# Conservation of energy

1. Oct 8, 2015

Hi guys! one quick question, if in a quantum system the hamiltonian of a particle evolves with time (let's say, the potential is a function of t), the energy is not conserved right? I just want to be sure about this, thanks!

2. Oct 8, 2015

### Daeho Ro

Then, where the energies gone? Whether the Hamiltonian is time-dependent, Schrodinger equation gives constant energy.

3. Oct 8, 2015

But if you think of the hamiltonian as a matrix, that means that the matrix has different matrix elements for every different time, therefore its eigenvalues are not the same as time evolves.

4. Oct 8, 2015

### Daeho Ro

Then, the wave function have to vary depending on time to keep the energy as constant.

5. Oct 8, 2015

Ok, but think about a non translationally invariant system. Momentum is not conserved in such a system because translation invariance is broken. If you break rotation invariance, angular momentum is not conserved in such a system, and if you break time invariance, then energy shouldnt be conserved.

6. Oct 8, 2015

### Daeho Ro

If it breaks the time translational invariance, then yes. It may come from
$$\dfrac{\partial}{\partial t} \int dV \psi^*(x,t) \hat{H}(x,t) \psi(x,t) ,$$
and it is not vanishing in general. Then, the energy can't conserved.

I confused with the stationary solution in quantum mechanics textbook. Sorry for that.