# About energy conservation in QM?

1. Sep 23, 2015

### 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?

2. Sep 23, 2015

### 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 $[H_S\otimes I_B, H_{int}] = [ I_S\otimes H_B, H_{int}] = 0$.

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

3. Sep 23, 2015

### Demystifier

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.

4. Sep 23, 2015

### fxdung

Thank you very much for your helpings!

Last edited: Sep 23, 2015