# Central Potential Scattering and Phase Shifts

### From FSUPhysicsWiki

We will now discuss scattering from a central potential in a different way. Recall that the wave function for an incident and scattered wave for a central potential is given by

where is the incoming wave and is the scattering amplitude.

To determine we start with the Schrödinger equation,

As before, this equation may be reduced to an effective one-dimensional equation,

with the full wave function given by

For a potential with a finite range we know that, for the problem reduces to that of a free particle, and thus

The solution of this equation is a linear combination of the spherical Bessel functions and the spherical Neumann functions,

When we use the asymptotic approximations of the spherical Bessel functions and the spherical Neumann functions, obtaining

Let us now define

The angle is known as the phase shift of the wave and it is the phase shift induced by scattering from the potential in the radial part of the wave function. Note that, in the absence of a scattering potential, the boundary condition that the wave function must be finite at the origin causes *B*_{l} to vanish for all values of *l*. Therefore, the magnitude of *B*_{l} compared to *A*_{l} is a meausre of the intensity of the scattering. We may rewrite the above expression as

Physically, we expect for repulsive potentials and for attractive potentials. Also, if then the classical impact parameter is much larger than the range of the potential and in this case we expect to be small.

Because the scattering amplitude has azimuthal symmetry (i.e., it is independent of ), we can write the full solution of the Schrödinger equation as a superposition of spherical harmonics only:

We now determine the coefficients by substituting in the wave function in terms of the scattering amplitude on the left-hand side:

Here, we assume that the incident wave propagates along the direction. This must hold for large We may show that

so that

Using the fact that

we may rewrite this as

By matching the coefficients of , we get

and doing the same for yields

Note that is a function of and therefore a function of the incident energy. If is known, then we can reconstruct the entire scattering amplitude and consequently the differential cross section. The phase shifts themselves must be determined by solving the Schrödinger equation.

The differential scattering cross section is

By integrating over the solid angle we obtain the total scattering cross section:

The last equality follows from the orthogonality of the Legendre polynomials,

Finally, note that since for all we obtain

If we take the imaginary part of the scattering amplitude,then

Therefore,

This relationship is known as the optical theorem. The optical theorem is a general law of wave scattering theory that relates the forward scattering amplitude to the total cross section of the scattering. It was originally discovered independently by Sellmeier and Lord Rayleigh in 1871.

Referring back to the formula for the scattering amplitude, one more important quantity can be discussed:

This quantity, for now referred to as the partial scattering for angular momentum is the ratio of the coefficients of the outgoing and incoming waves for a wave scattered on a potential of finite range

These ratios can simplify the problem of evaluating the continuity of the waveform at the boundary In general, if the interior wave function is known to be smoothly continuous across the boundary at then the phase shifts can be expressed in terms of the logarithmic derivatives evaluated at the boundary

Using the above equations for the form of *u*_{l}(*r*) beyond the region of scattering, the following relation is found:

Thus, after some algebra,

Note that, if which corresponds to then only the first portion of this expression survives. This is a special quantity corresponding to hard sphere scattering; we may define the phase angles known as the hard sphere phase shifts, as

Note that these phase shifts are present for any potential, not just that of a hard sphere.

## Scattering by Square Well Potential and Hard Sphere

Consider a beam of point particles of mass scattering from a finite spherical attractive well of depth and radius

The effective Schrödinger equation for is

Its solution is

where

In the region is

Here, and are arbitrary constants and

For large

where

We now apply the boundary conditions at which are continuity of and of its logarithmic derivative. We obtain

Let us now consider two limiting cases:

**(a)** and In this case, we find that, with some simplification, This behavior is a result of the centrifugal barrier that keeps waves of energy far below the barrier from feeling the effect of the potential.

**(b)** When the phase shift is the partial wave cross section is maximized. This is known as a resonant scattering.

From **(a)**, we see that the phase shift is small for small. However, when changes and passes through the resonance condition, the phase shift rises rapidly and has a sharp peak at resonant energy . This can be represented as

so that the partial wave cross section is

This is the Breit-Wigner formula for a resonant cross section.

Let us also consider a hard sphere, given by the potential,

For scattering state

For ,

Matching continuity boundary condition at , we get,

so the scattering phase shift of the wave is:

For ,

so

The term dominates in the scattering process, so the scattering amplitude and the cross section are:

Therefore, the total cross section is

## Problems

**(1)** Consider the scattering of a particle from a real spherically symmetric potential. If is the differential cross section and σ is the total cross section, then show that

for a general central potential using the partial-wave expansion of the scattering amplitude and the cross section.

**(2)** Consider an attractive delta shell potential () of the form,

**(a)** Derive the equation for the phase shift caused by this potential for arbitrary angular momentum.

**(b)** Obtain the expression for the wave phase shift.

**(c)** Obtain the wave scattering amplitude.