The Relativistic Quantum Double-Slit Experiment

Of all the experiments that physicists have performed to reveal the invisible nature of Reality, the double-slit experiment stands in the first rank. At the beginning of the Nineteenth Century Thomas Young used it to demonstrate that light has the nature of a wave. And in 1961 Claus Jönsson of the University of Tübingen performed the experiment with electrons, thereby proving and verifying the proposition that the wave-particle duality of the quantum theory gives us a correct description of Reality. Now I want to reconceive the experiment in a way that brings Relativity into play.

The theory framing the experiment comes from the classical theory of wave motion. As an analogue of an electromagnetic plane wave, imagine a gravity wave on water, one whose crests and troughs define straight lines parallel to the y-axis of our coordinate grid. The wave propagates in a direction parallel to the x-axis and passes through a flat screen that stands upright oriented parallel to the y-axis. The screen will thus be wetted to a height equal to the maximum amplitude of the wave.

Now imagine placing a straight barrier into the water some distance in front of the screen in such a way that the barrier lies parallel to the y-z plane. The barrier has two narrow gaps in it, so the wave is not completely prevented from reaching the screen. Thus the barrier’s gaps act as two sources of circular waves. Passing through any given point at the same frequency and speed, the two circular waves maintain between them a phase difference that’s determined by the waves’ length and the distances from the point to the gaps that act as the waves’ sources. That phase difference determines the degree to which the waves interfere with each other and whether the interference is constructive or destructive. If we consider the points where the screen meets the water’s base level, we calculate that the screen will be wetted in a pattern of humps whose maximum height comes at or near the point nearest being midway between the gaps and whose heights decline symmetrically the further the humps occur away from that point.

Electromagnetic waves display the same kind of behavior. A plane wave propagating in the x-direction and striking a barrier, into which two slits have been cut in the z-direction, will form on a screen behind the barrier an array of bands of varying brightness, the contiguous points of equal brightness defining straight lines oriented parallel to the z-axis.

In the quantum-theoretical interpretation of light the wave guides the motion of the photons, the square of the wave’s amplitude setting the probability density for the location of the momentum and energy associated with each photon. Likewise, other particles, such as electrons, are guided in their motions by ghostwaves, which are described mathematically by the quantum-mechanical state function. Thus, electrons shot at a barrier into which a pair of slits have been cut will fall onto a screen behind the barrier in proportions that define alternating light and dark bands.

To the extent that the electrons are guided in their motions by the ghostwave, to that same extent the ghostwave must pace the electrons. The electrons can thus be represented in this imaginary experiment by a plane ghostwave propagating in the x-direction with the speed of the electrons. That fact will raise problems that must be solved in order to create a properly relativistic quantum theory.

Assuming that the electrons move solely in the positive
x-direction at the speed v, we associate with them a ghostwave of wavelength
λ
and propagation vector k=2π/λ
that moves as a plane wave. Our apparatus consists of a barrier with slits cut
into it a distance y_{1} above the x-axis and y_{2} below the
x-axis (y_{0}=y_{1}+y_{2}) and at a distance +x_{0}
beyond the barrier a detector screen that displays/records where the particles
hit it. To begin we assume that y_{1}=y_{2} so that the central
bright band B_{0} in the interference pattern lies on the detector
screen on the x-axis (y=0). We calculate the distance from each slit to B_{0}
as

(Eq’n 1)

In accordance with our assumptions, the difference between the path lengths
is obviously r_{1}-r_{2}=0. With the same number of wavelengths
on the two paths we get constructive interference on the detector screen and,
thus, a bright band.

But other bright bands appear on the detector screen on
both sides of B_{0}. Because it comes from constructive interference, a
bright band only occurs on the detector screen where the path lengths from the
slits differ from each other by an integer number of wavelengths. For bright
bands in the positive y-direction from the x-axis we have

(Eq’n 2)

in which

(Eq’ns 3)

We can also carry out our calculations in terms of an angle that the wave path makes with the x-axis. We define a base angle θ through the statement that

(Eq’n 4)

In that case we have

(Eq’ns 5)

in which Δθ
is effectively the same for both path lengths if y_{0}<<x_{0}.

Using the relation cos(A±B)=cosAcosB sinAsinB, we can rewrite Equation 2 as

(Eq’n 6)

If I project a particle at a wall and if the particle is
big enough, I know where the particle will hit the wall. If the particle is
small, I don’t know where it will hit because I have made
δp_{y}
small, so I have thereby made δy
large. We might call that fact Heisenberg’s paradox: the more precisely I launch
my particle from a given point, the less accurately it hits the target. That
fact necessitates that between the time I launch it and the time it hits the
target, the particle has an indeterminate existence.

