On A New Formulation of Micro-phenomena: The Double-slit Experiment

Abstract

Based on the novel view that a micro-entity could be considered as a particle associated with a field partaking of the energy of particle which are both described by deterministic causal equations of motion, we examine the success of our new theory in elucidating the underlying physics of the double-slit experiment. Here, we explain with clear details how each micro-particle scatters from one of the slits at a given time. After the scattering through one of the slits, the particle shares some of its energy with its surrounding field and a particle-field system is again formed which its motion is governed by a deterministic dynamics during its flight towards the detecting screen. The interference pattern is then explained by showing how the final location of each particle-field system at the time of reaching the detecting screen is distributed according to an angular distribution (equal to the what quantum theory predicts for the fringe effects in a two-slit experiment). The probabilistic nature of such a distribution can be explained by considering the variations of the kinetic energy of the particle-field system at different local situations.

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On A New Formulation of Micro-phenomena: The Double-slit
Experiment

A. Shafiee1 (*), A. Massoudi2 (**)
and M. Bahrami3 (*)
*) Research Group On Foundations of Quantum Theory and Information,
Department of Chemistry, Sharif University of Technology,
P.O.Box 11365-9516, Tehran, Iran.
**) Center of Complex Systems Research, Department of Chemistry,
K N Toosi University of Technology,
P. O. Box 1587-4416, Tehran, Iran

Abstarct

Based on the novel view that a micro-entity could be considered as a particle associated with a field
partaking of the energy of particle which are both described by deterministic causal equations of motion, we
examine the success of our new theory in elucidating the underlying physics of the double-slit experiment.
Here, we explain with clear details how each micro-particle scatters from one of the slits at a given time.
After the scattering through one of the slits, the particle shares some of its energy with its surrounding field
and a particle-field system is again formed which its motion is governed by a deterministic dynamics during
its flight towards the detecting screen. The interference pattern is then explained by showing how the final
location of each particle-field system at the time of reaching the detecting screen is distributed according to
an angular distribution (equal to the what quantum theory predicts for the fringe effects in a two-slit
experiment). The probabilistic nature of such a distribution can be explained by considering the variations
of the kinetic energy of the particle-field system at different local situations.


1 Introduction

The wave-like nature of light was first proved by Thomas Young who observed the
interference pattern of light in his famous double-slit experiment in1802 [1]. However, if we think
of light as consisting of photons, the interference pattern of light will seem mysterious.
Nevertheless, this is not a characteristic feature of light, only. After Young's examination, we
know at present that the double-slit experiment demonstrates the strange wave behavior of
micro-entities, not merely the photons. Yet, there has been provided no physical theory thus far
which could explain the interference phenomenon in a two-slit experiment on the basis of a single
micro-entity foundation. For this reason, Feynman states that "[i]n reality, it contains the only
mystery" in the quantum domain [2]. The main mysterious point is that in the two-slit experiments,
it appears as though each single entity interferes with itself. Because, if only one micro-entity (say,
an electron) is sent at a time from the source towards the slits, the interference fringes are also
observed when a sufficiently large number of micro-entities has been counted. But, according to a
particle picture, e.g., an electron which has been sent from a source, either passes through one of
the slits or passes through another slit. If this were the correct description, we would be unable to

1Corresponding author: shafiee@sharif.edu
2a_massoudi@alborz.kntu.ac.ir
3mbahrami@mehr.sharif.ir

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account for the interference fringes. For, if the electron passes through one of the slits, the other slit
can be neglected and it should be possible for the electron to arrive at all points on the detecting
screen after the slits. No bright and dark fringes will happen afterwards. If one still challenges this
view by detecting through which slit the electron actually passes, the interference fringes will be
destroyed. Any possibility which enables us to know through which slit the micro-entity has been
passed perturbs the whole quantum state of the system to such an extent that no interference
pattern is detected. In effect, two complementary events can be recognized here: either we know
which slit the micro-entity passed through, or we observe the interference effect.
In this paper, we are going to give an explanation for the puzzling problem of interference
fringes by using the causal formulation of micro-incidents described in [3] and [4]. According to
our approach, each micro-entity constitutes a particle which is associated with a field. A
particle-field (PF) is an integrated system which can be imagined as a particle which its properties
is merged with its surrounding space so much so it cannot be realized as a classical particle. What
describes the surrounding space of a micro-particle is a probability field partaking of the energy of
the particle. Any micro-particle shares its properties with its enfolding probability field to create a
new meaning of a tiny substance. So, contrary to the de Broglie-Bohm depiction of micro-particles
and pilot waves [5], in our description there is no room for active waves which can act on the
particles (through a so-called quantum force) or guide their motion. A PF system should not be
envisaged as a particle guided by a field. Rather, it should be considered as a unified entity with its
own dynamics which is, however, affected by the physical conditions the particle is subjected to
[3]. Hence, the particle plays the key role here and the behavior of its associated field is contingent
upon what the particle sees and experiences. Nevertheless, the whole behavior of the entire system
is very different from what we may observe for an individual particle alone, due to the interplay of
the particle and its allied field.
In this way, we are going to elucidate that despite a PF system reaches one of the slits at a
given time according to a determinate trajectory, it will arrive at specific locations on the detecting
screen, causing the dark and bright interference fringes.
In the following section we will describe the physical situation of the experiment and
explain the problem in details. Then, we have a concluding section to sum up our results.

