advanced35 min studyLesson 399 in Physics

Bound states as scattering poles

Continue amplitudes into complex momentum and separate normalizable bound states from virtual states and resonances.

Before you begin

On this page

The question for today

What you’ll understand

Locate a bound-state pole on the positive imaginary momentum axis.
Derive the shallow-binding relation and its range correction.
Distinguish physical-sheet bound poles from virtual and resonance poles.

Scattering experiments use particles with positive real kinetic energy, while bound states have negative energy and normalizable wavefunctions. Analytic continuation links these two problems. A pole of the continued amplitude can signal that an outgoing solution becomes a decaying bound-state tail. Its location and sheet matter: a pole on the wrong imaginary axis is not a bound state, and a lower-half-plane resonance has a finite escape width rather than a normalizable stationary probability distribution.

Continue the radial boundary condition

At positive energy the external outgoing s wave contains $e^{ikr}$ . Continue wavenumber to $k=i\kappa$ with positive real $\kappa$ . That factor becomes exponentially decaying, and the corresponding free asymptotic energy becomes negative. A bound state requires the regular interior solution to match this decaying tail without any growing exponential. This extra matching condition selects discrete energies instead of the continuum of real scattering wavenumbers.

In the scattering amplitude or S matrix, the coefficient ratio becomes singular precisely when the incoming coefficient of that regular solution vanishes. For a short-range self-adjoint problem, such a pole on the positive imaginary momentum axis corresponds to a bound state on the physical sheet, provided it has the appropriate physical residue and normalizable wavefunction. The energy plane has a threshold branch point because energy is proportional to squared momentum, so specifying only a negative energy without its momentum sheet is insufficient.

k=i\kappa,\quad \kappa>0:\qquad e^{ikr}=e^{-\kappa r},\qquad E_b=-\frac{\hbar^2\kappa^2}{2\mu},\qquad \psi_b(r)\propto\frac{e^{-\kappa r}}r\quad(r>R)

κ has inverse-length units, Eb is the negative state energy relative to threshold and R is the interaction range. The exterior 1/r expression is not valid at the origin; a regular interior wavefunction completes it.↔ Scroll wide equations horizontally.

Extract a shallow pole from the effective-range form

Insert imaginary momentum into the low-energy inverse amplitude. With effective range neglected, the pole condition gives inverse decay length equal to inverse scattering length. Positive large scattering length therefore produces a shallow bound state in this controlled near-threshold model. Its binding energy magnitude is proportional to inverse squared scattering length. The adjective large compares that length with the potential range and any other scale governing the low-energy expansion.

This result does not mean every positive scattering length proves a bound state. A hard repulsive core has positive scattering length yet no bound state; continuing a zero-range truncation out to a momentum comparable with the core's inverse size would exceed its validity. With a small effective-range correction, solve the continued inverse denominator perturbatively near the shallow root. A second root of the truncated quadratic can lie outside the expansion regime and should not be promoted to an additional physical bound state without checking the full model.

-\frac1a-\frac12r_e\kappa^2+\kappa=0,\qquad r_e=0:\quad\kappa=\frac1a,\quad |E_b|=\frac{\hbar^2}{2\mu a^2}

The quadratic condition follows from a continued short-range effective-range expansion. The shallow-binding formula requires a > 0 and a much larger than range scales, with no relevant open inelastic decay channel.↔ Scroll wide equations horizontally.

Classify poles before assigning a state

For negative scattering length in the zero-range approximation, the pole lies at negative imaginary momentum. The nominal outgoing exponential then grows with radius; it is not a normalizable bound state. Such a near-threshold second-sheet pole is called a virtual state and can still strongly enhance low-energy scattering. A resonance pole instead typically has positive real momentum and negative imaginary momentum on an outgoing continuation, giving a complex energy with a negative imaginary part.

These analytic objects influence real-axis data differently, and none should be inferred from one intensity measurement alone. Multiple thresholds add several momentum square roots and a richer sheet structure. The pole equation must then specify which channel momenta are continued on which branches. The root-finding lab solves only a positive quadratic on a fixed bracket, so it can support numerical-bracketing intuition for a bound-state search but cannot determine a scattering pole, choose an analytic sheet or verify wavefunction normalizability.

k=-i\kappa\ (\kappa>0):\ e^{ikr}=e^{+\kappa r}\quad\text{(virtual-state outgoing continuation)},\qquad k=k_R-ik_I\ (k_R,k_I>0)\quad\text{(resonance candidate)}

Pole classification also requires the appropriate analytic sheet and full boundary conditions. A negative-energy virtual pole shares an energy value with a different momentum branch but lacks a physical decaying bound-state tail.↔ Scroll wide equations horizontally.

