Scattering Applications

Symmetry organizes a scattering calculation into channels that cannot mix and channels whose amplitudes are related. It does not determine the remaining phase shifts or reaction probabilities. The central example is spinless scattering from a central potential; adding spin or internal states preserves the method but can leave a matrix of coupled channels at each total angular momentum.

Helpful background. Angular-momentum algebra supplies multiplet labels; parity supplies inversion eigenvalues. Partial-wave expansion derives the asymptotic wave normalization used below.

Rotational invariance makes a block for each angular momentum

Consider a time-independent scattering Hamiltonian and its asymptotic channel Hamiltonian. A symmetry must act compatibly on both, including their domains and boundary conditions, so that it also acts on incoming and outgoing states. When the scattering operators exist, this compatibility makes the scattering matrix commute with the symmetry representation. Normalizable incoming packets give the physical interpretation; fixed-energy channel states use continuum normalization.

At fixed energy, use unit-flux channel states $|\alpha;JM,\pi\rangle$. The label $\alpha$ distinguishes internal states, orbital/spin couplings and other channels with the same total angular momentum and parity. Exact rotations and parity give

$$\langle\beta;J'M',\pi'|S(E)|\alpha;JM,\pi\rangle =\delta_{JJ'}\delta_{MM'}\delta_{\pi\pi'} S^{J\pi}_{\beta\alpha}(E).$$

The block is independent of $M$, but can mix its different $\alpha$ labels. Only energetically open channels carry asymptotic outgoing flux. Closed channels can nevertheless affect the open-channel amplitudes through virtual coupling. An external field or nonsymmetric boundary can remove some of these labels.

If all open channels are retained and there is no absorption outside the model, each block is unitary. Thus $\sum_\beta|S^{J\pi}_{\beta\alpha}|^2=1$ for a unit incoming flux in channel $\alpha$. Retaining only an elastic entry need not give a unit-modulus number when other channels are open. See multichannel scattering for the channel dynamics.

A spinless central potential leaves one phase shift per partial wave

For a spinless particle with $H=p^2/(2\mu)+V(r)$, take a real short-range potential and a rotation-invariant domain. Here $\mu$ is the mass, or the reduced mass in a two-body relative-coordinate problem. Rotations preserve every $L_i$. In a single elastic channel, unitarity then gives

$$S_\ell(E)=e^{2i\delta_\ell(E)},\qquad \delta_\ell\in\mathbb R.$$

For an incident wave along $z$, the convention $\psi\sim e^{ikz}+f(\theta)e^{ikr}/r$ gives

$$f(\theta)=\frac1{2ik}\sum_{\ell=0}^{\infty} (2\ell+1)(S_\ell-1)P_\ell(\cos\theta), \qquad E=\frac{\hbar^2k^2}{2\mu}.$$

The amplitude has units of length, and $d\sigma/d\Omega=|f|^2$. These formulas use distinguishable-particle scattering before any identical-particle symmetrization. A Coulomb tail needs modified asymptotic phases; simply using the free spherical waves above misses its long-range logarithmic phase. Tong’s scattering notes and the convention dictionary explain these boundaries.

As a simple consequence, if only $\ell=0$ contributes,

$$f=\frac{e^{i\delta_0}\sin\delta_0}{k}, \qquad \sigma=\frac{4\pi}{k^2}\sin^2\delta_0.$$

The angular distribution is isotropic, but its magnitude is still dynamical. Even a symmetry-allowed channel can have $\delta_0=0$ and no scattering. The bound $\sigma\leq4\pi/k^2$ is the elastic s-wave unitarity bound in this normalization.

Total angular momentum can be conserved while orbital channels mix

With spin, a rotational scalar preserves $\mathbf J=\mathbf L+\mathbf S$, but need not preserve $L^2$ and $S^2$ separately. A central spin–orbit term is a useful special case:

$$V_{LS}(r)\,\mathbf L\cdot\mathbf S =\frac{V_{LS}(r)}2(J^2-L^2-S^2).$$

It is diagonal in $\ell,S,J$ and can give different phase shifts without mixing $\ell$. By contrast, a tensor force can couple the triplet $\ell=0$ and $\ell=2$ channels with $S=1,J=1$. Both have positive orbital parity. Their common $J$ and parity allow a two-channel block; rotational symmetry does not force its off-diagonal amplitude to vanish.

For two spin-$1/2$ particles, a truly spin-independent interaction acts identically on singlet and triplet spin factors at the same orbital channel and threshold. Different observed behavior can still arise from exchange restrictions. Spin-dependent interactions can split the channels; whether $S$ itself remains conserved must be checked for the actual interaction.

Inversion and exchange impose different restrictions

An orbital partial wave has inversion parity $(-1)^\ell$. For particles with intrinsic parities, the two-particle channel parity includes their product as well. Parity invariance prohibits mixing channels of opposite total parity; it does not exclude odd partial waves from a plane-wave beam. Such a beam is not a parity eigenstate.

Exchange instead interchanges identical particles. For two identical spin-$1/2$ fermions with no further internal degrees of freedom, the singlet spin state is antisymmetric and the triplet is symmetric. Total antisymmetry therefore permits even $\ell$ with the singlet and odd $\ell$ with the triplet. If other internal labels are present, their exchange symmetry must also be included. This constraint is distinct from invariance under inversion of the laboratory coordinates.

Relativistic calculations often use helicity labels instead of fixed-axis spin projections. For nonzero momentum, dimensionful helicity is $h=\mathbf J\cdot\mathbf p/|\mathbf p|$; its dimensionless label is $h/\hbar$. Finite-helicity massless representations and the frame dependence of massive helicity are explained in the helicity bridge.

Exercises

1. Derive the s-wave cross section above from $S_0=e^{2i\delta_0}$. Does rotational invariance determine whether the unitarity bound is reached?

Solution

Use $e^{2i\delta}-1=2ie^{i\delta}\sin\delta$. Then $|f|^2=\sin^2\delta_0/k^2$ is constant over the sphere, whose solid angle is $4\pi$. The maximum occurs at $\delta_0=\pi/2$ modulo $\pi$. Symmetry fixes the angular form, while the potential and energy determine $\delta_0$.

2. In an $S=1,J=1$ problem, the triangle rule permits $\ell=0,1,2$. Which orbital sectors can a parity-even tensor interaction mix, assuming equal intrinsic parities for the channels? Does a central $\mathbf L\cdot\mathbf S$ term produce that mixing?

Solution

The $\ell=0,2$ sectors have the same parity and can mix. The $\ell=1$ sector has opposite parity and remains separate. A central spin–orbit term is diagonal in $L^2,S^2,J^2$, so it does not mix $\ell=0$ and $2$. “Allowed by symmetry” does not say that every symmetric interaction produces the coupling.

3. Two identical spin-$1/2$ fermions, with no other internal labels, are fully spin polarized. Is an s-wave collision channel available?

Solution

Their spin state lies in the symmetric triplet. The relative spatial state must therefore be antisymmetric, requiring odd $\ell$. The $\ell=0$ channel is absent, independently of the interaction’s radial strength. This is an exchange restriction, not a claim that all scattering vanishes.

References

• M. L. Goldberger and K. M. Watson, Collision Theory, Dover, 2004 reprint of the 1964 edition.
• R. G. Newton, Scattering Theory of Waves and Particles, 2nd ed., Springer, 1982.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover, 2006 reprint of the 1972 edition.
• D. Tong, Applications of Quantum Mechanics, University of Cambridge lecture notes, 2017, chapter 1. Author’s notes.