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E Fermi's Golden Rule
The time dependent Schrödinger Equation can be written as
where H_{1} is a small perturbation to the Hamiltonian H_{0}. The solution
of the unperturbed case is
The eigenvalues are
,
and the orthonormal eigenfunctions are
.
Under the assumption that the perturbation H_{1} 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 c_{m}(0)=1, while c_{n}(0)=0for all other states. Furthermore, it is assumed that the scattering out of the
initial state is small, so that c_{m}(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).
Figure:
The function
has the
property of a
function.

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, E_{m} = E_{n}.
Next: F Second Order Co
Up: Dissertation Christoph Wasshuber
Previous: D Integration of Fermi
Christoph Wasshuber