Thus far I have been tacitly assuming that the experiment is being observed by someone at rest relative to the apparatus. Now I want to ask how the same experiment looks to an observer moving in the positive y-direction at some speed u relative to the first observer.

Initially the first observer removes the barrier from the apparatus and illuminates the screen with plane waves of light that have been amplitude modulated with a square-wave pattern at some frequency ν. The screen will thus appear bright for short intervals separated by dark intervals of t=1/(2ν).

In the second observer’s view the first observer and the apparatus move in the negative y-direction at the speed u. If the first observer has established a series of synchronized clocks along the top of the screen, the second observer will see any two adjacent clocks out of synchrony by an interval

(Eq’n 7)

in which y represents the distance that the first observer measures between the clocks and L represents the Lorentz factor between the observers,

(Eq’n 8)

Thus the second observer will see on the screen a series of light and dark bands that move in the negative y-direction. That observer interprets that pattern as indicating that the light falling on the screen has been aberrated through some angle by the relative motion between their frame and the first observer’s frame.

The second postulate of Relativity tells the second observer that

(Eq’n 9)

The angle through which the wave is aberrated thus equals

(Eq’n 10)

For any sinusoidal component of the wave the first observer measures an
angular frequency ω=**k**•**c**,
in which **k** represents the wave vector and **c** gives us an actual
velocity (as distinct from speed) of light. The second observer measures the
same component and gets **k**’ and
ω’,
which correlate to the first observer’s measurements through the Lorentz
Transformation (see "The Photodynamics of Moving Bodies" on this website),

(Eq’ns 11)

If the first observer measures a distance y_{0}
between the slits in the barrier, then the second observer will measure the
distance to be y_{0}’=y_{0}/L, in which L represents the Lorentz
factor between the two observers. Further, if the first observer has set a
series of synchronized clocks along the barrier, then the second observer will
measure between any two clocks separated by a distance y in the first observer’s
frame a temporal displacement of Δt’=uyL/c^{2}
(Eq’n 7). That temporal displacement entails an important consequence for wave
mechanics.

The first observer projects electrons at the barrier with
a tightly defined momentum p= ħk=2πħ/λ
oriented only in the x-direction. The corresponding ghostwave, of length
λ,
travels in the x-direction at the speed v=p/m=ħk/m.
The angular frequency at which the ghostwave passes any given point is thus
ω=kv=(p/ħ)(p/m)=p^{2}/ħm
and the kinetic energy associated with the wave is E=ħω=p^{2}/m.
But that last equation is missing a factor of ½ on the right side. So how did we
go so wrong?

Note that in the above analysis we have used the relativistic mass,

(Eq’n 12)

If we have a velocity in the classical range, we can use the binomial theorem to create the approximation

(Eq’n 13)

But that still doesn’t solve the problem we discerned in the last equation in
the paragraph above. To do that we would have to have p=mc so that E=mc^{2}.
Although we have assumed that the energy in the wave consists solely of the
particle’s kinetic energy, it would actually have to include the mass-energy of
the particle at rest. As we shall see below, there’s a simple way to solve that
problem and the relativistic analysis of this experiment necessitates it.

If u lies in the classical range, then the second observer
measures the same k and ω
in the x-direction and measures an additional k_{y} and
ω_{y}
in the y-direction, with k_{y}=mu/ħ
and ω_{y}=mu^{2}/ħ.
Thus, where the first observer sees the wave propagating along a line that makes
a 90-degree angle with the barrier, the second observer sees it propagating
along a line that makes an angle θ
with the barrier with θ=90-arctan(u/v).
That shifted angle must be due to aberration of the wave.

Relativity gives us an easy way to understand and to
calculate the aberration of plane waves. The first observer sees each of the
wave’s crests striking every point along the barrier simultaneously, subsequent
crests striking at intervals of t=λ/v.
Because the second observer sees the first observer’s synchronized clocks
temporally displaced from each other, they will see any given crest striking the
barrier at only one point at a time, that crest proceeding down the barrier as
the clocks advance. That fact means that the next crest must be striking the
barrier that’s displaced by t=λ/v
from the clock under which the first crest is striking the barrier: the distance
between those two clocks is then y=λc^{2}/uLv.
In this semi-classical treatment, then, the angle between the line defined by a
wavecrest and the line of the barrier is
θ’=arctan(λ/y)=arctan(uLv/c^{2}),
which is inconsistent with the result obtained in the preceding paragraph.

