About energy conservation in QM?

[fxdung]

Does energy conservation law still hold if the system contact with varying source of energy?
Because in QM the Hamintonian of the system always commune with itself,so the conservation law still correct.But if it is,where is the exchange energy between the system and the enviroment?

 

[JorisL]

So you are thinking of a composite system which you split into a heat bath (general term) and the system you are interested in?

If we call the system S and the bath B we can write the total Hamiltonian as ##H = H_S\otimes I_B + I_S\otimes H_B + H_{int}## where ##I_{S/B}## are unit operators on the respective Hilbert spaces.
Clearly the total Hamiltonian is related to some conserved quantity, the total energy of the combined system.
As is the case for the parts S and B when ##H_{int} = 0##, here ##H_{int}## contains the interactions.

Here we used that the Hilbert space ##\mathcal{H}_{total} = \mathcal{H}_S\otimes \mathcal{H}_B## and that ##(A\otimes B)\cdot (C\otimes D) = (A\cdot C) \otimes (B\cdot D)##

You can see this by showing ##[H_S\otimes I_B, I_S\otimes H_B] = 0## (trivial).
However in the presence of interations, which is the case you are looking at we generally no longer have this conservation.
The conservation remains valid if ##[H_S\otimes I_B, H_{int}] = [ I_S\otimes H_B, H_{int}] = 0##.

Does this clarify your question?

Edit; Demystifier explained it in simpler terms if this is a bit overkill

 

[Demystifier]

Does energy conservation law still hold if the system contact with varying source of energy?
Because in QM the Hamintonian of the system always commune with itself,so the conservation law still correct.But if it is,where is the exchange energy between the system and the enviroment?

In general
$$\frac{dA}{dt}=\frac{\partial A}{\partial t} +\frac{i}{\hbar}[H,A]$$
so for ##A=H## we have
$$\frac{dH}{dt}=\frac{\partial H}{\partial t}.$$
So if ##H## has explicit dependence on time (due to coupling with the environment), then energy is not conserved.

 

[fxdung]

Thank you very much for your helpings!