2 The double-slit experiment

We prepare a source emitting a beam of micro-entities (electrons, neutrons, atoms or in
general PF systems) towards a plane including two slits each of finite width ? separated by a
distance ?. The slits are placed along the ?-direction and we ignore the thickness of them along
the ? and ? directions. So, the ? and ? coordinates of the position of the slits are fixed in the ??
plane, denoted by ?? and ??, respectively. The intensity of the incident beam is supposed to be so
low that the PF entities arrive at the slits one at a time. Every PF system sent from the source has a
definite amount of linear momentum ?? at ? ? 0 before reaching the slits, where (see Appendix
A):


???? ??

Here, ?? is a scaler function of the components of the de Broglie momentum ?? and the
particle's momentum ??
field ??. The plane field is defined as:
????
?? ??
?? (1)
?, both defined at ? ? 0, as well as the amplitude of the momentum (plane)

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???????,????,????? ? ??exp??????.? ?????; ? ? 0 (2)

In relation (2), ? ???? is the local vector of a particle of mass ? in the Euclidean space and
???
have no interaction with their environment, so the energy of each PF system is conserved and
equal to its kinetic energy defined as ???
If we assume that the momentum of each PF system has the same magnitude in all
directions at ? ? 0 (that is ?? is an isotropic momentum), using the relation ??
? ? ?, ?,? and ?? is a scaler function, we can conclude that


??,?
?

where ???
of particle at ? ? 0 (see also appendix A).
Now, we assume that each PF entity, encompassing a particle enfolded by its associated
plane field in (2), reaches the double-slit apparatus at ? ? 0, where the slits are placed along the
?-direction. A system of slits is a position-measuring device [6, 7]. Thus, as stated earlier in [4],
when a PF system reaches a slit, a discontinuity happens in the trajectory of the system and the
dynamics suddenly breaks at ? ? 0. Hence, at the position of the slits, the energy of field is
transferred to the particle under an irreversible conversion process resulting to an unfolding
particle having a null field with zero amplitude. The function-dependency of the field, however,
specifies the corresponding wave function of the particle at ? ? 0.
Accordingly, at the position of the slits, if we introduce the ?-component of the particle's
momentum as ??,?


??,?
? ??

where ??
important to note that at ? ? 0, ??? 0 and ??? 1 (see the Appendix A for the definition of
??).
At ? ? 0 (i.e., when a PF entity arrives at the slits), the uncertainty relation of position and
momentum only reflects our lack of knowledge about the exact values of ??,?
?-component of the position of particle, since the particle's properties are well-defined at the
hidden level. Assuming that a particle passing through the first slit, e.g., has a definite position ?
in the continuum ??
obtained, respectively, as

??
??
?, where ?? is the magnitude of the de Broglie momentum defined in (1). The PF systems
?? at ? ? 0.
?? ??,?
??? where
?
??
√????
??? ?
??
√?????
??? (3)
?
?? is the de Broglie wavelength interpreted in our theory as the associated wavelength
?, we have:
?
??
??
√???
??? (4)
? is the de Broglie momentum of the PF system at ? ? 0. In reaching the relation (4), it is
?
and the
?? ?/2 ? ? ? ??
?? ?/2, the average values of ??? and ???? can be

