further a clarification has to be made on “seeing” a particle at a given
position at a given time in respect of the double-slit experiment. In this experiment we are
concerned with particles released from a source with a given momentum and given
energy. As such according to the uncertainty principle, nothing can be said definitely on
the position of these particles. It can only be said that there is a certain probability that the
particle would be found in a certain position. Thus the particle is “everywhere” “until” it
is “caught” at some position such as a slit or a screen. Though we have used the word
“until”, time is not defined as far as the particle is concerned as it has a definite energy. It
can only be said that there is a certain probability that the particle could be “seen” at a
given place at a given time, with respect to the observer. The particle is not only
everywhere but also at “every instant”. Thus it is meaningless to say the particle is at a
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given slit at a given time as neither time nor position is defined for the particle with
respect to itself. The particle would meet the screen at some position on the screen at
some time but “before” that it was everywhere and at every instant.
EXPERIMENTS OF AFSHAR
Afshar5 has claimed that he was able to demonstrate that an electron or a photon would
exhibit both particle and wave properties (Figure 2 – on next page). He allowed light to
pass through two slits and to interact with a wire grid placed so that the nodes were at the
positions of zero probability of observing a photon. The photons were not affected by the
wire grid as the nodes were at the positions of zero probability and at those positions
there were no photons to interact with the grid. The photons were then intercepted by a
lens system that was able to identify the slit through which any single photon had passed.
According to Afshar the nodes of the grid at the positions of zero probability indicated
that the wave properties of the photons were observable while the lens system in
detecting the slit through which the photon had passed demonstrated the particle
properties of the photons.
However, in this experiment, assuming that the lens system detects the slit through which
the photon passed, what is observed is again the particle properties of the photons. The
wire grid with the nodes at the position of zero probabilities does not interact with the
photons, as there are no photons at positions of zero probability to interact with the grid.
No so called waves are observed, as there is no screen for the particles to strike. Thus the
wire grid has no effect in this experiment and with or without such a grid the lens system
would behave the same way.
Let us consider what would happen if the wire grid is shifted forwards towards the
source, backwards towards the lens system or laterally. As the nodes of the wire grid
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would be shifted from the positions of zero probability some photons would strike the
grid and they would not proceed towards the lens system. Thus the number of photons
that reach the lens system would be reduced and there would be a decrease in intensity of
light received at the lens. Though Afshar claims that wave properties are observed just by
placing a wire grid so that its nodes are at the positions of zero probability, it is not so.
Figure 2: The wire grid and the lens system of Afshra, and the corresponding images observed.
No interference patterns after the lenses and Afshra claims that the wire grid
demonstrates the wave property while the images correspond to the particle property.
(Courtesy Afshra)
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The so called wave properties could be observed only by placing a screen in between the
wire grid and the lens system. As we have mentioned above, even then what is observed
is a collection of images at the points where the photons strike the screen, and not wave
properties as such. In this case as all the photons would have been absorbed by the
screen, the lens system would not be able to detect any photons nor the “slit through
which the photons passed”. On the other hand if the screen is kept beyond the lens system
then there would not be any photons to strike the screen and hence no “wave properties”.
EXPERIMENTS AT KELANIYA
We at the University of Kelaniya have given thought to this problem, and Suraj
Chandana, one of my students has carried out a number of experiments, which may be
identified as extensions of the experiment of Afshar. Chandana and de Silva6 had
predicted that if we were to have a single slit and then a screen, instead of the wire grid
and the lens system, “after” the photons have passed through the two slits, then the
photons would pass through the single slit with the same probability as that of finding a
photon at the point where the slit was kept. This implied that if the slit was kept at a point
where the probability of finding the photon is zero, the photon would not pass through the
slit to strike the screen, but on the other hand, if the slit was kept at any other point there
was a non zero probability that the photon would pass through the slit, and striking the
screen. This implies that if a stream of photons is passed through two slits, and “then” a
single slit, to strike the screen, depending on the position of the single slit the intensity
with which the photons strike the screen would change. Further it implies that these
intensities should correspond to the intensities observed in connection with the
interference patterns observed in the case of the standard double-slit experiment, if the
positions of slit were along a line parallel to the double-slits and the screen. Chandana has
been successful in obtaining the results as predicted.
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In another experiment Chandana7 had an Aluminium sheet of very small thickness
joining the points or positions where the probability of finding a photon is zero (positions
of zero probability), stretching from the double-slits to the screen as illustrated in the
figure 3. As an obstacle placed at a position of zero probability would not affect the
photon the Aluminium sheet had no effect on the visible interference patterns on the
screen. This experiment was carried out by Chandana with number of Aluminium sheets
placed along lines joining the positions of zero probability stretching from the doubleslits
to the screen. We were not surprised to find that the Aluminium sheets did not
interfere with the interference patterns. However, even if one of the sheets is slightly
displaced the interference pattern is destroyed as the photons now interact with the sheets
at points where the probability of finding a photon is not zero.
Figure 3: The figure represents the aluminium sheet joining the positions of zero probability
from a position closer to the double slit to the screen.
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These observations are not consistent with the wave picture as a wave would not be able
to penetrate the Aluminium sheets without being affected. Even the pilot waves of Bohm8
are not known to go through a material medium undisturbed. As we have argued a single
electron emitted from the source would not exhibit a faint interference pattern on the
screen but a spot or an image having passed beyond the slits. The western Physicists are
interested in the wave picture to explain the interference patterns as they find it difficult
to believe that a particle would pass through both slits simultaneously. Thus they mention
of particle properties when they are interested in “capturing” particles and of wave
properties in explaining phenomena such as the interference pattern.
PRINCIPLE OF SUPERPOSITION IN QUANTUM PHYSICS
We consider the Quantum entities to be particles though of a nature different from that of
Classical Newtonian particles. We have no inhibition in believing that the Quantum
particles unlike the Newtonian particles could pass through both slits at the “same time”,
as the logic of the Sinhala Buddhist culture permits us to do so. Western Physics and in
general western Mathematics and sciences are based on Aristotelian two valued twofold
logic according to which a proposition and its negation cannot be true at the same time.
Thus if a particle is at the slit A, the proposition that the particle is at A is true and its
negation that the particle is not at A is not true, and vice versa. Therefore if the particle is
at A then it cannot be anywhere else as well, and hence cannot be at B. However, in
fourfold logic (catuskoti) a proposition and its negation can be both true, and hence in
that logic it is not a contradiction to say that a particle is at the slit A and at somewhere
else (say at the slit B) at the “same instant” or “every instant” Thus according to catuskoti
the particle can be at many places at the same time with respect to the observer.
In the case of the double-slit experiment, the momentum of a particle is known, as the
particles are fired with known energies, and hence the position is not known. In such a
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situation Heisenberg’s uncertainty principle demands that the position of the particle is
not known. The position of the particle is relieved only after a measurement is made to
determine the position. Before the measurement, the particle is in a superposition of
states corresponding to the positions in space the particle could be found. After the
measurement the particle would be found in a definite position (state), having collapsed
from the superposition of a number of states to that of the definite state. Before the
measurement what could have been said was that there was a certain probability of
finding the particle at a given position. Though the particle is in a superposition of states
before a measurement is made to find the position, it is in a definite state with respect to
the momentum.
In Quantum Mechanics unlike in Classical Mechanics, a state of a system, a particle or an
object is represented by a vector in a Mathematical space known as the Hilbert space. The
observables such as position, momentum, and spin are represented by what are known as
Hermitian operators. If a system is in a state represented by an eigenstate ׀> of a
Hermitian operator A, belonging to the eigenvalue a, then the system has the value a
corresponding to the observable represented by the Hermitian operator A. This is
expressed mathematically by A ׀> = a ׀>. If B is the conjugate operator of A, then the
value corresponding to the observable represented by B is not known. All that can be
said, according to the standard Copenhagen interpretation, is that if the value
corresponding to the observable represented by B is measured, then there is a certain
probability of obtaining an eigenvalue of B as that value. Before the measurement is
made nothing could be said of the value. In plain language this means that if the value of
a certain observable is known then the value of the conjugate observable is not known.
