Relativity, time, and quantum mechanics

[anuttarasammyak]

[Mentors’ note: Spun off from this thread]
I understand that you are interested in the concept of time in quantum mechanics. Here are examples of my questions, in case they may be of reference.

１.How can one calculate the proper time of an electron in the 1s orbital of a hydrogen atom? How should the Lorentz factor be applied to motion that is not classical?　Would that result in a superposition of different proper times?

2. To determine the invariant spacetime interval between events associated with a particle, one would obtain the spacetime coordinates (t, x, y, z). However, due to the uncertainty principle, measuring the spatial coordinates x, y, z imparts momentum to the particle, so its rest frame changes from the original one. It would be a mess. Since there is no measurement operator for t, does time play the role of reconciling this inconsistency?

3. It appears that, with respect to a measurement at time zero, when collapse takes place in interaction with apparatus or environment, the state before measurement and the state after measurement are asymmetric, e.g. entangled and disentangled. Could this be the origin of thermodynamic or arrow of time?

 

[Nugatory]

１.How can one calculate the proper time of an electron in the 1s orbital of a hydrogen atom?

How should the Lorentz factor be applied to motion that is not classical?[/quote]We don’t. Proper time is the length of a path through spacetime between two events; the electron has no path or position so the notion “the proper time of an electron” is meaningless. Instead we work with the time measured by our lab clock between observation events.

2. To determine the invariant spacetime interval between events associated with a particle, one would obtain the spacetime coordinates (t, x, y, z). However, due to the uncertainty principle, measuring the spatial coordinates x, y, z imparts momentum to the particle, so its rest frame changes from the original one.

X,y,z,t are the coordinates of the observation event using the frame in which the lab is at rest. The “rest frame of the electron” never enters into the calculation (unsurprisingly, because it’s not defined).

3. It appears that, with respect to a measurement at time zero, when collapse takes place in interaction with apparatus or environment, the state before measurement and the state after measurement are asymmetric, e.g. entangled and disentangled.

The global collapse interpretation you are trying to use here is non-relativistic so cannot be applicable here. Instead we have to use the methods of quantum field theory, in which there is no global collapse.

Could this be the origin of thermodynamic or arrow of time?

We’ll find that in statistical mechanics.

 

[anuttarasammyak]

Addeentum to 1 of post #1 on proper time in quantum mechanics :
Consider a double-slit experiment with an electron beam at fixed accelerating voltage. Suppose that each electron carries its own clock and reads the time when it is just before the slits and again when it hits the screen. The difference in the readings would reveal which of the two slits the electron passed through. It therefore seems that precice proper time reading (= which-path information) cannot be compatible with the presence of interference.

 

[anuttarasammyak]

the electron has no path or position so the notion “the proper time of an electron” is meaningless.

Let me confirm for my understanding. Even in the 1s state of exotic hydrogen atom consisting of proton and muon, does muon not collapse due to its no proper time? If it collapses, won't the long-life effect of motion take place ?

 

[renormalize]

How should the Lorentz factor be applied to motion that is not classical?

We don’t. Proper time is the length of a path through spacetime between two events; the electron has no path or position so the notion “the proper time of an electron” is meaningless.

Let me confirm for my understanding. Even in the 1s state of exotic hydrogen atom consisting of proton and muon, does muon not collapse due to its no proper time? If it collapses, won't the long-life effect of motion take place ?

The answer by @Nugatory is incorrect. As suggested by @anuttarasammyak, experiments with muonic atoms show that bound particles do experience time dilation, as reported in this paper:
Relativistic time dilatation of bound muons and the Lorentz invariance of charge
Abstract: The relativistic lengthening of the lifetime of a decaying elementary particle bound in a stationary state of an exotic atom provides evidence that the particle is actually in motion even though such motion cannot be visualized classically. This application of special relativity to the particles within an atom helps illuminate an argument frequently used for the Lorentz invariance of electric charge.
I don't have access to this paper, but an answer at https://physics.stackexchange.com/questions/730311/ by someone who has read it states:
Muonic atom decay lifetimes were measured back in the 1960s, and these lifetimes agreed with relativistic theoretical calculations. For example, the theoretically expected time dilation for lead was 16-20%, and the experimental value was 14±4%. Not a very precise test, but enough to strongly suggest the atomic muons are subject to time dilation.

 

[Nugatory]

The answer by @Nugatory is incorrect. As suggested by @anuttarasammyak, experiments with muonic atoms show that bound particles do experience time dilation, as reported in this paper:

I think you misread what I said.

 

[renormalize]

I think you misread what I said.

That's certainly possible! Given that you say "the electron has no path or position so the notion “the proper time of an electron” is meaningless" how then would you interpret those experiments that purport to demonstrate decay-time dilation for muons in atomic stationary states?