In a lot of conclusions and studies from quantum mechanics, we need to discuss whether the Hamiltonian is time-dependent or not. If the Hamiltonian is time-dependent, is it always the potential that causes the time-dependency? Is there a counterexample? Since $$\hat H=-\frac{\hbar^2}{2m}\nabla^2+U$$ If $U$ is not time-dependent but $\hat H$ is, then it is only possible that $m$ is time-dependent. Is this possible? What about a multiple-particle system?
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2 Answers
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I'm not aware of any cases where the mass is time dependent.
The potential is usually the time dependent part. For single-particle systems, could mean you are changing some external field. The most common example might be if you have a incoming electromagnetic (light) incident on an electron in the 1s state of a hydrogen atom. A mathematically easier problem would be a particle in an infinite square well and slowly ramping up a potential on just the left side of the well.
For many-body systems, there are a few different approaches. The most general is that you now have a multi-particle wavefunction $\psi ( \vec r_1, \vec r_2, ....)$ that is a function of all the positions of the particles. The interactions of the particles can be put into a potential $U(\vec r_1, \vec r_2,...)$. If there are no external time-dependent fields then this can be solved as a time-independent problem. But there could still be some time-depended external perturbations (like a field, for example).
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$\begingroup$ What about the case where the potential has a nonzero imaginary part. This corresponds to an unstable particle. Does mass change in this case? I am not sure how to interpret this case correctly. $\endgroup$– trisctCommented Apr 16, 2020 at 3:55
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$\begingroup$ @trisct but then the Hamiltonian wouldn’t be hermitian $\endgroup$ Commented Apr 16, 2020 at 8:03
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Quantum mechanics describes the interactions of charged particles with the electromagnetic field. Thus, the time dependence may come:
- Via the scalar potential (i.e. via the potential term)
- Via the vector potential (i.e. via the kinetic term)
- Via the magnetic field in the Zeeman term.
Moreover, one often deals with effective Hamiltonian, e.g., when working in the effective mass approximation, where the time dependence may appear even in the (effective) mass.
