- the electron is a light-speed particle or a wave, similar to the photon, bouncing inside a small "box".
In a wave packet, which is built from solutions to the Dirac equation, the expected position of the electron makes a small circle at the speed of light. The movement is called the zitterbewegung. The role of zitterbewegung, if any, is not currently understood. Anyway, in a sense, the electron is a light-speed particle in a box.
The double-slit experiment
________________________ screen
|
| \
|α \ angle
| \
| \
| \
| \
---------- | ----- \ ----------- two slits
| |
□ □
^ ^
| | v
electron
The boxes □ represent two alternative "paths" of an electron moving vertically at the speed v. The box contains a particle bouncing back and forth at the speed of light.
We should determine the relative phase of the rightmost box when it meets the leftmost box on the screen.
The phase difference determines the interference pattern on the screen. Are we able to reproduce the pattern predicted by the Schrödinger equation?
According to de Broglie, the wavelength of the electron is
λ = h / p,
where p is the momentum:
p = me v.
The crucial thing is what is the length of the path of the light-speed particle in the box.
-----------------
| |
| | v
-----------------
| |
| | v
-----------------
c
Let us assume that, before meeting the double-slit, the light-speed particle moves along the diagonals of a rectangle. It zigzags upward in the diagram above.
We assume that
v << c.
The sides of the rectangle have relative lengths c and v.
|\
|α\
What happens if we tilt the vertical lines in the diagram by some small angle α to the left?
No, we are not able to explain the de Broglie wavelength in this way.
The Schrödinger equation is Galilean covariant?
|
particle • --> p
|
|
double slit
^ y
|
------> x
---> v moving frame
The wavelength of the particle is
λ = h / p.
Now switch to a moving frame, in which the momentum of the particle is smaller. Its wavelength then can be much larger. Can it produce the same interference pattern as in the static frame?
Let us then replace the particle with a laser beam. In the moving frame, the Doppler effect makes the wavelength of the beam longer. Can the interference pattern remain the same? A moving double slit does produce an interference pattern which is different from a static double slit?
Special relativity probably makes the laser interference patterns to match in different moving frames. Electromagnetism is Lorentz covariant.
The Schrödinger equation is Galilean covariant, says the Physics Stack Exchange post. But is it?
Various people on the Internet claim that it is not Galilean covariant.
A particle flux reflected by a potential wall and Galilean covariance
v
| <--- p flux
|
| -p ---> reflected flux
-v
potential
wall
^ y
|
------> x
<--- v moving frame
The free particle solution for the Schrödinger equation is
where p is the momentum of the particle, r is the spatial coordinate of the wave, the energy
E = p² / (2 m),
m is the mass of the particle, and t is the time coordinate of the wave.
The velocity of the particle is
v = p / m.
The phase velocity of the wave ψ:
p Δr = E Δt
=>
Δr / Δt = E / p
= p² / (2 m) * 1 / p
= 1/2 v.
Let us switch to a frame where the momentum p of the flux arriving from the right is almost zero. Then the energy E is almost zero.
In the moving frame, the wave function of the incoming flux is almost constant with respect to the new time and spatial coordinates. The wave function at the wall is almost constant.
The wave function of the outgoing flux is approximately
ψ(r, t) ~ exp(i (2 p • r - 4 E t)).
The phase velocity of the outgoing wave is v, since the velocity of the particle is 2 v. The wall moves to the right at a speed v. Thus, the wave function at the wall is almost constant.
We can find a solution where the sum of the incoming and outgoing wave functions is constant at the wall. Galilean covariance is satisfied.
But is there a problem here? When we transform a wave function to a moving frame, we must transform also p and E. It does not suffice to transform r and t. What determines the phase of the transformed wave function? In the wall example, we were able to find a solution. The solution requires that the wave functions of the incoming flux and the outgoing flux have matching phases. Can we always find such matching phases? Probably yes. We have to check the proofs in the literature.
Does a gravity field or an accelerating potential break the Schrödinger equation?
On February 12, 2025 we were able to show that Maxwell's equations do not have a solution for an accelerating system. This is because linear equations cannot capture the accelerating process and conserve energy.
Can we do the same thing with the Schrödinger equation? The equation is linear and very simple. Can it handle accelerating systems?
Conclusions
We were not able to find a derivation of the Schrödinger equation from zitterbewegung.
Instead, our attention turned to Galilean covariance of the Schrödinger equation. Is it Galilean covariant?
In the next blog post we will investigate this, and also check if an accelerating system can have solutions for the Schrödinger equation.
Is it problematic if the Schrödinger equation is not Galilean covariant? Not really. Since the equation is not relativistic, we know that the equation is only approximate. We know that the equation matches extremely well various empirical tests. It works in practice.
Suppose that the Schrödinger equation does not have solutions for accelerating systems. That is not problematic, either. The equation is approximate, and it is enough to have approximate solutions.
There are problems in adding a potential V to the relativistic Dirac equation. The Klein paradox produces nonsensical results. We can say that we do not know an equation which would accurately describe quantum mechanics.
No comments:
Post a Comment