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E Fermi's Golden Rule

The time dependent Schrödinger Equation can be written as

where H1 is a small perturbation to the Hamiltonian H0. The solution of the unperturbed case is

The eigenvalues are , and the orthonormal eigenfunctions are . Under the assumption that the perturbation H1 is small, the wave function for the perturbed case can be expanded in a series of the orthonormal wave functions of the unperturbed case [29].

Inserting (E.3) into the time dependent Schrödinger equation (E.1), and using (E.2) gives

If this is multiplied by , and integrated over the volume one obtains

since orthonormality makes

which vanishes for and equals unity for m=n. With the definitions

one can write (E.5) as

In general, when the perturbation is turned on at t=0, it may be assumed that the system is in state m at t=0, and cm(0)=1, while cn(0)=0for all other states. Furthermore, it is assumed that the scattering out of the initial state is small, so that cm(t)=1 for all time. This assumption neglects the  conservation of particles. Considering these assumptions, one can solve (E.8).

The probability of occupying state n is then

For times , which are long enough such that the scattering process has been completed, the function

has the property of a -function (see Fig. E.1).

The transition probability per unit time, the   transmission rate, is then given by the so called Fermi Golden Rule

This means that a scattering process, in our case a tunnel event, only takes place if the energy of the particle is conserved, Em = En.

Next: F Second Order Co Up: Dissertation Christoph Wasshuber Previous: D Integration of Fermi

Christoph Wasshuber