Time-dependent Perturbation Theory in QM
04 Feb 2019
I spent a little time this morning going over time-dependent perturbation theory in ordinary quantum mechanics. Here I wanted to work it out.
Suppose we have a system with a time-dependent Hamiltonian \( H \). If we can diagonalize \( H \) then we have “solved” the theory. But most of the time, \( H \) is too complicated to solve directly.
Suppose we knew that the hamiltonian could be written in a special form
Here, is a hamiltonian that we know how to solve exactly, and is a time-dependent perturbation. The advantage of viewing the total hamiltonian in this form is that we can work in a basis that diagonalizes . As a result, the time evolution of basis states due to the known hamiltonian is only a phase since the states are eigenvectors. All of the non-trivial dependence on time is due to the perturbation . So, we define an “interaction picture” by factoring out these two different time dependences.
Let be a Schrodinger picture ket at time t. Define
Now, we need to work out what operators look like in the interaction picture.
should be the same as
We have that
for all states
For this to be true, we must have
This defines operators in the interaction picture. We can see that states and operators in the interaction picture already include time evolution due to the unperturbed hamiltonian. Next, we will find the rest of the time dependence comes completely from the perturbation, as we work out differential equations for these quantities.
Let’s naively compute the time derivative of our interaction picture ket
Using the product rule and the ordinary Schrodinger equation, we get
Since commutes with the exponential, we can cancel out those terms and get
Putting in the identity operator in a funny form we get
But check this out. This is just the definition of the interaction picture operators and kets! So finally,
This is exactly how we had hoped - the time dependence of the interaction picture kets is due entirely to the potential.
Time Evolution Operator and Dyson Series
Instead of writing a differential equation for the states, we will convert this into a differential equation for an operator that acts to time-evolve states. We will call this time evolution operator in the interaction picture and require it to satisfy the following properties:
Using the differential equation for interaction picture states, we can work out a differential equation satisfied by
Since this has to hold for all kets, it implies that
With the initial condition
Now, we may not be able to solve this equation directly, but we can approximate a solution using iterated integrals. This is similar to Picard iteration in the proof of existence and uniqueness to solutions of ordinary differential equations.
First, write as
Now, recursively apply this definition of to the term inside the integral. Here, only iterated once, we get:
We distribute to obtain
Nice. We can clean this expression up by introducing the notion of time ordering. In the th integral, we are integrating over the region defined by . Equivalently, we can integrate over the whole hypercube and divide by the number of such regions, . We just have to make sure that the operators are in the right order.
This is the Dyson series.
It expresses the interaction picture time evolution operator as a series solution in terms of the perturbation potential. With it, we can compute approximations to transition amplitudes in a perturbed problem.