However, the state ׀> can be expressed as a linear combination of the eigenstates ׀> of
B in the form F >=Σ > i i cy where Î i c C , the field of complex numbers. In other
words the coefficients of ׀>’s in the expansion of ׀> are complex numbers. The
Copenhagen interpretation tells us that when the observable corresponding to B is
measured it would result in a state corresponding to one of the ׀>’s with the
measurement yielding the eigenvalue b to which the particular ׀> belongs, the
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probability of obtaining the value b being given by the value of the relevant 2 c . Before
the measurement is made nothing can be said regarding the observable corresponding to
B. According to Bohr, it is meaningless to talk of the state of the system with respect to B
as nothing could be observed. There is no knowledge regarding the observable
corresponding to B as it has not been observed. The value or the knowledge of the
observable is “created” by the observer who sets up an experiment to measure the value
in respect of B. The observed depends on the observer and it makes no sense to talk of an
observable unless it has been observed. This interpretation is rooted in the positivism as
opposed to the realism in which the entire corpus of knowledge in Newtonian Physics is
based.
As a particular case one could refer to the conjugate Hermitian operators in respect of
position and momentum of a particle in Quantum Mechanics. When the position of a
particle is measured then its momentum is not known. According to the Copenhagen
Interpretation, it can only be said that if an apparatus is set up to measure the momentum,
the observer would observe one of the possible values for the momentum and that there is
a certain probability of observing the particular value. Before the measurement is made
the particle has no momentum, as such, and it is meaningless to talk of the momentum of
the particle. The observer by his act of observation gives or creates a value for the
momentum of the particle, so to speak of. Once the momentum is measured the observer
has knowledge of the momentum but not before it. However, after the momentum is
measured, the knowledge of the position of the particle is “washed off” and hence it
becomes meaningless to talk of the position of the particle. The observer could have
knowledge only of either the momentum or the position, but not of both. A version of
this conclusion is sometimes referred to as the uncertainty principle.
What we have been discussing in the proceeding paragraphs is the principle of
superposition. A particle or a system with its position known is represented by a vector
׀> in Hilbert space, which is an eigenvector of the Hermitian operator A corresponding
to the position. As the position of the particle or the system is known, the momentum is
not known. If B is the Hermitian operator corresponding to the momentum, then lF> is
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not an eigenvector of B. However, ׀> can be expressed as a linear combination of the
eigenvectors ׀>’s of B though the momentum is not observed. The superposition of the
׀>’s cannot be observed, and neither can be resolved into observable constituent parts.
This is different from the principle of superposition in Classical Physics, where the
resultant can be resolved into its constituent parts.
For example as we have mentioned in the introduction the resultant magnetic field due to
two magnets can be resolved into its two components and can be observed. One of the
magnets can be taken off leaving only one of the constituent magnetic fields. The
superposition is there to be observed and if the magnet that was taken off is brought back
to its original position the resultant magnetic field reappears. In Quantum Physics the
superposition cannot be observed without disturbing the system and when it is disturbed
to measure the conjugate variable, only one of the states in the superposition could be
observed and we would not have known in advance if that particular state were to appear
as a result of the disturbance induced by us.
COPENHAGEN INTERPRETATION
In Classical Physics, as we have already stated, superposition is there to be observed.
However, in Quantum Physics the superposition cannot be observed, and further unlike in
Classical Physics interpretations are required to “translate” the abstract Mathematical
apparatus and concepts into day to day language. In Classical Physics one knows what is
meant by the position or the momentum of a particle and those concepts can be observed
and understood without an intermediate interpretation. However, in Quantum Physics, the
state of a particle or a system is represented by a vector in Hilbert space and observables
are represented by Hermitian operators in Hilbert space. An interpretation or
interpretations are needed to express these and other concepts to build a concrete picture
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out of the abstract apparatus. Copenhagen interpretation is one such interpretation and it
is the standard interpretation as far as the west is concerned.
Bohr more than anybody else was instrumental in formulating the Copenhagen
interpretation, and he in turn was influenced by positivism and Chinese Ying - Yang
Philosophy. As a positivist he believed that only the sensory perceptible phenomena exist
and did not believe in the existence of that could not be “observed”. When a state of a
particle or system is represented by an eigenvector of an observable (Hermitian operator
in Hilbert space) the corresponding value of the observable can be measured and the
positivist school had no problem in accepting the existence of such state. For example if
the momentum of a particle is known then the state of the particle is represented by a
certain vector in Hilbert space, belonging to the particular eigenvalue that has been
measured. However, the problem arises when the conjugate Hermitian operator, in this
case the position, is considered, as in positivism the ontology is connected with
observations and sensory perceptions.
As we have seen a given eigenstate of a Hermitian operator that has been observed can be
expressed as a linear combination of the eigenstates of the conjugate operator. To a
positivist, though the given eigenstate exists as it is observed, the eigenstates of the
conjugate operator are not observable and it is meaningless for him to talk of such states.
Thus if the momentum of a particle has been measured, the eigenstates belonging to the
eigenvalues of the conjugate operator, which is the position, are not observed and the
positivist would not say anything regarding the existence of such states. As far as the
positivist is concerned, there is only a probability of finding the particle at some position,
and the particle will be at some position only after the relevant measurement is carried
out.
In the case of the double-slit experiment, this means that a positivist would not say
whether the particle passes through a particular slit as it is not observed. However he
assumes that it passes through one of the slits and not both as the western thinking
demands that the particle should be at one of the slits and not at both slits. If a
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measurement is made, that is if an experiment is carried out to find out the slit through
which the particle passes, then the particle would be found at one of the slits washing out
the “interference pattern”. Then superposition is collapsed and “decoherance” sets in
resulting “chaotic pattern”.
A realist differs from a positivist in that the former would want to know the slit through
which the particle passes even without observing it. He would say the particle would pass
through one of the slits whether one observes it or not, and that it is an integral property
of the particle independent of the observer. The Classical Physicists were realists. An
object in Classical Physics has a momentum whether it is measured or not. The observer
in Classical Mechanics measures the momentum that the particle already possesses. In
Quantum Physics the positivists would say that the particle has no momentum before it is
measured but acquires a momentum as a result of the measurement. We would not go
into further details on the differences between the realist position and the positivist
position as it is outside the scope of this article. However, what is relevant to us is that
both the realist and the positivist would agree that the particle goes through one slit,
meaning that at a “given time” the particle is found only at one of the slits. They would
also agree on the wave nature of the particles. They have to depend on the wave nature as
they assume that the particle passes through only one slit, and as such they would not be
able to explain the “interference patterns” without the wave properties of the particles.
A NEW INTERPRETATION
We differ from the positivists as well as the realists since we believe that the particle is
found at both slits and hence pass through both. In general we include the postulate that
the eigenstates ׀>’s in F >=Σciy i > exist in addition to ׀> (Postulate 3 below). We
have also introduced the concept of a mode. A mode of a particle or a system is
essentially a potential observable. A mode has the potential to be observed though it may
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not be observed at a particular instant. For example, position, momentum, spin are
modes. A particle or a system can be in both modes corresponding to two conjugate
Hermitian operators, though only one mode may be observed.
A revised version of the postulates of the new interpretation formulated by Chandana and
de Silva9 is given below.
1. A state of a Quantum Mechanical system is represented by a vector (ray) c in the
Hilbert space, where c can be expressed as different linear combinations of the
eigenvectors in the Hilbert space, of Hermitian operators, any operator
corresponding to a mode. In other words a state of a Quantum Mechanical system
can be represented by different linear combinations of eigenvectors of different
modes, each linear combination being that of the eigenvectors of one of the modes.
Thus a state could have a number of modes, each mode being a potential
observable.
2. If c is expressed as a linear combination of two or more eigenvectors of a
Hermitian operator, that is a mode, then the corresponding mode cannot be
observed (or measured) by a human observer with or without the aid of an
apparatus. In other words the particular mode cannot be observed and a value
cannot be given to the observable, which also means that no measurement has
been made on the observable.
