Time-Dependent Perturbation
One of the principal tasks in quantum mechanics is the evaluation of transition probabilities from say state Ф„ to another state Ф,,,. This takes place under the influence of a time-dependent perturbation v(q, t) that tailors the transition. When does this transition of a system from one state to another makes sense? It may make sense if the reason for the transition, i.e., v(q, t) acts only within a finite time period, say from t = t0 to t = T. With the exception of this time period, the total energy is a constant of motion that can be measured. For the case when the Hamiltonian is dependent on time f, there are no stationary solutions of the Schrodinger equation. Hence the identification of bound states with discrete energy levels and stationary eigenfunctions must be modified. The time-dependent perturbation theory or so-called method of variation of constants assures that.
The Hamiltonian H of the system:
where Ho is the unperturbed Hamiltonian and describes a conservative system i.e., that which is independent of time t. Tlie index 0 stands for the time independence. The perturbation v(f) is dependent on time t and has the effect of causing transitions between eigenstates of Ho that will be stationary in the absence of v(f).
Instead of decomposing the Hamiltonian in 11.43, the adiabatic approximation assures that H has parameters that change very slowly with time. This follows that the system can be described approximately by means of stationary eigenfunctions of the instantaneous Hamiltonian. We finally consider the sudden approximation in which H is constant in time except for very short time intervals when it changes from one form to another.
If
then the state of a system should be described by the time Schrodinger equation:
Example
Solve equation 11.45 considering that
Thus
where 4'!,0|(f )is the eigenfunction and E„ eigenvalue of the Hamiltonian Hu. Equation 11.47 is the solution of equation 11.45 for Hu. The functions 11.47 form a complete set of basis functions. The solution of 11.45 may be expanded in terms of these functions:
If the Hamiltonian H should have not been dependent on time t then C,„should also not have been dependent on time t. If we substitute 11.48 and 11.43 into 11.45 then
We consider this time-dependent perturbation v(q, t) described by the following transition amplitude K„:
We show that this formula mimics the generalized formula of the perturbation theory. We write the propagator K„ via the time-independent Schrodinger eigenstates |Ф„) [1]:
Here, the Heaviside step function:
entails compelling the system at a starting point to be evolving towards the future and
with £„ being the eigenenergies for the eigenstates n. Expression 7.43 might be viewed as if the particle moves around in all possible locations in the energy space (acquiring different phase factors) as it propagates from the selected initial position to the final position.
We substitute 11.51 into 11.50 so that in the second summand of 11.50 we have
The perturbed amplitude in the energy eigenbasis is then:
Letting the matrix element of the perturbation operator constructed on the eigenfunctions ) and Фm(qc):
The quantity v„m(fc) is the matrix element of the potential and is time dependent as the perturbation itself is dependent explicitly on time. The quantity A,wl, representing the expansion coefficient or the probability amplitude that at time moment tb the particle will be found in the state with m if at the initial time moment ta the particle was found in the state with я, is expressed as follows:
where,
is the Bohr angular frequency for the transition:
Then
Multiplying this relation by Ф,„(qb) and then integrate over tb we represent the perturbation as small series representation via the expansion coefficient A„m(tb, t„) as function of the perturbation v:
For the zero order correction, we assume that before the introduction of the perturbation the system is found at the я state. For the first order correction we have
Suppose that the wave function at the time moment t, is Ф„(^,) then the wave function at the time moment f, is:
This implies that the wave function at the time moment f, may have the form
FIGURE 11.7 Transition amplitude represented pictorially in higher orders of the perturbation theory.
This is the first order term with interest being the scattering probability from the state with quantum number n to one with m at the time interval dt expressed via
The transition amplitude in 11.60 can be represented pictorially in higher orders of the perturbation theory (Figure 11.7):
The first order describes the direct transition from the state with m to one with n and the second order together with higher orders transitions via intermediate states where all the intermediate virtual states k,l, ••• are summed over, and the interaction times
are integrated over. The evaluation of the transition amplitude via the sequence of the first order is valid for the case when the perturbation is sufficiently small, as indicated earlier. However, there exist cases when the velocity of variation of the perturbation v(f) is insignificant that v may not be small. This case may be observed from slow atomic collisions where the adiabatic approximation based on the eigenfunctions and eigenvalues of the full Hamiltonian is very appropriate, as shown at the start of this section.
