Is QM Phase Inherently Relativistic?

In summary: So, in summary, the phase of a pure momentum state for a single particle traveling freely is described by (p.r – Et) / h-bar, which also happens to be the Minkowski space-time dot product of the 4-momentum and the space-time 4-vector. This relation is not just a coincidence and has implications for relativistic quantum mechanics.
  • #1
LarryS
Gold Member
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The phase of a pure momentum state for a single particle traveling freely is

(p.r – Et) / h-bar.

This also happens to be the Minkowski space-time dot product of the 4-momentum and the space-time 4-vector (relativistic action).

Is that just a coincidence or is phase inherently relativistic?
 
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  • #2
Good question.

If you use a relativistic wave equation this phase somehow survives in the non-relativistic limit. But I guess that is not really the explanation you are expecting.
 
  • #3
In The Feynman Lectures Vol 3 Ch 7, Feynman begins by postulating that the wave function for a particle at rest is constant in space and varies as [itex]e^{-iE t/ \hbar}[/itex] in time. He then uses the relativistic dot product to derive the wave function for a moving particle.

In other words, a particle at rest can be thought of as being in a plane wave that is headed purely in the time direction. Its wavefronts are planes of constant time. However, with respect to a moving frame, those surfaces of constant time will now be "tilted" in spacetime, leading to the spatial variation of [itex]e^{ipx/\hbar}[/itex].
 
  • #4
matonski said:
In The Feynman Lectures Vol 3 Ch 7, Feynman begins by postulating that the wave function for a particle at rest is constant in space and varies as [itex]e^{-iE t/ \hbar}[/itex] in time. He then uses the relativistic dot product to derive the wave function for a moving particle.

In other words, a particle at rest can be thought of as being in a plane wave that is headed purely in the time direction. Its wavefronts are planes of constant time. However, with respect to a moving frame, those surfaces of constant time will now be "tilted" in spacetime, leading to the spatial variation of [itex]e^{ipx/\hbar}[/itex].

Interesting. I will have to read up on that.
 
  • #5
referframe said:
The phase of a pure momentum state for a single particle traveling freely is

(p.r – Et) / h-bar.

This also happens to be the Minkowski space-time dot product of the 4-momentum and the space-time 4-vector (relativistic action).

Is that just a coincidence or is phase inherently relativistic?

In non-relativistic QM, E=p^2/2m.
 
  • #6
Meir Achuz said:
In non-relativistic QM, E=p^2/2m.

Yes, and that expression normally occurs in operator form on left side of the Schrodinger equation. But the right side of that equation and also the definition of the momentum and energy operators is based on the expression (p.r - Et)/h-bar .
 

1. What is quantum mechanics (QM)?

Quantum mechanics is a branch of physics that studies the behavior and interactions of particles on a very small scale, such as atoms and subatomic particles.

2. What is the role of relativity in QM?

Relativity is not a fundamental part of quantum mechanics, but it is important in understanding the behavior of particles at high speeds or in strong gravitational fields. The principles of relativity help to reconcile the differences between classical mechanics and quantum mechanics.

3. Can QM be understood without considering relativity?

Yes, quantum mechanics can be understood without considering relativity. In most cases, the effects of relativity on particles are negligible and not relevant to the behavior being studied.

4. How does QM take into account the principles of relativity?

QM takes into account the principles of relativity by using the mathematical framework of special relativity to describe the behavior of particles at high speeds. This allows for the prediction and explanation of phenomena such as time dilation and length contraction.

5. Is the phase factor in QM inherently relativistic?

Yes, the phase factor in QM is inherently relativistic. The phase factor, also known as the wave function, is a fundamental concept in QM that describes the probability of a particle being in a certain state or location. It takes into account the principles of relativity to accurately predict the behavior of particles in different reference frames.

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