3. However, the non observation of a mode does not mean that the mode does not
“exist”. We make a distinction between the “existence” of a mode, and the
observation of a mode with or without the aid of an apparatus. A mode
corresponding to a given Hermitian operator could “exist” without being observed.
The knowledge of the “existence” of a mode is independent of its observation or
measurement. In other words the knowledge of the “existence” of a mode of a
Quantum Mechanical state is different from the knowledge of the value that the
observable corresponding to the relevant Hermitian operator would take.
20
4. If a mode of a Quantum Mechanical state is represented by a single eigenvector,
and not by a linear combination of two or more eigenvectors, of a Hermitian
operator, then the mode could be observed by a human observer with or without
the aid of an apparatus, and the value of the corresponding observable (or the
measured value) is given by the eigenvalue which the eigenvector belongs to. It
has to be emphasised that only those modes of a Quantum Mechanical state, each
represented by a single eigenvector, and not by a linear combination of
eigenvectors, of an Hermitian operator can be observed at a given instant.
5. If a mode of a Quantum Mechanical state is represented by an eigenvector of a
Hermitian operator then the mode corresponding to the conjugate operator cannot
be represented by an eigenvector of the conjugate Hermitian operator. It can be
expressed as a linear combination of two or more of the eigenvectors of the
conjugate operator. This means that the mode corresponding to the conjugate
operator cannot be observed, or in other words it cannot be measured. However,
the relevant mode “exists” though it cannot be observed.
6. It is not necessary that at least one of the modes corresponding to two conjugate
operators should be represented by a single eigenvector of the relevant operator. It
is possible that each mode is represented by linear combinations of two or more
eigenvectors of the corresponding operator. In such situations neither of the modes
could be observed.
7. A state of a Quantum Mechanical system can be altered by making an operation
that changes a mode or modes of the state. However, not all operations correspond
to measurements or observations. Only those operations that would result in a
mode being expressed as a single eigenvector, and not as a linear combination of
the eigenvectors of an operator would result in measurements.
21
8. A particle entangled with one or more other particles is in general represented by a
linear combination of eigenvectors of an Hermitian operator with respect to a
mode, while the whole system of particles is in general represented by a linear
combination of the Cartesian products of the eigenvectors. In the case of two
particles it takes the form S cij ׀φi› ׀φj›. If one of the particles is in a mode that is
observed, then the particles entangled with it are also in the same mode as an
observable. If a measurement is made on some other mode then instantaneously,
the corresponding values in the same mode of the entangled particles are also
determined. In such case, for two particles the whole system is represented by
vectors of the form ׀φi› ׀φ j›. If the number of entangled particles is less than the
dimension of the space of the eigenvectors of the Hermitian operator, then if a
measurement is made in the particular mode, the particle would be represented by
one of the eigenvectors, while the other particles entangled with it would be each
represented by a different eigenvector of the Hermitian operator. However, if the
number of entangled particles is greater than the dimension of the space of the
eigenvectors, then in some cases, more than one particle would be represented by a
given eigenvector.
According to this interpretation if the momentum of a particle is known then it has not
one position but several positions. In other words the particle can be at number of
positions in superposition though we are not able to observe it at any one of those
positions. The particle could be observed only if it is at one position. If an experiment is
carried out to determine the position of the particle the superposition or the wave function
would collapse, and the particle would be located at one of the positions where it was
before the measurement was made.
Similarly if the particle is in the position mode that is observed then it can have several
momenta in superposition but we would not be able to observe any one of them. If we
perform an experiment to determine the momentum, that is if a measurement is made,
22
then the superposition of momenta would collapse to one of them, enabling us to
determine the value of the momentum.
With respect to the double-slit experiment this implies that the particle is at both slits in
superposition without being observed and if we perform an experiment to determine the
slit through which the particle passes then the superposition collapses and the particle
would be found only at one of the positions. The positivists while assuming that the
particle passes through only one slit would not say anything on the slit through which the
particle passes as it cannot be observed. For the positivist it is meaningless to speculate
on something that cannot be observed. The realists too assume that the particle passes
through only one slit but would not be satisfied with the positivist position, and claim that
a theory that is not able to determine the slit through which the particle passes is
incomplete.
We make a distinction between existence and being observed. A particle or a system can
exist in a certain mode without being observed. In this case the state of the particle or the
state is expressed as a linear combination or superposition of the eigenstates of the
relevant Hermitian operator and the particle or the system exists in all the relevant
eigenstates without being observed. The mode is observed only when the state of the
particle or the system is expressed as a single eigenstate of the relevant Hermitian
operator.
The existence of modes with more than one eigenstates has been known for sometime.
Monroe10 and his colleagues in 1996 were able to demonstrate the existence of two spin
states of Beryllium cation simultaneously however without observing them. One could
say that the interference obtained by them could be understood on the basis of the
existence of simultaneous spin states of the Beryllium cation. Since then similar
experiments have been carried out and the existence of superposition of eigenstates
cannot be ruled out anymore.
23
SINHALA BUDDHIST ONTOLOGY AND LOGIC
In Sinhala Buddhist ontology no distinction is made of the existence of sensory
perceptible objects and of other entities. There is no absolute existence as such and all
existences are relative to the mind. It has been shown by de Silva11 that even the mind is
a creation of the mind a phenomenon not in contradiction with cyclic thinking or
cakreeya cintanaya. It is the mind that creates concepts including that of self, and as such
sensory perceptible objects do not have any preference over the others.
As we have mentioned the positivists find it difficult to take cognizance of entities that
are not sensory perceptible and it is this ontology that makes them not to commit on the
existence of unobserved “objects”. In Sinhala Buddhist ontology all existences are only
conventional or sammuti and not absolute or paramarta. Thus the existence of
simultaneous eigenstates or superposition of eigenstates is not ruled out in Sinhala
Buddhist ontology. The Sinhala Buddhist culture has no inhibition to postulate the
existence of such states and it is not in contradiction with catuskoti or fourfold logic that
may be identified as the logic of the Sinhala Buddhist culture.
As Jayatilleke12 has shown in fourfold logic unlike in twofold logic a proposition and its
negation can be both true or false. In twofold logic if a proposition is true then its
negation is false, and if a proposition is false, then its negation is true. Thus the
proposition that a particle is at A, and the proposition that a particle is not at A, can be
both true in fourfold logic. We may deduce from that a particle can be both at A and B at
the “same time”. In other words a particle can be at both slits in respect of the double-slit
experiment, and in general a mode represented by a superposition of two or more than
24
two eigenvectors can exist as the particle or the system can be at number of “positions”
simultaneously in fourfold logic.
In twofold Aristotelian logic of the west a particle has to be either at A or not at A. Thus
the western Physicists whether they are realists or positivists find it difficult to accept that
a particle can pass through both slits simultaneously, and they have to resort to so called
wave nature in order to explain the interference patterns.
The Sinhala language reflects the use of fourfold logic in expressions such gevi nogevi,
adu vediya, yana ena, where the opposites are combined as samasa without the use of a
word equivalent to or in English. This is a result of threefold logic where both a
proposition and its negation can be true. I understand that this usage is found in some
Indian languages as well, as threefold logic is present in the Vedic culture. In fourfold
logic we have another case where a proposition and its negation can be both false. The
famous statement na ca so na ca anno in the “Milinda Prashnaya” meaning that neither
oneself nor somebody else is reborn is an example for an instance where the fourth case
in fourfold logic has been used. Perhaps a more familiar example is the answer given by
King Devanampiya Tissa to Arhant Mahinda. The king answering the Arhant Thera said
that he was neither a relative nor a non relative of himself.
DISCUSSION
It is seen that both wave picture and the ordinary particle picture fail to explain the
interference patterns observed in the double-slit experiment. The wave picture fails as a
weak intensity stream of electrons (one electron at a time) exhibits no interference
patterns in the case of few electrons. The ordinary particle picture fails as a particle
passing through only one slit would not produce interference patterns. The western
25
Physicists had to resort to the wave picture as the logic in their culture would not permit a
particle to pass through both slits.
