Quantum Mechanics Course

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← Part VIII/S-Matrix Theory

6. S-Matrix Theory

Reading time: ~40 minutes | Pages: 10

Universal framework: relates asymptotic in and out states, foundation of modern scattering theory.

The S-Matrix Concept

Scattering matrix (S-matrix): Connects asymptotic states

$$|\psi_{out}\rangle = \hat{S}|\psi_{in}\rangle$$

where $|\psi_{in}\rangle$ is incoming state at $t \to -\infty$, $|\psi_{out}\rangle$ at $t \to +\infty$

Philosophy: Don't worry about details inside interaction region - only relate observable asymptotic states

Properties of S-Matrix

1. Unitarity: Probability conservation

$$\hat{S}^\dagger\hat{S} = \hat{S}\hat{S}^\dagger = \hat{I}$$

No particles created or destroyed in elastic scattering

2. Time-reversal invariance:

$$S_{fi} = S_{if}^*$$

Reciprocity: scattering amplitude same in both directions

3. Lorentz invariance: Transforms correctly under boosts/rotations

4. Analyticity: $S(E)$ is analytic function of energy (with poles/cuts)

S-Matrix Elements

Matrix elements between momentum eigenstates:

$$S_{fi} = \langle\vec{k}_f|\hat{S}|\vec{k}_i\rangle$$

Decompose into identity + scattering:

$$S_{fi} = \delta_{fi} - 2\pi i\delta(E_f - E_i)T_{fi}$$

where $T_{fi}$ is transition matrix (T-matrix)

$\delta_{fi}$: No scattering (particles pass through)

Second term: Actual scattering (energy-conserving)

Relation to Scattering Amplitude

For elastic scattering:

$$f(\theta) = -\frac{m}{2\pi\hbar^2}T_{fi}$$

Cross section from S-matrix:

$$\frac{d\sigma}{d\Omega} = |f(\theta)|^2 = \left|\frac{m}{2\pi\hbar^2}T_{fi}\right|^2$$

Partial Wave S-Matrix

For spherical potentials, diagonal in $\ell$:

$$S_\ell = e^{2i\delta_\ell}$$

Phase of $S_\ell$ encodes phase shift

Unitarity:

$$|S_\ell| = 1$$

$S_\ell$ lies on unit circle in complex plane

Unitarity and Optical Theorem

From $\hat{S}^\dagger\hat{S} = \hat{I}$:

$$\text{Im}[f(0)] = \frac{k}{4\pi}\sigma_{tot}$$

Optical theorem: forward scattering amplitude determines total cross section

Physical meaning: Interference between incident and scattered waves

Analytic Structure

S-matrix as function of complex energy/momentum:

  • Poles: Bound states (negative imaginary energy)
  • Poles: Resonances (complex energy $E_R - i\Gamma/2$)
  • Branch cuts: Thresholds for inelastic channels

Analytic continuation connects bound states to resonances to scattering states

Poles and Resonances

Bound state: Pole on negative imaginary $k$ axis

$$k = i\kappa, \quad E_b = -\frac{\hbar^2\kappa^2}{2m}$$

Resonance: Pole on unphysical sheet

$$k_R = k_0 - i\Gamma_k/2, \quad E_R - i\Gamma/2$$

Complex energy → exponential decay in time

Multichannel Scattering

Multiple reaction channels $\alpha, \beta, \gamma, \ldots$

S-matrix becomes matrix:

$$S_{\alpha\beta}$$

Probability to go from channel $\beta$ to channel $\alpha$

Unitarity:

$$\sum_\alpha |S_{\alpha\beta}|^2 = 1$$

Sum of all channel probabilities = 1

Elastic and Inelastic Cross Sections

Elastic: Return to same channel

$$\sigma_{el} = \int|f(\theta)|^2d\Omega = \frac{\pi}{k^2}\sum_\ell(2\ell+1)|1-S_\ell|^2$$

Reaction (inelastic): Go to different channel

$$\sigma_{reac} = \frac{\pi}{k^2}\sum_\ell(2\ell+1)(1-|S_\ell|^2)$$

Total:

$$\sigma_{tot} = \sigma_{el} + \sigma_{reac}$$

Dispersion Relations

From analyticity, Cauchy integral relates real and imaginary parts:

$$\text{Re}[f(E)] = \frac{1}{\pi}\mathcal{P}\int_{-\infty}^\infty\frac{\text{Im}[f(E')]}{E'-E}dE'$$

Kramers-Kronig relations

Application: Determine scattering amplitude from cross section data

S-Matrix Bootstrap

Historical program (1960s):

  • Determine S-matrix from general principles alone
  • Unitarity, analyticity, crossing symmetry
  • Avoid explicit Hamiltonian/Lagrangian

Led to dual resonance models → string theory

Lippmann-Schwinger Equation

Formal solution for scattering states:

$$|\psi^{(\pm)}\rangle = |\phi\rangle + \frac{1}{E - \hat{H}_0 \pm i\epsilon}\hat{V}|\psi^{(\pm)}\rangle$$

where $|\phi\rangle$ is free particle state, $\pm i\epsilon$ gives outgoing/incoming boundary conditions

T-matrix:

$$\hat{T} = \hat{V} + \hat{V}\frac{1}{E-\hat{H}_0+i\epsilon}\hat{V} + \cdots$$

Born series: successive orders of scattering

Green's Function Approach

Resolvent operator:

$$G^{(\pm)}(E) = \frac{1}{E - \hat{H} \pm i\epsilon}$$

Scattering amplitude from Green's function:

$$f(\vec{k}_f, \vec{k}_i) = -\frac{m}{2\pi\hbar^2}\langle\vec{k}_f|\hat{V}|\psi^{(+)}_i\rangle$$

Relativistic Generalization

In quantum field theory, S-matrix central object:

$$S_{fi} = \langle f, \text{out}| i, \text{in}\rangle = \delta_{fi} + i(2\pi)^4\delta^{(4)}(p_f-p_i)\mathcal{M}_{fi}$$

where $\mathcal{M}_{fi}$ is invariant amplitude

Feynman rules compute $\mathcal{M}$ diagrammatically

Modern Applications

  • Particle physics: LHC computes S-matrix elements for collision processes
  • Nuclear physics: R-matrix theory for reactions
  • Atomic collisions: Multichannel quantum defect theory
  • Condensed matter: Landauer-Büttiker formalism (transport)
  • Quantum information: Scattering approach to quantum computing
  • String theory: S-matrix bootstrap revived for conformal field theories
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