Partial Wave Analysis and Scattering Cross-Sections

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Hello! Welcome to the next lesson in our course on Quantum Scattering Theory.

In our previous lesson, we explored the Born approximation, a powerful tool for calculating scattering cross-sections when the interaction potential is weak or the incident energy is high. We saw that it provides a beautiful connection between the scattering amplitude and the Fourier transform of the potential. However, we also identified its crucial limitation: it fails for strong potentials at low energies, a regime highly relevant in many areas of physics, including condensed matter and nuclear physics.

To address this, we need a more general, non-perturbative approach. This lesson introduces partial wave analysis, a method that is particularly powerful for spherically symmetric potentials and works regardless of the potential's strength.

This lesson is designed to help you perform partial wave analysis and apply phase shifts to calculate scattering cross-sections. We will decompose the scattering problem by angular momentum, introduce the crucial concept of phase shifts ( $δ_{l}$ ), and see how these shifts, which encode the full effect of the potential, are used to determine experimental observables like the differential and total cross-sections. This method provides a fundamental framework for understanding low-energy scattering, from electron-impurity interactions in solids to neutron-proton scattering.

1. The Principle of Partial Wave Analysis

The core idea is rooted in a fundamental symmetry. For a central potential, $V (r)$ , the Hamiltonian commutes with the angular momentum operators, $L^^{2}$ and $L^_{z}$ . This means angular momentum is conserved during the scattering process. Consequently, it's natural to decompose the total wavefunction into states of definite angular momentum, known as partial waves.

An incident plane wave traveling along the z-axis, $ψ_{in c} = e^{ik z}$ , can be expanded in terms of Legendre polynomials $P_{l} (cos θ)$ and spherical Bessel functions $j_{l} (k r)$ :

e^{ik z} = l = 0 \sum \infty (2 l + 1) i^{l} j_{l} (k r) P_{l} (cos θ)

This is the Rayleigh formula. It shows that the incident plane wave is a superposition of infinitely many partial waves, each with a specific angular momentum $l$ and an amplitude determined by $j_{l} (k r)$ .

Matplotlib Plot of Bessel Functions J_0(x), J_1(x), and J_2(x) — The radial dependence of the first few partial waves in the incident plane wave is described by spherical Bessel functions, $j_l(x)$, shown here. Notice that for a small argument $x=kr$, the functions with higher $l$ are suppressed ($j_l(x) \sim x^l$). This means a low-energy particle (small $k$) predominantly interacts via the lowest angular momentum partial waves.

Because angular momentum is conserved, each partial wave scatters independently. The effect of the potential is to modify each partial wave separately. Our goal is to figure out precisely how each wave is modified.

2. Phase Shifts: The Fingerprint of the Potential

Let's consider the asymptotic form of the wavefunction far from the scattering center, where $V (r) \to 0$ . The radial Schrödinger equation for the $l$ -th partial wave simplifies, and its general solution $R_{l} (r)$ becomes a linear combination of spherical Bessel and Neumann functions. Asymptotically, for $k r \to \infty$ , this can be written as:

R_{l} (r) \propto \frac{sin ( k r - l π /2 + δ _{l} )}{k r}

Here, $δ_{l}$ is the phase shift of the $l$ -th partial wave.

What does this phase shift signify?

If there were no potential ( $V (r) = 0$ ), the solution would be the regular spherical Bessel function $j_{l} (k r)$ , which has the asymptotic form $\frac{s i n ( k r - l π /2 )}{k r}$ . In this case, $δ_{l} = 0$ .
The presence of the potential $V (r)$ shifts the phase of the oscillating radial wavefunction far away from the origin. This shift, $δ_{l}$ , encapsulates the entire effect of the scattering potential on the $l$ -th partial wave. An attractive potential typically pulls the wave in, leading to a positive phase shift, while a repulsive potential pushes it out, causing a negative one.

The beauty of this method is that instead of needing to know the full wavefunction, we only need to determine the set of phase shifts ${δ_{0}, δ_{1}, δ_{2}, \dots}$ to completely characterize the scattering process.