In the case of the experiments conducted by Chandana at the University of Kelaniya the
wave picture as well as the particle picture come across more problems as neither a wave
nor an ordinary particle would be able to penetrate the aluminium sheets without being
affected. These experiments justify our new interpretation involving modes of the particle
or the system and the particle picture presented here where a particle can pass through
both slits. In general we postulate that a particle or system can exist in a mode where
more than one eigenstates are in a superposition. The position where a particle is found
depends only on the relevant probability, and the so-called interference patterns are only
collections of images formed by such particles striking the screen at different positions
with the relevant probabilities.
The new postulates are based in the Sinhala Buddhist culture and are consistent with the
Sinhala Buddhist ontology and the fourfold logic. It appears that, unlike Classical Physics
with its twofold logic and realist ontology, Quantum Physics is rooted in the Sinhala
Buddhist ontology and logic and we should be able to develop new concepts in Quantum
Physics, especially regarding the motion of a Quantum particle from a point A to another
point B. It is not known how a particle “moves” from the double-slit to the screen in the
experiments carried out by Chandana, nor how a particle with less energy than the value
of a potential barrier “scales the walls”. It may be that it is neither the particle that left the
point A nor some other particle that reaches the point B.
26
1
References
Bagget Jim, 1997. The Meaning of Quantum Theory, Oxford University Press.
2 Afshar, S.S., 2005. Sharp complementary wave and particle behaviours in the same welcherweg experiment, Proc. SPIE 5866,
229-244.
3 Bagget Jim, 1997. The Meaning of Quantum Theory, Oxford University Press.
4 Afshar, S.S., 2005. Sharp complementary wave and particle behaviours in the same welcherweg experiment, Proc. SPIE 5866, 229-
244.
5 Afshar, S.S., 2005. Sharp complementary wave and particle behaviours in the same welcherweg experiment, Proc. SPIE 5866, 229-
244.
6 Chandana S. and de Silva Nalin, 2004. On the double-slit experiment, Annual Research Symposium, University of Kelaniya, 57-58.
7 Chandana S. and de Silva Nalin, 2007. Some experiments involving double-slits, Annual Research Symposium, University of
Kelaniya,133-134.
8 Bohm D, 1980. Wholeness and the implicate order, Routledge, London.
9 Chandana S. and de Silva Nalin, 2004. A new interpretation of Quantum Mechanics, Annual Research Symposium, University of
Kelaniya, 59-60.
10 Monroe C., Meekhof D. M., King B. E., Wineland D. J., 1996. A ``Schrödinger Cat'' Superposition State of an Atom, Science, 272,
1132.
11 de Silva Nalin, Sinhala Bauddha Manasa, www.kalaya.org/files/nps/070405.pdf.
12 Jayatilleke, K. N.,1963. Early Buddhist Theory of Knowledge, Motilal Banarsidass.
It is shown that neither the wave picture nor the ordinary particle picture
offers a satisfactory explanation of the double–slit experiment. The western
Physicists who have been successful in formulating theories in the
Newtonian Paradigm based in their culture find it difficult to interpret
Quantum Physics which deals with particles that are not sensory perceptible.
A different interpretation of Quantum Physics based in the Sinhala Buddhist
culture is presented in what follows. According to the new interpretation
Quantum particles have different properties from those of Classical
Newtonian particles. The interference patterns are explained in terms of
particles each of which passes through both slits.
2
Quantum Physics in the Sinhala Buddhist culture
INTRODUCTION
Planck introduced his ideas on quanta or packets of energy towards the end of the
nineteenth century. In that sense Quantum Physics is more than one hundred years old.
From the very beginning Quantum Physics came up with strange phenomena that made
the western Physicists to disbelieve what they themselves were proposing to understand
the new features that were being observed.
The so-called double-slit experiment1 continues to baffle the western Physicists who are
glued to twofold two valued logic that is behind western thinking. As it was one of the
most fundamental experiments that they could not understand in Quantum Physics the
Nobel Prize winning Physicist Richard Feynmann once declared that no body understood
Quantum Physics! This statement by Feynmann makes one to delve into the meaning of
understanding. In other words one has to understand what is meant by understanding.
However, it is clear that if one is confined to twofold formal logic, and linear thinking
one would be confused by a statement such as understanding what is meant by
understanding. A decade ago the western intellectuals who are only familiar with linear
thinking and not with cyclic thinking would have left deliberations into such statements
to whom they call mystics, as such statements did not come within their “rational" way of
thinking.
The principle of superposition which was familiar to Classical Physicists as well, has
taken an entirely different meaning with respect to Quantum Physics. The essence of the
principle can be explained as follows. If x and y are two solutions of what is called a
linear differential equation then x+y is also a solution of the same differential equation.
3
This is a simpler version of what is generally known as the principle of superposition. In
Classical Physics two magnets giving rise to two different magnetic fields would
combine to give one magnetic field, and a compass that is brought to the resulting
magnetic field would respond to the resulting field, and not to the field of any one of the
magnets. It has to be emphasised that a magnet is in only one state, corresponding to the
respective magnetic field and it is the two fields of the two magnets that combine to give
one field though one would not find a single magnet that gives rise to the resultant field.
We could describe this phenomenon as that of two or more becoming one. However, in
the Quantum world things are different, and the principle of superposition has an unusual
interpretation.
THE WAVE NATURE OF PARTICLES
In order to discuss the new interpretation of the principle of superposition we first
consider the so called double-slit experiment where a stream of electrons (in general
particles or photons) is made to pass through two slits and then to strike a screen. If both
slits are open an interference pattern is observed on the screen. Now in Quantum Physics
it is said that particles such as electrons posses wave properties and photons (light)
exhibit particle properties in addition to their normal properties. Interference patterns are
supposed to result from wave properties and according to the western Physicists the wave
theory successfully explains the formation of such patterns in the case of a stream of
particles fired from a source to strike the screen after passing through the slits. The
western Physicists would claim that the double-slit experiment demonstrates that particles
such as electrons do exhibit wave properties.
The double-slit experiment has been carried out with only one electron passing through
the slits one at a time2 (electrons at very low intensities) instead of a stream of particles
released almost simultaneously to pass through the two slits. Even at very low intensities
4
interference patterns have been observed after sufficiently large number of electrons had
been fired from the source (Figure 1). The western Physicists have been puzzled by this
phenomenon. In the case of several electrons passing through the slits simultaneously it
could be explained using the wave properties of the particles, in other words resorting to
the wave picture. Unfortunately in the case of electrons being shot one at a time this
explanation was not possible as what was observed on the screen was not a faint
interference pattern corresponding to one electron but an electron striking the screen at a
single point on the screen. These points in sufficiently large numbers, corresponding to a
large number of electrons, finally gave rise to an interference pattern. The wave nature is
only a way of speaking, as even in the case of large number of particles what is observed
is a collection of points and not waves interfering with each other.
5
Figure 1: The interference pattern produced by a single-photon source with (a) 30, (b) 300, and
(c) 3000 photons registered. In contrast, the decoherent distribution of (d) 30, (e)
300, and (f) 3000 photons lacks the dark fringes. (Courtesy Afshra)
The western Physicists also believe that an electron as a particle could pass through only
one of the slits and a related question that has been asked is whether it was possible to
find out the slit through which an electron passes on its way to the screen. Various
mechanisms, including “capturing” the electron using Geiger counters, have been tried to
“detect the path” of the electron, and it has been found that if the particular slit through
which the electron passed was detected then the interference patterns were washed out. In
other words determining the particle properties of the electron erased its wave properties.
Bohr, who was instrumental in formulating the Copenhagen interpretation3, was of the
view that one could observe either the particle properties or the wave properties but not
both, and the inability to observe both wave and particle properties simultaneously came
to be referred to as complementarity. The experiments that attempted to determine the slit
through which the electron passed were known as which-way (welcherweg) experiments
as they attempted to find the way or the path of the particle from the source to the screen.