Work it through

A shallow dimer scale

Assume a valid zero-range shallow-bound-state model with $a=10\,\mathrm{nm}$ and $\hbar^2/(2\mu)=2.0\,\mathrm{meV\,nm^2}$ . Find its pole momentum, energy and the ratio of exterior wavefunction amplitudes at radii differing by $10\,\mathrm{nm}$ after removing the geometric 1/r factor.

The positive scattering length gives $\kappa=1/a=0.10\,\mathrm{nm^{-1}}$ . The pole is at $k=+0.10i\,\mathrm{nm^{-1}}$ , on the decaying bound-state continuation.
The state energy is $E_b=-[\hbar^2/(2\mu)]\kappa^2=-(2.0)(0.10^2)\,\mathrm{meV}=-0.020\,\mathrm{meV}$ . The positive binding energy magnitude is therefore $0.020\,\mathrm{meV}$ .
The reduced exterior amplitude is proportional to $e^{-\kappa r}$ . Increasing radius by ten nanometers multiplies it by $e^{-1}=0.368$ . The full wavefunction also changes by its explicit inverse-radius geometric factor.

The result

The pole is +0.10i nm⁻¹ and the bound energy is −0.020 meV; the reduced exterior amplitude falls by e⁻¹ over an additional 10 nm.

The living notebook

Explore the model

Interactive lab

Interactive lab · root-finding

Bracketing as one ingredient of a state search

Reset and set BOTH Target c = 2 and Initial upper bound U = 3. Predict that bisection keeps √2 inside its bracket. Reset and set Target c = 4 and Initial upper bound U = 3; predict the positive root 2 remains bracketed and the reported bound shrinks with Step. This exact quadratic bisection instrument does not solve the effective-range pole equation, impose radial matching or select a physical scattering sheet.

Quadratic / midpointCurrent bracketPositive root

Midpoint after 0 refinements: 1.5
Bracket error bound: 1.5
Computed residual · x² − c: 0.25

Target c2Initial upper bound U3

t = 0 s

Bisection of x² − c on [0,U], capped at 50 iterations for floating-point resolution. The error bound is the maximum distance from the displayed midpoint to either bracket endpoint (half the width in exact arithmetic). Compensated square evaluation preserves the bracket when ordinary multiplication rounds a residual to zero. A truly exact endpoint or midpoint collapses the bracket. This is one monotone quadratic, not a general root solver. 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 shrinking numerical bracket establishes a root of the stated equation, while physical state identification needs additional boundary information.

Pause & reason

Check your understanding

Interactive checks need JavaScript. You can still read the explanation.

Read the checkpoint explanation

A repulsive hard core has positive scattering length without a bound state. The shallow-binding inference is controlled only when the continued pole lies inside the regime where the low-energy approximation is valid.

Your turn

Try it yourself

For the same zero-range coefficient $\hbar^2/(2\mu)=2.0\,\mathrm{meV\,nm^2}$ , change the scattering length to $a=-10\,\mathrm{nm}$ . Locate the near-threshold pole, compute its energy value and explain why it is not the bound state from the worked example.

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 the same algebraic denominator zero but inspect the sign of imaginary momentum and the outgoing radial exponential.

Compare with the solution

The pole is $k=i/a=-0.10i\,\mathrm{nm^{-1}}$ . Squaring k gives the same negative energy value $-0.020\,\mathrm{meV}$ , but the outgoing factor now grows as $e^{+0.10r}$ when r is measured in nanometers. It is a virtual-state pole on the continued unphysical sheet, not a normalizable physical bound-state tail. The equality of energy values demonstrates why energy alone is an incomplete pole label. The scattering amplitude on the real axis remains finite and respects elastic unitarity within the model.

Keep the idea

Take it with you

A bound state requires a decaying tail matched to a regular interior solution.
The universal shallow-binding relation needs a large positive length within a valid range expansion.
Virtual states and resonances occupy different analytic continuations from physical bound poles.

Read further

Original sources and deeper explanations to follow at your own pace.

Optional progress, saved in this browser when available. Revisit any lesson at any time.

Lesson not marked complete.