$$\require{cancel}$$

# 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 $$|n\rangle$$.

5.1: Time–Dependent Hamiltonians is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Graeme Ackland via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.