The outcome of these experiments made it clear that the which-way experiments washed
out the interference patterns. It was believed that at any given time the electrons exhibited
either the particle properties or wave properties but not both.
However, what the western Physicists failed to recognize was that in the case of one
electron shot at a time there was no weak interference pattern observed on the screen for
each electron thus illustrating that a single electron did not exhibit any wave properties.
The electron strikes the screen at one point, and it is the collection of a large number of
such points or images on the screen that gave the interference pattern. In the case of a
stream of electrons fired to strike the screen each electron would have met the screen at
one point and the collection of such points or images would have given rise to an
interference pattern. Thus we could say that the interference patterns are obtained not as a
result of the “wave nature” of electrons but due to the collectiveness of a large number of
electrons that strike the screen. The “wave nature” arises out of “particle” properties and
not due to “wave properties”. Afshar4 comes closer to this view when he states “in other
words, evidence for coherent wave-like behavior is not a single particle property, but an
6
ensemble or multi-particle property”. We are of the opinion that in the double-slit
experiments no wave properties are observed contrary to what is generally believed. It is
the particle properties that are observed, though not necessarily those of ordinary
classical particles.
As a case in point this does not mean that a particle in Quantum Physics has a definite
path from the source to the screen through one of the slits, as could be expected in the
case of classical particles. For a particle to have a path it should posses both position and
momentum simultaneously. A path at any point (assuming that it is a continuous path)
should have a well defined tangent. In the case of a particle moving, the direction of the
velocity (and the momentum) of the particle at any given point defines the unit tangent
vector to its path. Conversely the tangent to the path at any point defines the direction of
the velocity and the momentum of the particle at that point. However, according to the
Uncertainty Principle, both the momentum and the position of a particle cannot be
determined simultaneously, and if the position is known then the momentum cannot be
determined. Without the momentum the direction of the velocity of the particle and hence
the tangent vector cannot be known implying that a continuous curve is not traced by a
particle in space. On the other hand if the momentum of the particle is known then only
the direction and magnitude of the velocity (momentum) and properties of other non
conjugate observables such as spin of the particle are known, without the position being
known. Thus the particle can be everywhere, with variable probabilities of finding the
particle at different points, but at each point the particle being moving in parallel
directions with the same speed. However, as will be explained later, this does not mean
that we could observe the particle everywhere.
In the light of the uncertainty principle it is futile to design experiments to find out the
path of a particle. The so-called which-way experiments have been designed to detect the
slit through which the particle moves, on the assumption that the particle moves through
one slit only. The which-way experiment actually stops the particle from reaching the
screen and hence there is no possibility of obtaining any interference pattern. It is not a
case of observing particle properties destroying the wave properties of matter, but an
7
instance of creating a situation where the particle is either not allowed to strike the screen
or pass through only one slit deliberately. In effect it is the particle properties exhibited
at the screen that are cut off.
What is important is to note that interference patterns are observed only if both slits are
kept open, and if the particles are free to reach the screen. If one slit is closed or obstacles
are set up in the guise of which-way experiments or otherwise, so as not to allow the
particles to reach the screen then no interference patterns are observed. The most
important factor is the opening of the two slits. In the case of which-way experiments as
well, what is effectively done is to close one of the slits as particles through that slit are
not allowed to reach the screen. With only one slit open while the other slit is effectively
closed with the which-way experiment apparatus, no interference patterns are observed.
The probability of an electron striking the screen at a given point with only one slit open,
is not the same as that when both slits are open. Thus when a large number of particles
strike the screen, the different probabilities give rise to different “patterns” which are
essentially collection of points where the particles meet the screen. The “interference
patterns” observed when both slits are open is replaced by “chaotic patterns” when one of
the slits is closed (Figure 1 – interference and decoherence). The “interference patterns”
as well as the “chaotic patterns” are the results of particle properties, the difference being
due to the number of slits that are open. If both slits are closed there is no pattern at all as
no particle would reach the screen under such conditions. When one of the slits is open
the particle can be only at that position where the slit is whereas when both slits are open
there is a probability that the particle could be at both the slits “before” reaching the
screen.
The western Physicists are obsessed with the idea that a particle can be only at one
position at a given time. While this may be the experience with our sensory perceptible
particles (objects) or what we may call ordinary Newtonian classical objects such as
billiard balls, it need not be the case with Quantum particles. However, from the
beginning of Quantum Physics, it appears that the western Physicists have been of the
8
view that a particle can be at one position at a given time whether it is being observed or
not. Hence they seem to have assumed that on its “journey to the screen from the source”
a particle could pass through only one of the slits. They have worked on the assumption
that even if both slits are open the particle passes through only one of the slits but
behaves differently to create interference patterns as if the particle is “aware” that both
slits are open. According to the view of the western Physicists if only one slit is open the
particle having “known” that the other slit is closed passes through the open slit and
“decides” not to form any interference patterns. It is clear that the explanation given by
the western Physicists for the formation of interference patterns on the basis of the
particle picture is not satisfactory. We saw earlier that the explanation given in the wave
picture is also not satisfactory as a single electron fired from the source does not form a
faint interference pattern on the screen. If the particles behave like waves then even a
single particle should behave like a wave and produce a faint interference pattern, having
interfered with itself. What is emphasised here is that the final interference pattern is not
the sum of faint interference patterns due to single particles, but an apparent pattern
formed by a collection of images on the screen due to the particles. There is no
interference pattern as such but only a collection of the points where the particles strike
the screen, or of the images formed by the particles that were able to reach the screen.
The images finally depend on the probability that a particle would be at a given position.
Before we proceed further a clarification has to be made on “seeing” a particle at a given
position at a given time in respect of the double-slit experiment. In this experiment we are
concerned with particles released from a source with a given momentum and given
energy. As such according to the uncertainty principle, nothing can be said definitely on
the position of these particles. It can only be said that there is a certain probability that the
particle would be found in a certain position. Thus the particle is “everywhere” “until” it
is “caught” at some position such as a slit or a screen. Though we have used the word
“until”, time is not defined as far as the particle is concerned as it has a definite energy. It
can only be said that there is a certain probability that the particle could be “seen” at a
given place at a given time, with respect to the observer. The particle is not only
everywhere but also at “every instant”. Thus it is meaningless to say the particle is at a
9
given slit at a given time as neither time nor position is defined for the particle with
respect to itself. The particle would meet the screen at some position on the screen at
some time but “before” that it was everywhere and at every instant.
EXPERIMENTS OF AFSHAR
Afshar5 has claimed that he was able to demonstrate that an electron or a photon would
exhibit both particle and wave properties (Figure 2 – on next page). He allowed light to
pass through two slits and to interact with a wire grid placed so that the nodes were at the
positions of zero probability of observing a photon. The photons were not affected by the
wire grid as the nodes were at the positions of zero probability and at those positions
there were no photons to interact with the grid. The photons were then intercepted by a
lens system that was able to identify the slit through which any single photon had passed.
According to Afshar the nodes of the grid at the positions of zero probability indicated
that the wave properties of the photons were observable while the lens system in
detecting the slit through which the photon had passed demonstrated the particle
properties of the photons.
However, in this experiment, assuming that the lens system detects the slit through which
the photon passed, what is observed is again the particle properties of the photons. The
wire grid with the nodes at the position of zero probabilities does not interact with the
photons, as there are no photons at positions of zero probability to interact with the grid.
No so called waves are observed, as there is no screen for the particles to strike. Thus the
wire grid has no effect in this experiment and with or without such a grid the lens system
would behave the same way.
Let us consider what would happen if the wire grid is shifted forwards towards the
source, backwards towards the lens system or laterally. As the nodes of the wire grid
10
would be shifted from the positions of zero probability some photons would strike the
grid and they would not proceed towards the lens system. Thus the number of photons
that reach the lens system would be reduced and there would be a decrease in intensity of
light received at the lens. Though Afshar claims that wave properties are observed just by
placing a wire grid so that its nodes are at the positions of zero probability, it is not so.