The following video provides an exceptionally clear and intuitive derivation of how these phase shifts arise and relate to the total wavefunction.

L20.3 Scattering amplitude in terms of phase shifts

In this MIT OpenCourseWare lecture by Prof. Allan Adams (filling in for Prof. Zwiebach), the lecturer masterfully uses physical intuition to introduce phase shifts. He argues that the potential can only affect the outgoing part of each partial wave, modifying it by a phase factor that directly defines the phase shift.

Please watch from the beginning to 09:59. Focus on the core argument: the total wavefunction is a superposition of partial waves. The incoming part of each partial wave is fixed by the incident plane wave, so the scattering potential can only alter the phase of the outgoing part.

3. From Phase Shifts to Cross-Sections

Now we connect the phase shifts $δ_{l}$ to the measurable quantities: the scattering amplitude $f (θ)$ and the cross-section $σ$ .

The total wavefunction $ψ (r)$ must have the general asymptotic form for a scattering solution:

ψ (r) r \to \infty e^{ik z} + f (θ) \frac{e ^{ik r}}{r}

At the same time, it is a superposition of our phase-shifted partial waves:

ψ (r) = l = 0 \sum \infty (2 l + 1) i^{l} e^{i δ_{l}} R_{l} (r) P_{l} (cos θ) r \to \infty l = 0 \sum \infty (2 l + 1) i^{l} e^{i δ_{l}} \frac{sin ( k r - l π /2 + δ _{l} )}{k r} P_{l} (cos θ)

The logic is to equate these two forms. By expressing the sine function and the plane wave $e^{ik z}$ in terms of incoming ( $\propto e^{- ik r} / r$ ) and outgoing ( $\propto e^{ik r} / r$ ) spherical waves, we can isolate the scattered part and extract $f (θ)$ . The video you just started continues with this exact derivation.

L20.3 Scattering amplitude in terms of phase shifts

Let's continue with the same video, which now performs the mathematical steps to derive the expression for the scattering amplitude in terms of the phase shifts.

Please watch from 09:59 to 14:41. Follow the algebraic steps that lead from the two expressions for the asymptotic wavefunction to the final formula for the scattering amplitude f_k( heta).

The crucial results from this derivation are:

1. Scattering Amplitude:
The scattering amplitude is given by a sum over all partial waves:

f (θ) = l = 0 \sum \infty f_{l} (θ) = \frac{1}{k} l = 0 \sum \infty (2 l + 1) e^{i δ_{l}} sin (δ_{l}) P_{l} (cos θ)

where $f_{l} (θ)$ is the partial wave scattering amplitude.

2. Differential Cross-Section:
The differential cross-section is simply the squared magnitude of the scattering amplitude:

\frac{d σ}{d Ω} = ∣ f (θ) ∣^{2} = \frac{1}{k ^{2}} l = 0 \sum \infty (2 l + 1) e^{i δ_{l}} sin (δ_{l}) P_{l} (cos θ)^{2}

Note the interference between different partial waves in the sum.

3. Total Cross-Section:
To find the total cross-section $σ_{t o t}$ , we integrate $d σ / d Ω$ over all solid angles ( $d Ω = 2 π sin θ d θ$ ). Thanks to the orthogonality of the Legendre polynomials ( $\int_{- 1}^{1} P_{l} (x) P_{l^{'}} (x) d x = \frac{2}{2 l + 1} δ_{l l^{'}}$ ), the interference terms vanish, and we get a remarkably simple result:

σ_{t o t} = l = 0 \sum \infty σ_{l} where σ_{l} = \frac{4 π}{k ^{2}} (2 l + 1) sin^{2} (δ_{l})

The total cross-section is an incoherent sum of contributions from each partial wave.

Test your understanding!

What is the maximum possible scattering cross-section, $σ_{l}$ , for a single partial wave $l$ ? When does this maximum occur? What is the physical interpretation of this phenomenon?

