Time dependent Hamiltonians: features

[abhinavd]

What is the definition of energy for quantum systems with time dependent Hamiltonians? Is it the eigenvalue of the Hamiltonian? (The eigenvalue is, in general, time dependent). However, the eigenstates of the Hamiltonian (even if it is time dependent) are stationary states, and hence no quantities must change with time. What is the reason for this inconsistency?

This leads us to this general question in classical mechanics: What is the general definition of energy when non-conservative forces are present? We'd defined energy as a quantity that remains unchanged with the time translational invariance of the Lagrangian, but that does not hold when time dependent potentials are present.

 

[juanrga]

What is the definition of energy for quantum systems with time dependent Hamiltonians? Is it the eigenvalue of the Hamiltonian? (The eigenvalue is, in general, time dependent). However, the eigenstates of the Hamiltonian (even if it is time dependent) are stationary states, and hence no quantities must change with time. What is the reason for this inconsistency?

This leads us to this general question in classical mechanics: What is the general definition of energy when non-conservative forces are present? We'd defined energy as a quantity that remains unchanged with the time translational invariance of the Lagrangian, but that does not hold when time dependent potentials are present.

\left\langle E \right\rangle \equiv Tr \{ \hat{H} \hat{\rho} \}

 

[abhinavd]

\left\langle E \right\rangle \equiv Tr \{ \hat{H} \hat{\rho} \}

You did not address the question of whether the energy is a function of time, and how it reconciles with the fact that it is a stationary state. So before going on to mixed states, let's answer the question for pure states, the eigenstates of the time dependent Hamiltonian.

 

[juanrga]

You did not address the question of whether the energy is a function of time, and how it reconciles with the fact that it is a stationary state. So before going on to mixed states, let's answer the question for pure states, the eigenstates of the time dependent Hamiltonian.

The expression for the energy is valid when H=H(t), and deals with both pure and mixed states. The equation for dE/dt is derived from it in the usual way (just derive both sides of the equation)

 

[xepma]

When the Hamiltonian is time-dependent, time translational invariance is absent. Since the symmetry is absent, the corresponding Noether charge is not a conserved quantity. Put differently: energy is not conserved in systems with a time-dependent Hamiltonian. This applies to both classical and quantum mechanical systems.

For classical systems we can still define the energy, and this quantity will violate conservation laws (i.e. be time-dependent). For quantum mechanical systems the energy is no longer a proper quantum number and cannot be used to label states. The best thing we can do is to use the formula given by juanrga -- it applies to both pure and mixed states.