Figure 2: The wire grid and the lens system of Afshra, and the corresponding images observed.
No interference patterns after the lenses and Afshra claims that the wire grid
demonstrates the wave property while the images correspond to the particle property.
(Courtesy Afshra)
11
The so called wave properties could be observed only by placing a screen in between the
wire grid and the lens system. As we have mentioned above, even then what is observed
is a collection of images at the points where the photons strike the screen, and not wave
properties as such. In this case as all the photons would have been absorbed by the
screen, the lens system would not be able to detect any photons nor the “slit through
which the photons passed”. On the other hand if the screen is kept beyond the lens system
then there would not be any photons to strike the screen and hence no “wave properties”.
EXPERIMENTS AT KELANIYA
We at the University of Kelaniya have given thought to this problem, and Suraj
Chandana, one of my students has carried out a number of experiments, which may be
identified as extensions of the experiment of Afshar. Chandana and de Silva6 had
predicted that if we were to have a single slit and then a screen, instead of the wire grid
and the lens system, “after” the photons have passed through the two slits, then the
photons would pass through the single slit with the same probability as that of finding a
photon at the point where the slit was kept. This implied that if the slit was kept at a point
where the probability of finding the photon is zero, the photon would not pass through the
slit to strike the screen, but on the other hand, if the slit was kept at any other point there
was a non zero probability that the photon would pass through the slit, and striking the
screen. This implies that if a stream of photons is passed through two slits, and “then” a
single slit, to strike the screen, depending on the position of the single slit the intensity
with which the photons strike the screen would change. Further it implies that these
intensities should correspond to the intensities observed in connection with the
interference patterns observed in the case of the standard double-slit experiment, if the
positions of slit were along a line parallel to the double-slits and the screen. Chandana has
been successful in obtaining the results as predicted.
12
In another experiment Chandana7 had an Aluminium sheet of very small thickness
joining the points or positions where the probability of finding a photon is zero (positions
of zero probability), stretching from the double-slits to the screen as illustrated in the
figure 3. As an obstacle placed at a position of zero probability would not affect the
photon the Aluminium sheet had no effect on the visible interference patterns on the
screen. This experiment was carried out by Chandana with number of Aluminium sheets
placed along lines joining the positions of zero probability stretching from the doubleslits
to the screen. We were not surprised to find that the Aluminium sheets did not
interfere with the interference patterns. However, even if one of the sheets is slightly
displaced the interference pattern is destroyed as the photons now interact with the sheets
at points where the probability of finding a photon is not zero.
Figure 3: The figure represents the aluminium sheet joining the positions of zero probability
from a position closer to the double slit to the screen.
13
These observations are not consistent with the wave picture as a wave would not be able
to penetrate the Aluminium sheets without being affected. Even the pilot waves of Bohm8
are not known to go through a material medium undisturbed. As we have argued a single
electron emitted from the source would not exhibit a faint interference pattern on the
screen but a spot or an image having passed beyond the slits. The western Physicists are
interested in the wave picture to explain the interference patterns as they find it difficult
to believe that a particle would pass through both slits simultaneously. Thus they mention
of particle properties when they are interested in “capturing” particles and of wave
properties in explaining phenomena such as the interference pattern.
PRINCIPLE OF SUPERPOSITION IN QUANTUM PHYSICS
We consider the Quantum entities to be particles though of a nature different from that of
Classical Newtonian particles. We have no inhibition in believing that the Quantum
particles unlike the Newtonian particles could pass through both slits at the “same time”,
as the logic of the Sinhala Buddhist culture permits us to do so. Western Physics and in
general western Mathematics and sciences are based on Aristotelian two valued twofold
logic according to which a proposition and its negation cannot be true at the same time.
Thus if a particle is at the slit A, the proposition that the particle is at A is true and its
negation that the particle is not at A is not true, and vice versa. Therefore if the particle is
at A then it cannot be anywhere else as well, and hence cannot be at B. However, in
fourfold logic (catuskoti) a proposition and its negation can be both true, and hence in
that logic it is not a contradiction to say that a particle is at the slit A and at somewhere
else (say at the slit B) at the “same instant” or “every instant” Thus according to catuskoti
the particle can be at many places at the same time with respect to the observer.
In the case of the double-slit experiment, the momentum of a particle is known, as the
particles are fired with known energies, and hence the position is not known. In such a
14
situation Heisenberg’s uncertainty principle demands that the position of the particle is
not known. The position of the particle is relieved only after a measurement is made to
determine the position. Before the measurement, the particle is in a superposition of
states corresponding to the positions in space the particle could be found. After the
measurement the particle would be found in a definite position (state), having collapsed
from the superposition of a number of states to that of the definite state. Before the
measurement what could have been said was that there was a certain probability of
finding the particle at a given position. Though the particle is in a superposition of states
before a measurement is made to find the position, it is in a definite state with respect to
the momentum.
In Quantum Mechanics unlike in Classical Mechanics, a state of a system, a particle or an
object is represented by a vector in a Mathematical space known as the Hilbert space. The
observables such as position, momentum, and spin are represented by what are known as
Hermitian operators. If a system is in a state represented by an eigenstate ׀> of a
Hermitian operator A, belonging to the eigenvalue a, then the system has the value a
corresponding to the observable represented by the Hermitian operator A. This is
expressed mathematically by A ׀> = a ׀>. If B is the conjugate operator of A, then the
value corresponding to the observable represented by B is not known. All that can be
said, according to the standard Copenhagen interpretation, is that if the value
corresponding to the observable represented by B is measured, then there is a certain
probability of obtaining an eigenvalue of B as that value. Before the measurement is
made nothing could be said of the value. In plain language this means that if the value of
a certain observable is known then the value of the conjugate observable is not known.
However, the state ׀> can be expressed as a linear combination of the eigenstates ׀> of
B in the form F >=Σ > i i cy where Î i c C , the field of complex numbers. In other
words the coefficients of ׀>’s in the expansion of ׀> are complex numbers. The
Copenhagen interpretation tells us that when the observable corresponding to B is
measured it would result in a state corresponding to one of the ׀>’s with the
measurement yielding the eigenvalue b to which the particular ׀> belongs, the
15
probability of obtaining the value b being given by the value of the relevant 2 c . Before
the measurement is made nothing can be said regarding the observable corresponding to
B. According to Bohr, it is meaningless to talk of the state of the system with respect to B
as nothing could be observed. There is no knowledge regarding the observable
corresponding to B as it has not been observed. The value or the knowledge of the
observable is “created” by the observer who sets up an experiment to measure the value
in respect of B. The observed depends on the observer and it makes no sense to talk of an
observable unless it has been observed. This interpretation is rooted in the positivism as
opposed to the realism in which the entire corpus of knowledge in Newtonian Physics is
based.
As a particular case one could refer to the conjugate Hermitian operators in respect of
position and momentum of a particle in Quantum Mechanics. When the position of a
particle is measured then its momentum is not known. According to the Copenhagen
Interpretation, it can only be said that if an apparatus is set up to measure the momentum,
the observer would observe one of the possible values for the momentum and that there is
a certain probability of observing the particular value. Before the measurement is made
the particle has no momentum, as such, and it is meaningless to talk of the momentum of
the particle. The observer by his act of observation gives or creates a value for the
momentum of the particle, so to speak of. Once the momentum is measured the observer
has knowledge of the momentum but not before it. However, after the momentum is
measured, the knowledge of the position of the particle is “washed off” and hence it
becomes meaningless to talk of the position of the particle. The observer could have
knowledge only of either the momentum or the position, but not of both. A version of
this conclusion is sometimes referred to as the uncertainty principle.
What we have been discussing in the proceeding paragraphs is the principle of
superposition. A particle or a system with its position known is represented by a vector
׀> in Hilbert space, which is an eigenvector of the Hermitian operator A corresponding
to the position. As the position of the particle or the system is known, the momentum is
not known. If B is the Hermitian operator corresponding to the momentum, then lF> is
16
not an eigenvector of B. However, ׀> can be expressed as a linear combination of the
eigenvectors ׀>’s of B though the momentum is not observed. The superposition of the
׀>’s cannot be observed, and neither can be resolved into observable constituent parts.