??? ?
?
?
?
???/?
???/?
??
?? ? ? ??
? (5)
and

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???? ?
?
?
?
???/?
??
???/?
??
?? ??? ??
?????
?? (6)

where ??
uniform probability of being found through the slit. From the relations (5) and (6), one concludes
that Δ? ?
? denotes the middle point of the first slit. We have also supposed that the particle has a
?
?√?. At the same time, according to (4), we have ???,?
shows that Δ??,?
?
that



which states that the width of the slit cannot be smaller than about the half magnitude of the de
Broglie momentum of the particle, in accordance with all theoretical as well as experimental
considerations reported so far (see below).
At the position of the slits, the wave function of the particle can be defined as

?
√?∑
???

where ? is small enough to assume that a little after the slits, the spatial distribution of the particle
along the ?-direction can still be nearly given by |????|?. This assumption is essential, because
for obtaining the field representation of the system at latter times in our approach, we need the
trajectory equation of the particle after the slits which can be traced back to an initial time
? ? ? ? 0, due to the discontinuity occurred at the position of the slits at ? ? 0.
In (8), we have

?
√?
0 elsewhere

where ??
the amplitude of the field becomes zero, when the particle passes through the slit. So, the
associated field of the particle at ? ? 0 is zero, demonstrating that our lack of knowledge about
the physical situation of the particle is only a subjective matter with no objective feature [4]. After
the passage of the particle through the slit at ? ? 0 (including ? ? ?), however, the field will
partake of some of energy of the particle and its amplitude will be not zero.
?? ? 0 and ???,?
??? ?
??
???
? which
?
?
√???.4 Therefore, from the indeterminacy relation Δ?Δ??,?
?
? ?/2, one infers
? ?
?
???? (7)

??????? ?
?
????????; 0 ? ? ? ? (8)

???????? ? ?
??
?? ?/2 ? ? ? ??
?? ?/2
; 0 ? ? ? ? (9)
? is the middle point of the ?th slit (? ? 1 or 2). According to what argued before in [4],

4Each slit can be imagined as an one-dimensional box along the ?-direction. Hence, the mean value of ???,?
?2?2
4?2 at ? ? 0, where ? is the energy quantum number. Assuming an uniform probability
distribution for each slit along the ?-direction, it is legitimate to consider that the magnitude of ? is so large that the
?2
2 as a continuous quantity. For example, for the values of ?0? 2?? and
? ? 22?? in a double-slit neutron experiment [8], the value of ? can be approximated as 2 ? 10??.
??? is,
strictly speaking, equal to
value of ???,?
??? can be given by
3?0

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At ? ? ?, the time evolution of the wave function in the ?-direction can be described in
terms of a free-particle propagator ???????,?;????,?? defined as [4]:


???????,?;????,?? ??
?
???????exp???????????????
???????
? (10)

where ???? is the ?-component of the position of the particle at ? ? ?. The wave function of the
PF system at ? ? ? can now be defined as:

??
????? ???????,?;????,??????????; ? ? ?

Ψ??????,?? ?
?
√?∑
???
?

?
???/?
???/?
??
(11)

At ? ? ?, the propagator in (10) tends to ??? ? ????? and the wave function in (8) is reproduced
for ? ? ?. As stated above, however, the associated field of the particle at ? ? 0 is endowed with
energy. Thus, the wave function in (8) is indeed an approximation for 0 ? ? ? ?.
After passing through a given slit, the equation of motion of the particle along the
?-direction can be expressed as:


???? ? ???? ? ?? ? ??

where ??,? is the ? component of the momentum of the particle for ? ? ?. Regardless of what
slit the particle has passed through, we also assume that the particle's momentum ??,? is the same
value.
Using the relations (10)-(12), the relation (11) can now be given as follows:

??
??,?
?; ? ? ? (12)

Ψ??????,?? ?
?
√??∑
???
?
?
?
???????
?
???/?
???/?
??
????? exp?
???,????????????
??
? (13)

Then, Ψ??????,?? in (13) can be represented as:


Ψ??????,?? ? ?????
??
????
???,?????
??
√???
?
∑
???
?
exp??
???,???
??
?
?; ? ? ? (14)
where


? ?
???,?
?? (15)

Now, for ? ? ?, one can introduce the field representation of the PF system in the
following form5:

5Here, while we have a continuous change of state from ????0?? to Ψ?????,?? (including ? ? ?), the corresponding
field of the system changes discontinuously from a null field at ? ? 0 to ??????,?? ? 0 for ? ? ?. For this reason,
we have traced the time evolution of the wave function according to the time-dependent dynamics of the system, but
assign the field representations at two different time stages ? ? 0 and ? ? ?, inferentially.