Phase Shifts

5.3 Phase Shifts

Summary: Scattering experiements in the real world yield complicated differential cross sections. If the behaviour for (imagined) incoming particles of definite angular momentum can be deduced from the angular dependence, it is easier to recognise what the results are telling us about the potential.

An important concept in scattering is that of phase shifts. The basic idea is that since angular momentum is conserved by a spherically symmetric potential $V (r)$ , if the incoming wave were an eigenfunction of angular momentum, so would the outgoing wave. This would be the case independently of the potential, and the only influence of the potential would be on the relative phase of the outgoing wave compared to the incoming wave. Since this is a set-up which cannot be experimentally realised it might seem a pointless observation, until you recognise that a plane wave must be expressible as a sum of such angular momentum eigenstates which are called partial waves. In practice it is often the case, for a short-ranged potential, and an incoming wave which is not too high energy, that only the lowest partial waves are significantly scattered. Classically, for a particle with momentum $p$ , the closest approach is $d = L ∕ p$ . If $a$ is the range of the potential, then only particles with $L ≲ p a$ (or $l ≲ k a$ ) will be scattered. Thus the first few phase shifts can be an efficient way of describing low-energy scattering data.

Let us recall that for a free particle in spherical coordinates the seperable solutions are $ψ = R_{l} (r) Y_{l m} (θ, ϕ)$ , where the radial wavefunction satisfies

\frac{1}{r} \frac{d^{2}}{d r^{2}} r R_{l} (r) - \frac{l (l + 1)}{r^{2}} R_{l} (r) + k^{2} R_{l} (r) = 0

whose solutions are spherical Bessel functions, the regular ones $j_{l} (k r)$ and the irregular ones $n_{l} (k r)$ ; the latter blow up at the orgin. For example $j_{0} (z) = sin z ∕ z$ , $n_{0} (z) = - cos z ∕ z$ and $j_{1} = sin z ∕ z^{2} - cos z ∕ z$ . Note that all tend to either $\pm sin z ∕ z$ or $\pm cos z ∕ z$ at large $z$ . Specifically,

j_{l} (z) \overset{z \to \infty}{\to} \frac{sin (z - π l ∕ 2)}{z} n_{l} (z) \overset{z \to \infty}{\to} - \frac{cos (z - π l ∕ 2)}{z}

It can be shown that the expansion of a plane wave in spherical coordinates is

e^{i k \cdot r} = e^{i k r cos θ} = \sum_{l = 0}^{\infty} i^{l} (2 l + 1) j_{l} (k r) P_{l} (cos θ) \overset{r \to \infty}{\to} \frac{1}{2 i k} \sum_{l = 0}^{\infty} (2 l + 1) (\frac{e^{i k r}}{r} - {(- 1)}^{l} \frac{e^{- i k r}}{r}) P_{l} (cos θ)

where the $P_{l} (cos θ)$ are Legendre polynomials (proportional to $Y_{l}^{0}$ ). Fairly obviously, for each partial wave the plane wave consists of both outgoing and incoming waves with equal and opposite flux. (Further notes on this expression and on spherical Bessel functions can be found in A.6; see also Gasiorowicz Supplement 8-B.)

Now consider the case in the presence of the potential. Close to the centre, the wave is not free and can’t be described by spherical Bessel functions. But beyond the range of the potential, it can. Furthermore $n_{l} (k r)$ can enter since the origin isn’t included in the region for which this description can hold. The general form will be

\begin{array}{l} ψ_{k} (r, θ) & = \sum_{l = 0}^{\infty} (C_{l} j_{l} (k r) + D_{l} n_{l} (k r)) P_{l} (cos θ) = \sum_{l = 0}^{\infty} A_{l} (cos δ_{l} j_{l} (k r) - sin δ_{l} n_{l} (k r)) P_{l} (cos θ) \\ \overset{r \to \infty}{\to} \sum_{l = 0}^{\infty} A_{l} \frac{sin (k r - π l ∕ 2 + δ_{l})}{k r} P_{l} (cos θ) = \frac{1}{2 i k} \sum_{l = 0}^{\infty} {(- i)}^{l} A_{l} (\frac{e^{i k r + i δ_{l}}}{r} - {(- 1)}^{l} \frac{e^{- i k r - i δ_{l}}}{r}) P_{l} (cos θ) . \end{array}

The coeffients

C_{l}

and

D_{l}

can be taken to be real (since the wave equation is real),

A_{l} = \sqrt{C_{l}^{2} + D_{l}^{2}}

and

tan δ_{l} = - D_{l} ∕ C_{l}

. The magnitudes of the incoming and outgoing waves are thus again equal.

Now we need to match the incoming waves with the plane waves, since that’s the bit we control; hence ${(- i)}^{l} A_{l} e^{- i δ_{l}} = (2 l + 1)$ . So finally

\begin{array}{l} ψ_{k} (r, θ) & \overset{r \to \infty}{\to} \frac{1}{2 i k} \sum_{l = 0}^{\infty} (2 l + 1) (e^{2 i δ_{l}} \frac{e^{i k r}}{r} - {(- 1)}^{l} \frac{e^{- i k r}}{r}) P_{l} (cos θ) \\ = e^{i k r cos θ} + \sum_{l = 0}^{\infty} (2 l + 1) (e^{2 i δ_{l}} \frac{e^{i k r}}{r} + \frac{e^{i k r}}{r}) P_{l} (cos θ) \\ \Rightarrow & f (k, θ) = \sum_{l = 0}^{\infty} (2 l + 1) \frac{e^{i δ_{l}} sin δ_{l}}{k} P_{l} (cos θ) \end{array}

Because of the orthogonality of the

P_{l} (cos θ)

, the cross section can also be written as a sum over partial cross sections

σ = \int | f (k, θ) |^{2} d Ω = \sum_{l = 0}^{\infty} σ_{l} = \sum_{l = 0}^{\infty} \frac{4 π}{k^{2}} (2 l + 1) {sin}^{2} δ_{l}

As argued at the start, this depends only on the phase shifts $δ_{l}$ .

The significance of $δ_{l}$ can be appreciated if we compare the asymptotic form of the radial wavefunction in the presence and absence of the potential; without we have $sin (k r - l π l ∕ 2) ∕ r$ but with the potential we have $sin (k r - π l ∕ 2 + δ_{l}) ∕ r$ . So $δ_{l}$ is just the phase by which the potential shifts the wave compared with the free case.

Gasiorowicz ch 19.2
Mandl ch 11.5
Shankar ch 19.5
Townsend ch 13.4,5

5.3.1 Hard sphere scattering
5.3.2 Scattering from a finite square barrier or well

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