Deflection of the ghostwave, as seen by the second observer relative to the first observer’s measurements, must be due to relativistic aberration of the wave. That aberration can be due to one of two effects: it can be due to the relative temporal displacement of clocks, as I assumed above, or it can be due to the Doppler shift of a pre-existing y-ward component of the ghostwave. Because the particles in question have no momentum in the y-direction in the first observer’s frame, they would be associated with ghostwave components that have infinite wavelength. But no amount of Doppler shifting would give such waves the wavelengths that correctly represent the y-ward momenta that the particles would have in the second observer’s frame of reference.

There is a way, though, in which a particle can have a
wave that represents no momentum in one frame and yet has a wavelength that can
be Doppler shifted into a correct representation of the particle’s momentum in
another frame. That way is marked by the use of the relativistically correct
representation of the particle’s energy. In that representation
ħω
represents the particle’s kinetic energy __plus__ its rest energy; that is,

(Eq’n 14)

For a particle at rest we have a standing ghostwave, so the state function must take the form

(Eq’n 15)

in which ω_{0}=k_{0}c;
that is, the ghostwave comprises two waves of equal frequency and wavelength
traveling in opposite directions at the speed of light. The quantum theory thus
gives us the energy and momentum of the particle as

(Eq’n 16)

and

(Eq’n 17)

The opposing waves must each propagate at the speed of light because: if an observer could pace one of them, its wavelength would go to infinity and the wave is thus Doppler shifted out of existence and we’re back where we started. Therefore, there is no inertial frame in which one or the other of the component waves stands motionless. The only speed that cannot be cancelled by any delta-vee is the speed of light.

In a frame in which that particle moves in the positive x-direction at a speed v the frequencies and the wave numbers of the waves are Doppler shifted by (1+v/c)L and (1-v/c)L respectively, so

(Eq’n 18)

The corresponding energy and momentum thus come out as

(Eq’n 19)

and

(Eq’n 20)

Noting that ħω_{0}=m_{0}c^{2},
we have E’ consistent with E=m_{0}c^{2}L. We also have k_{0}=ω_{0}/c,
which leads to

(Eq’n 21)

which is what we expect the description of the momentum to be.

However, the transformation of the angular frequency looks wrong, even though it gives us the correct relativistic increase of the particle’s rest mass. It appears that the period of the traveling wave is not properly time dilated. The resolution of the dilemma lies in the fact that the relativistic Doppler shift inverts; that is,

(Eq’n 22)

For a wave traveling in the negative x-direction, then, the Doppler shift in
the period as seen by an observer traveling in the negative x-direction at a
speed v (as we must do if the particle is to appear to us to be traveling in the
positive x-direction) becomes t’=t_{0}(1-v/c)L. The inverse thus
transforms as

(Eq’n 23)

which is what we used above.

This inherent agreement between Relativity and the quantum theory carries a strong implication that there can be no truly Newtonian quantum theory. That implication is further strengthened by temporal displacement of clocks arrayed along the y-axis are seen by an observer moving in the positive y-direction at a speed u.

As before, successive crests of the waves strike the
barrier at intervals of t=2π/ω.
The temporal displacement of clocks by t=u_{y}L_{y}/c^{2}
can thus be substituted into the phase function of the composite wave, so we
have

(Eq’n 24)

The y-ward momentum represented by that composite wave is thus

(Eq’n 25)

which is what we require. Note that in a purely Newtonian universe there would be no temporal displacement of the clocks and, thus, no aberration of the wave. Note also that the composite wave is automatically deflected by the correct angle: the momentum, represented in the wave number, guarantees it.

So far I’ve been using plane waves to represent the particles in this imaginary experiment. That may be a good approximation for a phalanx of particles or for a particle for which we can pretend that the y-ward momentum is determined precisely, but a proper description of a single particle requires something else. Symmetry requires that the ghostwave of a stationary particle be spherical and that fact entails a strange consequence.

What we require for a spherically symmetric standing wave is a spherical wave radiating away from the particle and one converging upon it. The expanding component seems entirely natural, expanding away from a single point, but the contracting component, converging upon a single point, seems unnatural and impossible. The contracting wave would have to originate at the right times on a widely arrayed set of points encompassing the particle and that distresses causality too much to be acceptable. The contracting wave might be interpreted as a reflection of the expanding wave, but the time lag before the reflected wave could act as the contracting wave (which time lag also prevents the contracting wave from keeping pace with changes in the particle’s motion) falsifies that interpretation. The only interpretation that works properly, that keeps the contracting wave with the particle regardless of how the particle is accelerated, is the one that claims that the contracting wave is a wave that is emitted by the particle and then expands backward in time.

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