This is different from the principle of superposition in Classical Physics, where the
resultant can be resolved into its constituent parts.
For example as we have mentioned in the introduction the resultant magnetic field due to
two magnets can be resolved into its two components and can be observed. One of the
magnets can be taken off leaving only one of the constituent magnetic fields. The
superposition is there to be observed and if the magnet that was taken off is brought back
to its original position the resultant magnetic field reappears. In Quantum Physics the
superposition cannot be observed without disturbing the system and when it is disturbed
to measure the conjugate variable, only one of the states in the superposition could be
observed and we would not have known in advance if that particular state were to appear
as a result of the disturbance induced by us.
COPENHAGEN INTERPRETATION
In Classical Physics, as we have already stated, superposition is there to be observed.
However, in Quantum Physics the superposition cannot be observed, and further unlike in
Classical Physics interpretations are required to “translate” the abstract Mathematical
apparatus and concepts into day to day language. In Classical Physics one knows what is
meant by the position or the momentum of a particle and those concepts can be observed
and understood without an intermediate interpretation. However, in Quantum Physics, the
state of a particle or a system is represented by a vector in Hilbert space and observables
are represented by Hermitian operators in Hilbert space. An interpretation or
interpretations are needed to express these and other concepts to build a concrete picture
17
out of the abstract apparatus. Copenhagen interpretation is one such interpretation and it
is the standard interpretation as far as the west is concerned.
Bohr more than anybody else was instrumental in formulating the Copenhagen
interpretation, and he in turn was influenced by positivism and Chinese Ying - Yang
Philosophy. As a positivist he believed that only the sensory perceptible phenomena exist
and did not believe in the existence of that could not be “observed”. When a state of a
particle or system is represented by an eigenvector of an observable (Hermitian operator
in Hilbert space) the corresponding value of the observable can be measured and the
positivist school had no problem in accepting the existence of such state. For example if
the momentum of a particle is known then the state of the particle is represented by a
certain vector in Hilbert space, belonging to the particular eigenvalue that has been
measured. However, the problem arises when the conjugate Hermitian operator, in this
case the position, is considered, as in positivism the ontology is connected with
observations and sensory perceptions.
As we have seen a given eigenstate of a Hermitian operator that has been observed can be
expressed as a linear combination of the eigenstates of the conjugate operator. To a
positivist, though the given eigenstate exists as it is observed, the eigenstates of the
conjugate operator are not observable and it is meaningless for him to talk of such states.
Thus if the momentum of a particle has been measured, the eigenstates belonging to the
eigenvalues of the conjugate operator, which is the position, are not observed and the
positivist would not say anything regarding the existence of such states. As far as the
positivist is concerned, there is only a probability of finding the particle at some position,
and the particle will be at some position only after the relevant measurement is carried
out.
In the case of the double-slit experiment, this means that a positivist would not say
whether the particle passes through a particular slit as it is not observed. However he
assumes that it passes through one of the slits and not both as the western thinking
demands that the particle should be at one of the slits and not at both slits. If a
18
measurement is made, that is if an experiment is carried out to find out the slit through
which the particle passes, then the particle would be found at one of the slits washing out
the “interference pattern”. Then superposition is collapsed and “decoherance” sets in
resulting “chaotic pattern”.
A realist differs from a positivist in that the former would want to know the slit through
which the particle passes even without observing it. He would say the particle would pass
through one of the slits whether one observes it or not, and that it is an integral property
of the particle independent of the observer. The Classical Physicists were realists. An
object in Classical Physics has a momentum whether it is measured or not. The observer
in Classical Mechanics measures the momentum that the particle already possesses. In
Quantum Physics the positivists would say that the particle has no momentum before it is
measured but acquires a momentum as a result of the measurement. We would not go
into further details on the differences between the realist position and the positivist
position as it is outside the scope of this article. However, what is relevant to us is that
both the realist and the positivist would agree that the particle goes through one slit,
meaning that at a “given time” the particle is found only at one of the slits. They would
also agree on the wave nature of the particles. They have to depend on the wave nature as
they assume that the particle passes through only one slit, and as such they would not be
able to explain the “interference patterns” without the wave properties of the particles.
A NEW INTERPRETATION
We differ from the positivists as well as the realists since we believe that the particle is
found at both slits and hence pass through both. In general we include the postulate that
the eigenstates ׀>’s in F >=Σciy i > exist in addition to ׀> (Postulate 3 below). We
have also introduced the concept of a mode. A mode of a particle or a system is
essentially a potential observable. A mode has the potential to be observed though it may
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not be observed at a particular instant. For example, position, momentum, spin are
modes. A particle or a system can be in both modes corresponding to two conjugate
Hermitian operators, though only one mode may be observed.
A revised version of the postulates of the new interpretation formulated by Chandana and
de Silva9 is given below.
1. A state of a Quantum Mechanical system is represented by a vector (ray) c in the
Hilbert space, where c can be expressed as different linear combinations of the
eigenvectors in the Hilbert space, of Hermitian operators, any operator
corresponding to a mode. In other words a state of a Quantum Mechanical system
can be represented by different linear combinations of eigenvectors of different
modes, each linear combination being that of the eigenvectors of one of the modes.
Thus a state could have a number of modes, each mode being a potential
observable.
2. If c is expressed as a linear combination of two or more eigenvectors of a
Hermitian operator, that is a mode, then the corresponding mode cannot be
observed (or measured) by a human observer with or without the aid of an
apparatus. In other words the particular mode cannot be observed and a value
cannot be given to the observable, which also means that no measurement has
been made on the observable.
3. However, the non observation of a mode does not mean that the mode does not
“exist”. We make a distinction between the “existence” of a mode, and the
observation of a mode with or without the aid of an apparatus. A mode
corresponding to a given Hermitian operator could “exist” without being observed.
The knowledge of the “existence” of a mode is independent of its observation or
measurement. In other words the knowledge of the “existence” of a mode of a
Quantum Mechanical state is different from the knowledge of the value that the
observable corresponding to the relevant Hermitian operator would take.
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4. If a mode of a Quantum Mechanical state is represented by a single eigenvector,
and not by a linear combination of two or more eigenvectors, of a Hermitian
operator, then the mode could be observed by a human observer with or without
the aid of an apparatus, and the value of the corresponding observable (or the
measured value) is given by the eigenvalue which the eigenvector belongs to. It
has to be emphasised that only those modes of a Quantum Mechanical state, each
represented by a single eigenvector, and not by a linear combination of
eigenvectors, of an Hermitian operator can be observed at a given instant.
5. If a mode of a Quantum Mechanical state is represented by an eigenvector of a
Hermitian operator then the mode corresponding to the conjugate operator cannot
be represented by an eigenvector of the conjugate Hermitian operator. It can be
expressed as a linear combination of two or more of the eigenvectors of the
conjugate operator. This means that the mode corresponding to the conjugate
operator cannot be observed, or in other words it cannot be measured. However,
the relevant mode “exists” though it cannot be observed.
6. It is not necessary that at least one of the modes corresponding to two conjugate
operators should be represented by a single eigenvector of the relevant operator. It
is possible that each mode is represented by linear combinations of two or more
eigenvectors of the corresponding operator. In such situations neither of the modes
could be observed.
7. A state of a Quantum Mechanical system can be altered by making an operation
that changes a mode or modes of the state. However, not all operations correspond
to measurements or observations. Only those operations that would result in a
mode being expressed as a single eigenvector, and not as a linear combination of
the eigenvectors of an operator would result in measurements.
