Time-Dependent Hamiltonians

Almost everything interesting in quantum mechanics — atoms absorbing light, qubits being driven by microwave pulses, particles decaying — happens because the Hamiltonian changes in time. Up to now we mostly studied systems whose energy operator $\hat H$ was fixed. This module is about what happens when it is not.

Splitting off a time-dependent piece

The standard setup writes the Hamiltonian as a known, time-independent part plus a small time-dependent perturbation:

\hat H(t) = \hat H_0 + \hat H'(t).

Here $\hat H_0$ is the unperturbed Hamiltonian whose eigenstates and energies we already know,

\hat H_0 \, |n\rangle = E_n \, |n\rangle ,

and $\hat H'(t)$ is a perturbation that switches on at some time and is assumed small compared with the spacings of $\hat H_0$ . The key new feature is that $\hat H'(t)$ carries explicit time dependence — for example an oscillating electric field $\hat H'(t) = -q\,\hat x\,\mathcal{E}_0\cos(\omega t)$ acting on a charged particle.

Stationary states are no longer stationary

For a time-independent Hamiltonian, an energy eigenstate evolves only by an overall phase, $|n\rangle \to e^{-iE_n t/\hbar}|n\rangle$ , so the probabilities of all observables stay frozen — that is why they are called stationary states. Once $\hat H'(t)$ is present this is no longer true. The exact state obeys the full time-dependent Schrödinger equation

i\hbar \, \frac{\partial}{\partial t}\,|\Psi(t)\rangle = \big(\hat H_0 + \hat H'(t)\big)\,|\Psi(t)\rangle .

Because $\hat H'(t)$ couples different eigenstates of $\hat H_0$ , a system that starts in $|i\rangle$ can acquire amplitude in other states $|f\rangle$ . The central question of the module is: what is the probability of finding the system in a different state $|f\rangle$ after the perturbation has acted?

Expanding in the unperturbed basis

Since the eigenstates $\{|n\rangle\}$ of $\hat H_0$ form a complete orthonormal basis, we may write the exact state at any time as

|\Psi(t)\rangle = \sum_n c_n(t)\, e^{-iE_n t/\hbar}\,|n\rangle .

The factor $e^{-iE_n t/\hbar}$ is pulled out deliberately: it is the trivial evolution that would happen even without the perturbation. All the interesting physics is then packed into the coefficients $c_n(t)$ . If $\hat H'(t) = 0$ , the $c_n$ are constant and we recover ordinary stationary-state evolution. When $\hat H'(t) \neq 0$ , the $c_n(t)$ drift, and $|c_n(t)|^2$ is the probability of finding the system in state $|n\rangle$ at time $t$ .

What "small" buys us

Treating $\hat H'(t)$ as small lets us solve the equations of motion for the $c_n(t)$ order by order in the strength of the perturbation, just as in time-independent perturbation theory. To lowest (first) order, the amplitude to leave the initial state is linear in $\hat H'$ , and the transition probability is quadratic in $\hat H'$ . Over the next lessons we will derive that first-order amplitude, square it to get transition probabilities, specialize to harmonic (sinusoidal) driving, find the resonance condition, and finally arrive at Fermi's golden rule for transition rates into a continuum of final states.

Common physical perturbations

A few perturbations recur throughout atomic, molecular, and condensed-matter physics:

Sudden switch-on: $\hat H'(t) = \hat V$ for $t > 0$ and zero before — a constant perturbation turned on at $t=0$ .
Harmonic drive: $\hat H'(t) = \hat V\cos(\omega t)$ — models a monochromatic light field or a microwave tone driving a qubit.
Pulse: $\hat H'(t)$ localized in time, such as a Gaussian envelope — models a laser pulse hitting an atom.

Each of these is just a specific choice of $\hat H'(t)$ slotted into the same framework. Mastering the general setup here means every later result is a special case rather than a new derivation.