I have a fundamental question about Quantum Mechanics or even mechanics in general. I am aware that there are stationary solutions and non-stationary solutions. The stationary solutions solve Schrödinger equation for time independent Hamiltonians, whereas non-stationary solutions are used to solve for time-dependent Hamiltonians. However, my question is whether strictly speaking there shouldn't be any time dependent Hamiltonians at all. The whole approach of time-dependent Hamiltonians seems artificial and premeditated to me, because one has to assume dynamics or evolution at a certain layer, which should actually be a prediction of the theory.
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2$\begingroup$ I don't understand the issue. For example, if we had a particle in a magnetic field that changes with time we would have a time dependent Hamiltonian. How is this artificial? $\endgroup$– BioPhysicistJun 22, 2018 at 20:07
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$\begingroup$ I think it is in a sense artificial, because the evolution of the magnetic field is contrived. Instead of explaining both the evolution of the magnetic field as well as the particle one is assuming the evolution of the magnetic field and calculating accordingly. $\endgroup$– eeqesriJun 22, 2018 at 20:10
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Based on your comment it seems like the issue is that we should make our system large enough so that our system is closed. While there is nothing wrong with this, you might be making the problem way harder than it needs to be.
For example, if we are looking at a particle in a time varying magnetic field, if we are only interested in the behavior of that specific particle then we would not want to describe everything else that is causing this magnetic field. We could include everything, but it does not help us to do so.
So if you definition of contrived is that we are not including the behavior of everything that makes the changing magnetic field, then you can call it contrived. However, in most situations this is a lot better than keeping track of many more things.
answered Jun 22, 2018 at 20:16
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$\begingroup$ I totally agree with you and I am convinced about the predictions of modern physics in such cases. It is just that the concept of a time varying Hamiltonian bugs me because in such cases for example the back reaction of the system on the time dependent component of the system is neglected. But I understand that in most cases it is probably negligible. $\endgroup$– eeqesriJun 22, 2018 at 20:21
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2$\begingroup$ @user139383 I understand what you are saying. The point is that in all calculations and models we use there always has to be some type of simplification. We just have to make sure we are justified in the simplifications we make. So, like you say, in a system where our single particle has little influence on the applied field then we are justified in looking at just the particle. If we have reason to suspect that our particle has a larger influence, we must make our system larger to account for this. $\endgroup$ Jun 22, 2018 at 20:24
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$\begingroup$ @user139383, You are sort of contradicting yourself. The same "back reaction" is ignored in the time-independent case. You are assuming that some mechanism creates a quadratic potential or a -1/r (e.g. Hydrogen Atom) potential and that source is never affected by the particle moving in it. If you can believe that you should have no problem accepting a time dependent potential controlled by an external mechanism that is not affected by the particle under observation. $\endgroup$– user196418Jun 22, 2018 at 21:22