5.1: Time–Dependent Hamiltonians

Recall that for a system described by a Hamiltonian, $\hat{H}_0$, which is time–independent, the most general state of the system can be described by a wavefunction $|\Psi , t \rangle$ which can be expanded in the energy eigenbasis $\{|n \rangle \}$ as follows:

$$|\Psi , t\rangle = \sum_n c_n \text{ exp}(−iE_nt/\hbar )|n\rangle \nonumber$$

where the coefficients, $c_n$, are time-independent, and $E_n$ denotes the eigenvalue corresponding to the energy eigenstate $|n \rangle$ of $\hat{H}_0$.

When we generalise to the case where the Hamiltonian is of the form

$$\hat{H} = \hat{H}_0 + \hat{V} (t) \nonumber$$

we can again expand in $|n \rangle$, the time-independent eigenbasis of $\hat{H}_0$

$$|\Psi , t\rangle = \sum_n c_n(t) \text{ exp}(−iE_nt/\hbar )|n\rangle \nonumber$$

but the coefficients, $c_n$, will now in general be time-dependent.

The wavefunction satisfies the time-dependent Schrödinger equation;

$$i\hbar \frac{\partial}{\partial t} |\Psi , t \rangle = \hat{H} |\Psi , t\rangle \nonumber$$

so that we can substitute the expansion of $|\Psi , t\rangle$ to determine the equations satisfied by the coefficients $c_n(t)$. Writing $E_n = \hbar \omega_n$ and denoting the time derivative of $c_n$ by $\dot{c}_n$ we obtain

$$i\hbar \sum_n (\dot{c}_n − i\omega_n c_n) \text{ exp}(−i\omega_n t)|n \rangle = \sum_n (c_n \hbar \omega_n + c_n\hat{V} ) \text{ exp}(−i\omega_nt)|n \rangle \nonumber$$

which simplifies immediately to give

$$\sum_n (i\hbar \dot{c}_n − c_n\hat{V} ) \text{ exp}(−i\omega_n t)|n \rangle = 0 \nonumber$$

We now premultiply this equation with another eigenstate of $\hat{H} 0, \langle m|$, to give

$$i\hbar \dot{c}_m \text{ exp}(−i\omega_m t) − \sum_n c_n V_{mn} \text{ exp}(−i\omega_nt) = 0 \nonumber$$

giving the following set of coupled, first–order differential equations for the coefficients:

$$\boxed{i\hbar \dot{c}_m = {\sum}_n c_nV_{mn} \text{ exp}(i\omega_{mn}t)} \nonumber$$

where $\omega_{mn} = \omega_m − \omega_n$ and $V_{mn} = \langle m|\hat{V} |n \rangle$.

This tells us how the coefficient $c_m$ varies with time, i.e. the probability that a measurement will show the system to be in the $m^{th}$ eigenstate. It is exact, but not terribly useful because we must, in general, solve an infinite set of coupled differential equations.

It is worth dwelling on the importance of the quantity $V_{mn}$. This ‘matrix element’ is an integral which tells us how much the potential $\hat{V}$ mixes states $|m \rangle$ and $|n\rangle$. If it is zero (which it often is, by symmetry) then $\hat{V}$ cannot induce a transition between states $|m \rangle$ and $|n\rangle$.