Metadata
shorthands: {}
aliases: [Interaction picture]
created: 2022-01-09 02:59:10
modified: 2022-01-16 13:48:42
We consider a Hamiltonian operator, which in the Schrödinger picture is composed as the sum of a time independent part,
Within the Dirac picture, the operators evolve in time according to the time evolution operator related to
With the boundary condition
The time evolution of the operators is described by:
The equation of motion of operators looks formally similar as in the Heisenberg picture (see Quantum mechanical time derivative):
Where
In particular, the time independent part of the Hamiltonian operator remains time independent:
Equation of motion for the wave function within the Dirac picture:
Where
is the perturbation operator in the Dirac picture.
The time dependence of the matrix elements of observables:
Which is formally the same as in the Schrödinger and Heisenberg pictures.
The time dependence of the wave function can be expressed in terms of a new time evolution operator:
With
Using the perturbation operator, we get the differential equation for the time evolution operator in the Dirac picture:
Which yields the integration:
This integral equation can be solved by successive approximation as in Schrödinger picture#The time evolution operator.
Where we introduced the time-ordering operator:
Using the Dirac picture, we can get the results of time-dependent perturbation theory. We take the time evolution operator up to first order in the Dirac picture:
And the wave function in the Dirac picture based on this:
The wave function in the Schrödinger picture can be written as:
Now we assume that the initial state at
And we will use the spectral resolution of
The wave function can then be evaluated as:
Where
And
Thus, the wave function in first order
Where
And for