Loading [MathJax]/jax/output/HTML-CSS/jax.js
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Physics LibreTexts

5.1: Time–Dependent Hamiltonians

( \newcommand{\kernel}{\mathrm{null}\,}\)

Recall that for a system described by a Hamiltonian, ˆH0, which is time–independent, the most general state of the system can be described by a wavefunction |Ψ,t which can be expanded in the energy eigenbasis {|n} as follows:

|Ψ,t=ncn exp(iEnt/)|n

where the coefficients, cn, are time-independent, and En denotes the eigenvalue corresponding to the energy eigenstate |n of ˆH0.

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

ˆH=ˆH0+ˆV(t)

we can again expand in |n, the time-independent eigenbasis of ˆH0

|Ψ,t=ncn(t) exp(iEnt/)|n

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

The wavefunction satisfies the time-dependent Schrödinger equation;

it|Ψ,t=ˆH|Ψ,t

so that we can substitute the expansion of |Ψ,t to determine the equations satisfied by the coefficients cn(t). Writing En=ωn and denoting the time derivative of cn by ˙cn we obtain

in(˙cniωncn) exp(iωnt)|n=n(cnωn+cnˆV) exp(iωnt)|n

which simplifies immediately to give

n(i˙cncnˆV) exp(iωnt)|n=0

We now premultiply this equation with another eigenstate of ˆH0,m|, to give

i˙cm exp(iωmt)ncnVmn exp(iωnt)=0

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

i˙cm=ncnVmn exp(iωmnt)

where ωmn=ωmωn and Vmn=m|ˆV|n.

This tells us how the coefficient cm varies with time, i.e. the probability that a measurement will show the system to be in the mth 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 Vmn. This ‘matrix element’ is an integral which tells us how much the potential ˆV mixes states |m and |n. If it is zero (which it often is, by symmetry) then ˆV cannot induce a transition between states |m and |n.


This page titled 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 the style and standards of the LibreTexts platform.

Support Center

How can we help?