Thermodynamics, Statistical Physics, and Quantum Mechanics

QUANTUM MECHANICS 319

where we have written the transverse components of momentum in terms
of spin raising and lowering operators. The initial spin is pointing along the
direction of the initial wave vector which we define as the
Let us quantize the final spins along the same axis. Now consider how the
three factors scatter the spins:

a)

b)

c)

d)

The term A is spin independent. It puts the final spin in the same

state as the initial spin.

is a diagonal operator, so the final spin is also along the initial

direction, and this term has a value of

flips the spinfrom to and contributesa matrixelement

of to the final state with the spin reversed.

gives a matrix element of zero since the initial spin cannot be

raised.

When we take the magnitude squared of each transition and sum over final
states, we get the factors for spins of

The differential cross section is written as

We have used the fact that energy is conserved, so to set
(see Problem 5.65 and Figure S.5.65).