Thermodynamics, Statistical Physics, and Quantum Mechanics

316 SOLUTIONS

5.65 Two-Delta-Function Scattering (Princeton)

Let us take anunperturbedwavefunction of the particle of the form

Suppose that, afterscattering, thewave vectorbecomes In the Born

approximation, thescattering amplitude is

(see, forinstance, Landau and Lifshitz,QuantumMechanics, Sect. 126),

where and (seeFigureS.5.65).Substituting the

potential into(S.5.65.2), we obtain

where is the projection of the vector
on thez axis. Thescatteringcross section

In order to apply the Born approximation, i.e., to useperturbationtheory,

we mustsatisfy atleast one of two conditions: