advanced35 min studyLesson 396 in Physics

S-matrix unitarity and eigenphases

Normalize open channels by flux and distinguish full unitary scattering from loss in a selected elastic channel.

Before you begin

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The question for today

What you’ll understand

State unitarity for a complete flux-normalized channel basis.
Interpret elastic inelasticity parameters and reaction cross sections.
Distinguish closed thresholds, eigenchannels and laboratory channel populations.

A real interaction can move probability from one internal channel to another without destroying it. The full scattering matrix remains unitary when every open outgoing channel is included with correct flux normalization. A single elastic element can nevertheless have magnitude below one. Understanding this distinction connects phase shifts to reactions, prevents false claims of probability loss, and explains why multichannel scattering cannot always be summarized by one real phase.

Normalize channels by current

An asymptotic channel specifies internal states and the relative translational motion left after their threshold energy is paid. Above its threshold the relative wavenumber is real and it carries traveling flux. Below threshold its asymptotic wavenumber is imaginary, so the physical solution decays and carries no independent outgoing flux at infinity. Such a closed channel can still affect the open-channel dynamics through virtual coupling or a nearby bound level.

Use incoming and outgoing basis waves normalized to unit flux rather than equal coordinate amplitude. The squared magnitude of an S-matrix element then gives an outgoing-channel fraction for a unit incoming channel. Without that normalization, channel speeds appear explicitly in the conversion from amplitude squared to current. Different thresholds can make those speeds substantially different. Simply adding raw squared coordinate amplitudes across channels of unequal speed is therefore not a valid test of probability conservation.

k_a=\frac{\sqrt{2\mu_a(E-E_a)}}{\hbar},\quad v_a=\frac{\hbar k_a}{\mu_a},\qquad \left.\frac{d\sigma_{b\leftarrow a}}{d\Omega}\right|_{\rm coordinate\ amplitude}=\frac{v_b}{v_a}|f_{ba}|^2

Ea is the channel threshold energy; μa is a reduced mass and the displayed real ka applies only for E > Ea. The cross-section formula assumes matched coordinate-amplitude conventions; flux-normalized S elements already include the speed factors.↔ Scroll wide equations horizontally.

Separate complete unitarity from elastic survival

For a self-adjoint Hamiltonian with all asymptotically open channels retained, incoming and outgoing total currents agree. In a unit-flux basis this becomes a unitary matrix relation. Its columns are orthonormal, not merely normalized individually. A proposed collection of transition probabilities can sum to one while failing the phase-sensitive orthogonality condition, so column probability sums alone are not enough to certify a quantum S matrix.

For a central problem in a selected elastic angular-momentum channel, write its elastic element as a nonnegative survival magnitude times a phase. A magnitude below one signals flux transferred to other channels that the reduced description does not display. The missing elastic flux produces a reaction cross section. The full matrix need not be nonunitary merely because this one element is attenuated. A phenomenological absorptive potential is one way to summarize eliminated channels, but its loss must be interpreted in that reduced sense.

S^\dagger S=I,\quad \sum_{b\ {\rm open}}|S_{ba}|^2=1,\quad S_{\ell,aa}=\eta_\ell e^{2i\delta_\ell},\quad \sigma_{{\rm reaction},\ell}=\frac{\pi}{k_a^2}(2\ell+1)(1-\eta_\ell^2)

The complete S matrix is unitary in a flux basis. For one chosen elastic channel 0 ≤ ηℓ ≤ 1; δℓ is real and the reaction cross section has area units in the distinguishable spinless convention.↔ Scroll wide equations horizontally.

Diagonalize to expose eigenphases

A finite-dimensional unitary matrix is normal and can be diagonalized by a unitary change of channel basis. Its eigenvalues have unit magnitude and can be written as twice-phase exponentials. These eigenchannels are coherent superpositions of physical channels, not necessarily the internal states prepared in an experiment. Mixing between them can make one laboratory transition probability depend strongly on relative eigenphases even when every eigenchannel separately preserves flux.

The basis change can depend on energy, particularly near thresholds or avoided resonances, and a single eigenphase cannot replace the full multichannel response. A proper calculation must specify which channels are open and which conserved angular-momentum and spin sector is being diagonalized. The pure two-level lab illustrates a normalized coherent state and its basis populations. It is not an S-matrix computation, does not solve channel thresholds and does not show scattering events or measurement collapse.

S=U\operatorname{diag}(e^{2i\delta_1},\ldots,e^{2i\delta_n})U^\dagger,\qquad \sigma_{{\rm el},\ell}=\frac{\pi}{k_a^2}(2\ell+1)|1-\eta_\ell e^{2i\delta_\ell}|^2

U is unitary and the eigenphases are real for a complete finite unitary block. The elastic cross section uses its incident wavenumber; it should be combined with reaction channels before applying the total optical theorem.↔ Scroll wide equations horizontally.

Work it through

A unitary two-channel mixer

In a unit-flux basis consider $S=\begin{pmatrix}1/2&i\sqrt3/2\\i\sqrt3/2&1/2\end{pmatrix}$ . For incidence in channel one, find the outgoing fractions. Then find its eigenvalues and eigenphases.

The first incoming channel selects the first column. The outgoing fractions are $|1/2|^2=1/4$ and $|i\sqrt3/2|^2=3/4$ , whose sum is one. Channel conversion preserves total probability.
The two normalized eigenvectors are $(1,1)/\sqrt2$ and $(1,-1)/\sqrt2$ . Their eigenvalues are $1/2+i\sqrt3/2=e^{i\pi/3}$ and $1/2-i\sqrt3/2=e^{-i\pi/3}$ .
Because an eigenvalue is $e^{2i\delta}$ , the continuous eigenphases at this energy can be chosen as $\delta_+=\pi/6$ and $\delta_-=-\pi/6$ . The off-diagonal imaginary entries also make the two columns orthogonal, confirming full unitarity.

The result

Outgoing fractions are 1/4 and 3/4; the eigenphases are ±π/6 modulo π. Attenuation of one elastic element does not mean the full S matrix loses probability.

The living notebook

Explore the model

Interactive lab

Interactive lab · quantum

Channel-population analogy

Reset and set BOTH Mixing angle θ = 60° and Relative phase φ = 0°. Predict basis probabilities 3/4 and 1/4. Reset and set θ = 120° and φ = 90°; predict 1/4 and 3/4. The instrument displays a pure two-state vector with optional coherent phase precession. It does not calculate S, impose channel thresholds, model scattering loss or perform collapse.

Bloch vector / P(0)P(1)Phase latitude

Probability |0⟩: 71.1%
Probability |1⟩: 28.9%
Bloch (x, y, z): 0.78, 0.45, 0.42

Mixing angle θ65°Relative phase φ30°

t = 0 s

A normalized two-level pure state, not an interacting field simulation. Playback adds one radian per second to the relative phase; basis probabilities stay fixed. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forA normalized pair of probabilities is useful, but an S matrix also requires phase-sensitive relations between all incoming channels.

Pause & reason

Check your understanding

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Read the checkpoint explanation

With unit-flux channel normalization the elastic survival fraction is the squared magnitude. The remaining fraction can exit through other channels while the full S matrix remains unitary.

Your turn

Try it yourself

For an s wave at $k=1.0\,\mathrm{nm^{-1}}$ , let the elastic element have $\eta=0.50$ and $\delta=0$ . Compute elastic, reaction and total cross sections and check the forward optical theorem.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Use π|1 − S|²/k² for elastic area and π(1 − |S|²)/k² for reaction area, with S = 0.50.

Compare with the solution

The elastic area is $\pi(1-0.50)^2=0.25\pi\,\mathrm{nm^2}$ . The reaction area is $\pi(1-0.25)=0.75\pi\,\mathrm{nm^2}$ , so the total is $\pi\,\mathrm{nm^2}$ . The elastic forward partial amplitude is $(S-1)/(2ik)=0.25i\,\mathrm{nm}$ , which gives $(4\pi/k)\operatorname{Im}f(0)=\pi\,\mathrm{nm^2}$ . Thus the optical theorem counts both elastic and reaction removal, even though its input is the elastic forward amplitude. A zero elastic phase does not mean zero scattering when η is below one.

Keep the idea

Take it with you

Unitarity applies to the complete set of flux-normalized open channels.
An elastic survival magnitude below one can represent channel conversion.
Eigenphases describe coherent eigenchannels and need their mixing matrix for laboratory predictions.

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