Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 11.15.
is the Green’s function and is a free particle wave function. The first order Born approximation assumes that the wave function isn’t changed much by the scattering process, so we can approximate the integral equation by replacing the full wave function in the integrand by the incident wave function:
We can generate a second order approximation by inserting 3 into the integral in 1. When we do this, it’s important to keep track of the position vectors that apply to each integration. We’ll relabel as in 1, and then insert 3 into it:
The first two terms just repeat the first order Born approximation, so to calculate the second order approximation we need to work out the third term involving the double integral. Using 2 we have for this term (which we’ll call )
In the original derivation, we simplified things by taking to point to the detector, which is assumed to be very far from the scattering region. This allows us to approximate the Green’s function:
This approximation is still valid for the first factor in the integrand in 6, but not for the second factor, since and are position vectors that both refer to locations within the scattering region (that is, where ). Therefore we can write
Example Soft-sphere scattering. A soft sphere is defined by the potential
where is a constant. Griffiths works out the first order Born approximation for the scattering amplitude in his example 11.4, so we’ll consider the second order term here. We’ll also consider only the case of low energy scattering, which is defined by the condition . We start by writing out 8 for this case.
To do the integral, we use an incident plane wave for :
If we do the integration first, we can take the polar axis to be along . Then
We can’t really make much headway with this integral without invoking the low energy assumption. In that case, all the exponentials are approximately 1, so we get
Doing the first, we have
The integral over is now easy, and we get
The second order correction to the scattering amplitude is the coefficient of so the correction is
Combining this with the first order correction taken from Griffiths’s Example 11.4 we get