Time-dependent perturbation theory

The time-dependent perturbation theory is a method for finding an approximate solution to a problem that is characterised by the time-varying properties of a quantum mechanical system. For example, the electronic state of a molecule exposed to electromagnetic radiation is constantly perturbed by the oscillating electromagnetic field. Time-dependent perturbation theory is key to finding solutions to such problems, and is used to calculate transition probabilities in spectroscopy.

Consider the unperturbed eigenvalue equation  given by eq262, where is the unperturbed Hamiltonian, is a complete set of orthonormal eigenfunctions of in a Hilbert space, and represents the exact eigenvalue solutions. Furthermore, let’s rewrite eq57


is the solution of the time-dependent Schrodinger equation. If it describes a stationary state, then

Otherwise, can be expressed generally as the linear combination , which is also a solution of eq281a.

In the presence of radiation, we add a correction term to the Hamiltonian in eq281a:

where is the perturbed wavefunction.

Since belong to the same Hilbert space, which is spanned by , we can express as

where the coefficient is now a function of time because we expect , which is the probability that a measurement of a system will yield an eigenvalue associated with , to change with time since the Hamiltonian is now time-dependent.


Show that .


Using ,


If we assume that the perturbation was turned on at time , then the system before is characterised by a stationary state. Substituting in eq281d, we have , which must be equal to eq281b at . This implies that

Substituting eq281d in eq281c and using the unperturbed eigenvalue equation, we get

Multiplying the above equation by , integrating over all space and using , we have

Assuming that the variation of  is insignificant because the perturbation is small and over a short time from to , then . Substituting eq281e in eq281f, we have

Integrating the above equation from to and using eq281e,

Eq281g, when substituted in eq281d, gives the approximate perturbed wavefunction  at :

where is the probability that a measurement of a system will yield an eigenvalue associated with  after perturbation at .

Let’s suppose the perturbation on an atom or molecule is caused by a plane-polarised electromagnetic wave with an electric component  oscillating in the -direction. According to classical physics, the force on a charge on the atom or molecule is . It is also the negative gradient of potential energy  of interaction between the field and the charge, i.e. . Therefore, , whose integrated form is , which is consistent with the classical interaction energy between the electric field and a charged particle. If we consider multiple charges and an oscillating electric field, the perturbation can be expressed approximately as

where is the maximum amplitude of the electric field in the -direction, and is the operator for the -component of the atom or molecule’s dipole moment, i.e. .


Why did we not consider the magnetic component of the electromagnetic wave?


This is because interaction between the magnetic component of the electromagnetic wave and the charges of an atom or molecule is more than a hundred times weaker than interaction between the electric component of the electromagnetic wave and the charges of the atom or molecule.


Substituting eq281h, and in eq281g for ,

Since is a function of time and not position, it is a constant with regard to the integral with respect to position and we have

When , we have , which implies that corresponds to the transition frequency of an atom or molecule from state to . Therefore, is the probability that a measurement of a system will yield an eigenvalue associated with after perturbation at from the state at ( is not infinite when , see Q&A below). In other words, the transition probability from state to of an atom or molecule irradiated by an electromagnetic wave oscillating in the -direction is proportional to , which is denoted by .


Evaluate when .


Let , with . As , we have .  Using L’Hopital‘s rule,


For an electromagnetic wave that is not plane-polarised, and . We called the transition dipole moment. In general, transitions are allowed when , while transitions are forbidden when .


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