Scattering, Bound States, and Tunneling

Bound states, scattering states, virtual states, resonances, and tunneling are related but not interchangeable. Boundary conditions define the physical problem, conserved flux fixes probabilities, and analytic continuation reveals which poles belong to which class. This page develops those distinctions for short-range, single-particle potential scattering and then states the limited bridge to black-hole wave equations.

Required background. Calculus and Linear Algebra supplies complex amplitudes, inner products, and orthogonal mode expansions; Ordinary Differential Equations supplies second-order boundary-value problems, matching, and Wronskians; Quantum States, Observables, and Measurement supplies stationary states, self-adjoint Hamiltonians, and probability. Helpful background. Waves, Green Functions, and Geometric Optics supplies incoming and outgoing waves, Green functions, and barrier propagation.

Flux fixes reflection and transmission probabilities

Consider a particle of mass $M$ in a real one-dimensional potential,

$$-\frac{\hbar^2}{2M}\psi''+V(x)\psi=E\psi,$$

with time dependence $e^{-iEt/\hbar}$. The probability current is

$$j=\frac{\hbar}{M}\operatorname{Im}(\psi^*\psi').$$

The stationary equation and its complex conjugate give $j'=0$. Suppose $V(x)$ approaches constants $V_L$ and $V_R$, and take $E$ above both asymptotic thresholds. Define

$$k_L=\frac{\sqrt{2M(E-V_L)}}{\hbar}, \qquad k_R=\frac{\sqrt{2M(E-V_R)}}{\hbar}.$$

A unit-amplitude wave incident from the left obeys

$$\psi(x)\sim \begin{cases} e^{ik_Lx}+r(E)e^{-ik_Lx},&x\to-\infty,\\ t(E)e^{ik_Rx},&x\to+\infty. \end{cases}$$

Since $Ae^{\pm ikx}$ carries current $j=\pm\hbar k|A|^2/M$, the flux ratios are

$$R=|r|^2, \qquad T=\frac{k_R}{k_L}|t|^2.$$

Current conservation therefore gives

$$R+T=1.$$

The factor $k_R/k_L$ is essential when the asymptotic channels have different velocities. If instead $E\lt V_R$, the right-hand wavenumber is imaginary and that closed channel carries no asymptotic transmitted flux. These statements assume a real potential and no absorptive or additional inelastic channel. Tong derives the equal-threshold version from the conserved current in Tong 2017, §6.1.1, Eqs. (6.3)–(6.4), printed pp. 190–192.

Scattering states occur for a continuum of real energies. They are delta-normalized rather than square-integrable and contain a prescribed incoming component together with the allowed outgoing components.

Boundary conditions and sheets classify poles

For a short-range potential with continuum threshold $E=0$, write $E=\hbar^2k^2/(2M)$. The variable $k$ uniformizes the two-sheeted energy surface: the physical branch has $k\gt0$ just above the continuum cut and $\operatorname{Im}k\gt0$ away from it, while continuation through the cut enters $\operatorname{Im}k\lt0$.

The pole-bearing object must also be named. For a self-adjoint Hamiltonian $H$, the resolvent

$$\mathcal R(E)=(H-E)^{-1}$$

is initially defined away from the real spectrum. Meromorphic continuation requires decay and regularity hypotheses stronger than simply labeling a potential short range: compactly supported bounded potentials provide a standard sufficient class, while slower decay can restrict the continuation domain. In such a class, suitably localized resolvent matrix elements can be continued meromorphically in $k$; the explicitly solvable point interaction below has the same continuation directly from its amplitudes. The one-channel scattering amplitudes exhibit the same relevant bound-state and resonance pole positions in these examples. The following table summarizes that setting. Threshold $k=0$ is a special point and must be analyzed separately.

Class	Complex momentum or energy	Boundary condition	Hilbert-space status
Scattering state	$k\gt0$ real	Prescribed incoming wave plus outgoing waves	Delta-normalized continuum state, not a pole
Bound state	$k=i\kappa$, $\kappa\gt0$	Exponential decay at both ends	Square-integrable discrete eigenstate; physical-sheet pole
Virtual state	$k=-i\kappa$, $\kappa\gt0$	Exponential growth on the continued solution	Non-normalizable pole on the continued sheet
Decaying resonance	$k=k_0-i\gamma$, $k_0,\gamma\gt0$	Purely outgoing at both ends	Non-normalizable continued-sheet pole

For a resonance it is often clearer to use

$$E_{\rm pole}=E_R-\frac{i\Gamma}{2}, \qquad \Gamma\gt0.$$

Its time factor is

$$e^{-iE_{\rm pole}t/\hbar} =e^{-iE_Rt/\hbar}e^{-\Gamma t/(2\hbar)}.$$

Thus the amplitude decays on scale $2\hbar/\Gamma$ and the probability on the lifetime scale $\tau=\hbar/\Gamma$ in a regime dominated by this isolated pole. A normalizable wave packet also contains non-pole spectral contributions, so this exponential need not describe its very early or very late evolution. The purely outgoing spatial solution grows at infinity and is not an eigenvector in the self-adjoint Hamiltonian’s Hilbert space. It is useful because it controls the decay and the nearby real-energy line shape.

The sheet and boundary condition are part of the definition: a negative-energy pole is not automatically a bound state, and not every pole is a decaying resonance. As a rigorous model, Zworski treats $P_V=-\Delta+V$ with compactly supported $V$ on $\mathbb R^3$ and defines its resonances through meromorphic continuation of the resolvent in Zworski 2017, §2.2, Theorem 2 and Definition 1, printed p. 18. The associated outgoing resonant-state condition appears in Definition 2 and Eq. (2.24), printed p. 19. Other dimensions, long-range potentials, and black-hole radial operators require their own continuation theorems.

