Energy Conservation in Quantum Systems - Is it Possible?

[gonadas91]

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!

 

[Daeho Ro]

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!

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

 

[gonadas91]

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.

 

[Daeho Ro]

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.

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

 

[gonadas91]

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 shouldn't be conserved.

 

[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.