Show answer

The partial cross-section is $σ_{l} = \frac{4 π}{k ^{2}} (2 l + 1) sin^{2} (δ_{l})$ . This is maximized when $sin^{2} (δ_{l}) = 1$ , which occurs when the phase shift $δ_{l}$ is an odd multiple of $π /2$ (e.g., $π /2, 3 π /2, \dots$ ). This phenomenon is called resonance. At a resonant energy, the scattering for that particular angular momentum channel is maximized. Physically, it corresponds to the incident particle being temporarily "trapped" by the potential, forming a quasi-bound state before being re-emitted.

The Optical Theorem

A profound consequence of this formalism is the optical theorem. If we evaluate the scattering amplitude in the forward direction ( $θ = 0$ ), where $P_{l} (1) = 1$ , we get:

f (0) = \frac{1}{k} l = 0 \sum \infty (2 l + 1) (sin δ_{l} cos δ_{l} + i sin^{2} δ_{l})

Taking the imaginary part gives:

Im [f (0)] = \frac{1}{k} l = 0 \sum \infty (2 l + 1) sin^{2} δ_{l}

Comparing this to the formula for the total cross-section, we arrive at:

σ_{t o t} = \frac{4 π}{k} Im [f (0)]

This theorem relates the total scattering rate to the imaginary part of the forward scattering amplitude. It is a fundamental consequence of the conservation of probability (unitarity) in quantum mechanics.

4. A Computational Approach: Finding the Phase Shifts

The theory is elegant, but how do we find the phase shifts $δ_{l}$ for a given potential $V (r)$ ?

For each $l$ , solve the radial Schrödinger equation numerically from $r = 0$ out to a matching radius $a$ where the potential is negligible.
At $r = a$ , the exterior solution is a known combination of spherical Bessel and Neumann functions involving $δ_{l}$ : $u_{l} (r) = C r [cos δ_{l} j_{l} (k r) - sin δ_{l} n_{l} (k r)]$ .
By matching the value and derivative of the numerical interior solution to the analytical exterior solution at $r = a$ , we can solve for $δ_{l}$ .

This procedure is perfect for a computational approach. Your background in Python and numerical methods will be very useful here. The following Google Colab notebook implements exactly this process for scattering off a Yukawa potential.

Quantum Scattering.ipynb

This interactive Google Colab notebook provides a full implementation of partial wave analysis for a Yukawa potential. It uses Numerov's method to solve the radial equation, calculates the phase shifts, and then computes and plots the scattering cross-section. This is a fantastic opportunity to see the theory put into practice.

Open the notebook and run the cells sequentially. Pay close attention to the code and comments in 'Step 3: Computation'. This cell is the heart of the calculation. Identify where the radial wavefunction u and its logarithmic derivative g are calculated using the wf function (which calls the Numerov integrator). Find the line where the phase shift ps[L] (which is δ_l) is calculated. The formula uses the value g, the spherical Bessel functions (jl, nl) and their derivatives (dj, dn) at the matching radius a. See how the scattering amplitude pieces (fl) for each l are calculated using the phase shifts. Finally, see how these pieces are summed up to get the total scattering amplitude f and then the cross-section sigma. In the notebook, sigma.append(??) is incomplete. Based on our formulas, complete this line. It should be sigma.append(abs(sum(fl)/k)**2). Feel free to explore the subsequent plots of the wavefunctions, phase shifts, and the final cross-section.

5. An Analytic Example: Low-Energy Hard-Sphere Scattering

To solidify the concepts, let's analyze the simplest possible scattering problem: a hard, impenetrable sphere of radius $R$ .

Potential: $V (r) = \infty$ for $r < R$ , and $V (r) = 0$ for $r > R$ .
Boundary condition: The wavefunction must vanish at the sphere's surface, so $R_{l} (r) = 0$ for $r \leq R$ .
The exterior solution ( $r > R$ ) is $R_{l} (r) \propto cos δ_{l} j_{l} (k r) - sin δ_{l} n_{l} (k r)$ .
Applying the boundary condition at $r = R$ gives $cos δ_{l} j_{l} (k R) - sin δ_{l} n_{l} (k R) = 0$ , which immediately determines the phase shifts: $tan δ_{l} = \frac{j _{l} ( k R )}{n _{l} ( k R )}$

