Subatomic Physics

10.1. The Golden Rule 283
Schrödinger equation gives (˙an ≡ dan/dt)
i� �
� �
−iEnt
˙anun exp +
�
n
�
� �
−iEnt
Enanun exp
�
n
= �
� �
−iEnt
an(H0 + Hint)un exp .
�
n
With equation (10.4), the second term on the left-hand side and the first term on
the right-hand side cancel. Multiplying by u∗ N from the left, integrating over all
space, and using the orthonormality relation, produce the result
i�˙aN = �
� �
i(EN − En)t
〈N|Hint|n〉an exp
. (10.8)
�
n
Here, a convenient abbreviation for the matrix element of Hint has been introduced:
�
〈N|Hint|n〉 = d 3 xu ∗ N (x)Hintun(x). (10.9)
The set of relations (10.8) for all N is equivalent to the Schrödinger equation (10.6)
and no approximation is involved.
A useful approximate solution of Eq. (10.8) is obtained if it is assumed that
the interacting system is initially in one particular state of the unperturbed system
and if the perturbation Hint is weak. In Fig. 10.1, the initial state is |α〉; itcan,
for instance, be a well-defined excited level. In terms of the expansion (10.7), the
situation is described by
aα(t) =1, all other an(t) =0, for t