Undergrad Time dependent Hamiltonian and energy

[hokhani]

TL;DR

Meaning of energy in time dependent Hamiltonians

What is the concept of energy for a general time dependent Hamiltonian? Is there a time dependent energy $E(t)$?

 

[div_grad]

Even in classical mechanics the Hamiltonian $H$ may be interpreted as the energy only when it does not depend explicitly on time. Otherwise, $H(t)$ is just that - the Hamiltonian, a function (obtained through the Legendre transform of a Lagrangian) which enters into Hamilton's equations of motion, the Hamilton-Jacobi equation, etc.

Likewise, in quantum mechanics the Hamiltonian can be interpreted as the energy operator only when it does not depend explicitly on time, because in this case the Schrödinger equation $\mathrm{i}\hbar\partial_t \Psi = H\Psi$ separates and one reduces it to the eigenvalue problem $H\psi = E\psi$. Otherwise, the Hamiltonian operator $H(t)$ is just a differential operator which enters into the Schrödinger equation $\mathrm{i}\hbar\partial_t \Psi = H(t)\Psi$ and now you cannot reduce it to the eigenvalue problem, because it is a genuine evolution equation for $\Psi$.

That being said, you can treat quantum-mechanical problems with time-dependent Hamiltonians using various approximate methods, the 3 most popular of them being: (i) the time-dependent perturbation theory, (ii) the adiabatic approximation, and (iii) the "sudden" approximation. In approach (i), you write the time-dependent Hamiltonian as $H(t) = H_0 + V(t)$, where the time-dependence is assumed to be included solely in some perturbation $V(t)$. Then for $H_0$ you can formulate the eigenvalue problem, and the role of $V(t)$ is then to induce transitions between various eigenstates of $H_0$. In approach (ii), you assume that the time-dependence of the Hamiltonian does not induce transitions to other states, and that for each particular instance of time $t_0$ you write the "eigenvalue" problem as $H(t_0)\psi = E(t_0)\psi$, so that the eigenvalues $E(t_0)$ depend parametrically on time. In approach (iii), you assume that the Hamiltonian is actually time-independent during some period of time $(t_0, t_1)$; then it suddenly changes into another time-independent Hamiltonian and it stays that way during the period of time $(t_1, t_2)$, and so on. But all of these are approximate methods, which may or may not work well in particular situations, and they exist predominantly to treat the genuine evolution equation $\mathrm{i}\hbar\partial_t \Psi = H(t)\Psi$ in as simple way as possible (though not always exact).

 

[div_grad]

It should also be remembered that, in general, the Hamiltonian is not an observable, because it is gauge-dependent. It is expressed in terms of canonical momenta, and not in terms of the kinetic momenta (which are equal to $m\mathbf{v}$, mass times velocity), and these two kinds of momenta do not always coincide. In particular, in the presence of electromagnetic fields, the canonical momenta in the Hamiltonian depend on the choice of gauge for the electromagnetic potentials, and so $H$ depends on the choice of gauge as well. In this case, there is a question of the interpretation of the eigenvalues of the Hamiltonian operator, because then it may not be obvious that they represent any energies of the system. Of course, all this is not an issue if one pays attention in carefully following the various mathematical steps along the solution of the problem. But it sure can lead to wrong results if one relies too much on "hand-waving" and (incorrect) guessing of what the solution or the next particular mathematical step should be, without actually following the first principles.