Now consider the low-energy limit, where $k R ≪ 1$ . We can use the small-argument approximations for the spherical Bessel functions:

s-wave ( $l = 0$ ): $j_{0} (x) \approx x$ and $n_{0} (x) \approx - 1/ x$ .
$tan δ_{0} \approx \frac{k R}{- 1/ ( k R )} = - (k R)^{2}$
Whoops, that's not right. Let's re-check the standard forms. $j_{0} (x) = sin (x) / x \approx 1 - x^{2} /6 + \dots$ . Wait, $j_{0} (x) = sin x / x$ , asymptotic is $sin (x) / x$ , small argument is $1 - x^{2} /6$ . Oh, my memory is failing me. Let's use the standard definitions: $j_{0} (x) = sin (x) / x$ and $n_{0} (x) = - cos (x) / x$ .
So for $x ≪ 1$ , $j_{0} (x) \approx x / x = 1$ and $n_{0} (x) \approx - 1/ x$ .
$tan δ_{0} = \frac{j _{0} ( k R )}{n _{0} ( k R )} \approx \frac{k R s i n ( k R ) / s i n ( k R )}{- k R c o s ( k R ) / s i n ( k R )} \to \frac{s i n ( k R )}{- c o s ( k R )} = - tan (k R)$
So $δ_{0} \approx - k R$ .
p-wave ( $l = 1$ ): $j_{1} (x) \approx x /3$ and $n_{1} (x) \approx - 1/ x^{2}$ .
$tan δ_{1} \approx \frac{( k R ) /3}{- 1/ ( k R ) ^{2}} = - \frac{( k R ) ^{3}}{3}$
You can see that for $k R ≪ 1$ , the phase shift $δ_{l} \propto (k R)^{2 l + 1}$ . The higher partial waves are strongly suppressed. At low energies, scattering is dominated by the s-wave ( $l = 0$ ).

The total cross-section is approximately the s-wave contribution:

σ_{t o t} \approx σ_{0} = \frac{4 π}{k ^{2}} sin^{2} δ_{0} \approx \frac{4 π}{k ^{2}} (δ_{0})^{2} \approx \frac{4 π}{k ^{2}} (- k R)^{2} = 4 π R^{2}

This is a remarkable result. The quantum low-energy cross-section is four times the classical geometric cross-section ( $π R^{2}$ ). The incident wave "bends" around the sphere, feeling its presence over a much larger area. This is a signature wave phenomenon. This classic example is worked out in detail in the provided notes.

Phase shift analysis

These notes from Prof. Masatsugu Sei Suzuki provide a concise derivation for hard-sphere scattering, a textbook application of partial wave analysis.

Please read sections 2 ('Hard sphere scattering (I)') and 3 ('Hard sphere scattering (II): Low energy case') on pages 3-7. It shows the derivation for tan(δl) and confirms the low-energy cross-section result we just discussed.

Conclusion

Today we have developed the powerful formalism of partial wave analysis, which is indispensable for treating scattering problems where the Born approximation is insufficient.

Key Takeaways:

For central potentials, scattering can be analyzed independently for each angular momentum partial wave $l$ .
The entire effect of the potential on the $l$ -th partial wave is encoded in a single number: the phase shift $δ_{l}$ .
The scattering amplitude $f (θ)$ and the total cross-section $σ_{t o t}$ can be expressed as sums over partial waves, weighted by their phase shifts.
The total cross-section is an incoherent sum of partial cross-sections, $σ_{t o t} = \sum_{l} σ_{l}$ , where $σ_{l} = \frac{4 π}{k ^{2}} (2 l + 1) sin^{2} δ_{l}$ .
At low energies ( $k R ≪ 1$ ), scattering is typically dominated by the s-wave ( $l = 0$ ), a significant simplification.

Preview of the Next Lesson:
We have seen that partial wave analysis is powerful, especially at low energies. The behavior of the phase shifts can be quite rich. In the next lesson, we will analyze resonance phenomena, where the phase shift rapidly changes by $π$ near a specific energy, leading to a sharp peak in the cross-section. This is crucial for understanding the formation of unstable particles and quasi-bound states in many areas of physics.