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8. A particle entangled with one or more other particles is in general represented by a
linear combination of eigenvectors of an Hermitian operator with respect to a
mode, while the whole system of particles is in general represented by a linear
combination of the Cartesian products of the eigenvectors. In the case of two
particles it takes the form S cij ׀φi› ׀φj›. If one of the particles is in a mode that is
observed, then the particles entangled with it are also in the same mode as an
observable. If a measurement is made on some other mode then instantaneously,
the corresponding values in the same mode of the entangled particles are also
determined. In such case, for two particles the whole system is represented by
vectors of the form ׀φi› ׀φ j›. If the number of entangled particles is less than the
dimension of the space of the eigenvectors of the Hermitian operator, then if a
measurement is made in the particular mode, the particle would be represented by
one of the eigenvectors, while the other particles entangled with it would be each
represented by a different eigenvector of the Hermitian operator. However, if the
number of entangled particles is greater than the dimension of the space of the
eigenvectors, then in some cases, more than one particle would be represented by a
given eigenvector.
According to this interpretation if the momentum of a particle is known then it has not
one position but several positions. In other words the particle can be at number of
positions in superposition though we are not able to observe it at any one of those
positions. The particle could be observed only if it is at one position. If an experiment is
carried out to determine the position of the particle the superposition or the wave function
would collapse, and the particle would be located at one of the positions where it was
before the measurement was made.
Similarly if the particle is in the position mode that is observed then it can have several
momenta in superposition but we would not be able to observe any one of them. If we
perform an experiment to determine the momentum, that is if a measurement is made,
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then the superposition of momenta would collapse to one of them, enabling us to
determine the value of the momentum.
With respect to the double-slit experiment this implies that the particle is at both slits in
superposition without being observed and if we perform an experiment to determine the
slit through which the particle passes then the superposition collapses and the particle
would be found only at one of the positions. The positivists while assuming that the
particle passes through only one slit would not say anything on the slit through which the
particle passes as it cannot be observed. For the positivist it is meaningless to speculate
on something that cannot be observed. The realists too assume that the particle passes
through only one slit but would not be satisfied with the positivist position, and claim that
a theory that is not able to determine the slit through which the particle passes is
incomplete.
We make a distinction between existence and being observed. A particle or a system can
exist in a certain mode without being observed. In this case the state of the particle or the
state is expressed as a linear combination or superposition of the eigenstates of the
relevant Hermitian operator and the particle or the system exists in all the relevant
eigenstates without being observed. The mode is observed only when the state of the
particle or the system is expressed as a single eigenstate of the relevant Hermitian
operator.
The existence of modes with more than one eigenstates has been known for sometime.
Monroe10 and his colleagues in 1996 were able to demonstrate the existence of two spin
states of Beryllium cation simultaneously however without observing them. One could
say that the interference obtained by them could be understood on the basis of the
existence of simultaneous spin states of the Beryllium cation. Since then similar
experiments have been carried out and the existence of superposition of eigenstates
cannot be ruled out anymore.
23
SINHALA BUDDHIST ONTOLOGY AND LOGIC
In Sinhala Buddhist ontology no distinction is made of the existence of sensory
perceptible objects and of other entities. There is no absolute existence as such and all
existences are relative to the mind. It has been shown by de Silva11 that even the mind is
a creation of the mind a phenomenon not in contradiction with cyclic thinking or
cakreeya cintanaya. It is the mind that creates concepts including that of self, and as such
sensory perceptible objects do not have any preference over the others.
As we have mentioned the positivists find it difficult to take cognizance of entities that
are not sensory perceptible and it is this ontology that makes them not to commit on the
existence of unobserved “objects”. In Sinhala Buddhist ontology all existences are only
conventional or sammuti and not absolute or paramarta. Thus the existence of
simultaneous eigenstates or superposition of eigenstates is not ruled out in Sinhala
Buddhist ontology. The Sinhala Buddhist culture has no inhibition to postulate the
existence of such states and it is not in contradiction with catuskoti or fourfold logic that
may be identified as the logic of the Sinhala Buddhist culture.
As Jayatilleke12 has shown in fourfold logic unlike in twofold logic a proposition and its
negation can be both true or false. In twofold logic if a proposition is true then its
negation is false, and if a proposition is false, then its negation is true. Thus the
proposition that a particle is at A, and the proposition that a particle is not at A, can be
both true in fourfold logic. We may deduce from that a particle can be both at A and B at
the “same time”. In other words a particle can be at both slits in respect of the double-slit
experiment, and in general a mode represented by a superposition of two or more than
24
two eigenvectors can exist as the particle or the system can be at number of “positions”
simultaneously in fourfold logic.
In twofold Aristotelian logic of the west a particle has to be either at A or not at A. Thus
the western Physicists whether they are realists or positivists find it difficult to accept that
a particle can pass through both slits simultaneously, and they have to resort to so called
wave nature in order to explain the interference patterns.
The Sinhala language reflects the use of fourfold logic in expressions such gevi nogevi,
adu vediya, yana ena, where the opposites are combined as samasa without the use of a
word equivalent to or in English. This is a result of threefold logic where both a
proposition and its negation can be true. I understand that this usage is found in some
Indian languages as well, as threefold logic is present in the Vedic culture. In fourfold
logic we have another case where a proposition and its negation can be both false. The
famous statement na ca so na ca anno in the “Milinda Prashnaya” meaning that neither
oneself nor somebody else is reborn is an example for an instance where the fourth case
in fourfold logic has been used. Perhaps a more familiar example is the answer given by
King Devanampiya Tissa to Arhant Mahinda. The king answering the Arhant Thera said
that he was neither a relative nor a non relative of himself.
DISCUSSION
It is seen that both wave picture and the ordinary particle picture fail to explain the
interference patterns observed in the double-slit experiment. The wave picture fails as a
weak intensity stream of electrons (one electron at a time) exhibits no interference
patterns in the case of few electrons. The ordinary particle picture fails as a particle
passing through only one slit would not produce interference patterns. The western
25
Physicists had to resort to the wave picture as the logic in their culture would not permit a
particle to pass through both slits.
In the case of the experiments conducted by Chandana at the University of Kelaniya the
wave picture as well as the particle picture come across more problems as neither a wave
nor an ordinary particle would be able to penetrate the aluminium sheets without being
affected. These experiments justify our new interpretation involving modes of the particle
or the system and the particle picture presented here where a particle can pass through
both slits. In general we postulate that a particle or system can exist in a mode where
more than one eigenstates are in a superposition. The position where a particle is found
depends only on the relevant probability, and the so-called interference patterns are only
collections of images formed by such particles striking the screen at different positions
with the relevant probabilities.
The new postulates are based in the Sinhala Buddhist culture and are consistent with the
Sinhala Buddhist ontology and the fourfold logic. It appears that, unlike Classical Physics
with its twofold logic and realist ontology, Quantum Physics is rooted in the Sinhala
Buddhist ontology and logic and we should be able to develop new concepts in Quantum
Physics, especially regarding the motion of a Quantum particle from a point A to another
point B. It is not known how a particle “moves” from the double-slit to the screen in the
experiments carried out by Chandana, nor how a particle with less energy than the value
of a potential barrier “scales the walls”. It may be that it is neither the particle that left the
point A nor some other particle that reaches the point B.
26
1
References
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3 Bagget Jim, 1997. The Meaning of Quantum Theory, Oxford University Press.
4 Afshar, S.S., 2005. Sharp complementary wave and particle behaviours in the same welcherweg experiment, Proc. SPIE 5866, 229-
244.
5 Afshar, S.S., 2005. Sharp complementary wave and particle behaviours in the same welcherweg experiment, Proc. SPIE 5866, 229-
244.
6 Chandana S. and de Silva Nalin, 2004. On the double-slit experiment, Annual Research Symposium, University of Kelaniya, 57-58.
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Kelaniya,133-134.
8 Bohm D, 1980. Wholeness and the implicate order, Routledge, London.
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10 Monroe C., Meekhof D. M., King B. E., Wineland D. J., 1996. A ``Schrödinger Cat'' Superposition State of an Atom, Science, 272,
1132.
11 de Silva Nalin, Sinhala Bauddha Manasa, www.kalaya.org/files/nps/070405.pdf.
12 Jayatilleke, K. N.,1963. Early Buddhist Theory of Knowledge, Motilal Banarsidass.
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