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/ℏ)|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/ℏ)|n⟩
but the coefficients, cn, will now in general be time-dependent.
The wavefunction satisfies the time-dependent Schrödinger equation;
iℏ∂∂t|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=ℏωn and denoting the time derivative of cn by ˙cn we obtain
iℏ∑n(˙cn−iωncn) exp(−iωnt)|n⟩=∑n(cnℏωn+cnˆV) exp(−iωnt)|n⟩
which simplifies immediately to give
∑n(iℏ˙cn−cnˆV) exp(−iωnt)|n⟩=0
We now premultiply this equation with another eigenstate of ˆH0,⟨m|, to give
iℏ˙cm exp(−iωmt)−∑ncnVmn exp(−iωnt)=0
giving the following set of coupled, first–order differential equations for the coefficients:
iℏ˙cm=∑ncnVmn exp(iωmnt)
where ωmn=ωm−ωn 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⟩.