Partial waves make elastic unitarity visible

For a real, short-range central potential in three dimensions, choose an incident plane wave along the $z$ axis. The far field is

$$\psi(\mathbf r)\sim e^{ikz}+f(\theta)\frac{e^{ikr}}{r}, \qquad \frac{d\sigma}{d\Omega}=|f(\theta)|^2.$$

The angular-momentum page explains the rotation eigenfunctions. Here axial symmetry about the incident direction selects $m=0$, for which $Y_{\ell0}$ is proportional to the Legendre polynomial $P_\ell(\cos\theta)$. Write the reduced radial solution as

$$u_\ell(r)\sim\frac{1}{2i} \left[ S_\ell e^{i(kr-\ell\pi/2)}-e^{-i(kr-\ell\pi/2)} \right].$$

The second term is incoming and the first is outgoing; this equation fixes their relative sign and defines the channel S-matrix $S_\ell$. In a purely elastic channel, equal incoming and outgoing fluxes imply $|S_\ell|=1$, so one may write

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

with real phase shift $\delta_\ell$. The corresponding angular amplitude is

$$f(\theta) =\frac1k\sum_{\ell=0}^{\infty}(2\ell+1) e^{i\delta_\ell}\sin\delta_\ell\, P_\ell(\cos\theta),$$

Orthogonality of the Legendre polynomials then gives

$$\sigma_{\rm tot} =\sum_{\ell=0}^{\infty}\sigma_\ell, \qquad \sigma_\ell =\frac{4\pi}{k^2}(2\ell+1)\sin^2\delta_\ell.$$

Each elastic channel obeys the unitarity bound

$$\sigma_\ell\le\frac{4\pi}{k^2}(2\ell+1),$$

and the same expansion yields the optical theorem

$$\sigma_{\rm tot}=\frac{4\pi}{k}\operatorname{Im}f(0).$$

Tong derives the scattering amplitude, partial-wave expansion, optical theorem, and bound in Tong 2017, §§6.2.2–6.2.4, Eqs. (6.23)–(6.37), printed pp. 209–215. Long-range potentials such as the unscreened Coulomb potential require modified asymptotics and are outside this short-range formula.

Exact transfer check: one attractive delta well

Let

$$V(x)=-g\delta(x), \qquad g\gt0, \qquad \alpha=\frac{Mg}{\hbar^2}.$$

The wavefunction is continuous at the origin. Integrating the stationary equation across it gives

$$\psi'(0^+)-\psi'(0^-)=-2\alpha\psi(0).$$

For left incidence with the same wavenumber on both sides, continuity gives $t=1+r$. Combining it with the derivative jump yields

$$t(k)=\frac{k}{k-i\alpha}, \qquad r(k)=\frac{i\alpha}{k-i\alpha}.$$

For real $k$,

$$|t|^2=\frac{k^2}{k^2+\alpha^2}, \qquad |r|^2=\frac{\alpha^2}{k^2+\alpha^2},$$

so the flux check gives $R+T=1$. Both amplitudes have a pole at $k=i\alpha$. Directly imposing decay at both ends gives

$$\psi_b(x)=\sqrt{\alpha}\,e^{-\alpha|x|}, \qquad E_b=-\frac{\hbar^2\alpha^2}{2M} =-\frac{Mg^2}{2\hbar^2}.$$

The pole’s positive imaginary momentum and its decaying wavefunction identify a bound state, not a resonance. Tong derives the jump condition and bound state in Tong, Quantum Mechanics, undated, §2.3.2, Eqs. (2.39)–(2.40), printed pp. 48–49.

Tunneling is a mechanism, not a spectral class

In a forbidden region, define

$$\kappa(x)=\frac{\sqrt{2M[V(x)-E]}}{\hbar}.$$

For a smooth, semiclassically broad barrier between simple turning points $x_1$ and $x_2$, matching the WKB branches gives

$$T\approx C\exp\!\left[-2\int_{x_1}^{x_2}\kappa(x)\,dx\right],$$

where $C$ is a nonexponential connection prefactor in the controlled regime. The approximation-methods page derives the local WKB control and its turning-point failure. Tong obtains the tunneling exponent by connecting the decaying and growing branches in Tong 2017, §2.2.5, Eqs. (2.29)–(2.31), printed pp. 47–49.

An exact barrier calculation checks the exponent independently. Set the common exterior potential to zero and take a rectangular barrier of height $V_0$ and width $L$, with $0\lt E\lt V_0$. Put

$$\kappa=\frac{\sqrt{2M(V_0-E)}}{\hbar}.$$

Direct matching gives

$$T=\left[ 1+\frac{V_0^2\sinh^2(\kappa L)}{4E(V_0-E)} \right]^{-1}.$$

In the broad-barrier limit $\kappa L\gg1$,

$$T\sim \frac{16E(V_0-E)}{V_0^2}e^{-2\kappa L}.$$

The prefactor depends on the sharp interfaces, but the exponential agrees with the WKB action because $\int_0^L\kappa\,dx=\kappa L$. Tong derives the exact coefficient and this limit in Tong, Quantum Mechanics, undated, §2.4.2, Eqs. (2.51)–(2.54), printed pp. 60–62.

This estimate does not justify continuing an outer WKB branch through a turning point. Thin or discontinuous barriers require direct matching, while multiple barriers can produce interference and resonant transmission. A classically allowed well separated from infinity by a controlled barrier can produce a shape resonance whose small width is set by barrier leakage. Zworski analyzes this mechanism and its exponentially narrow resonances in Zworski 2017, §1.3, printed pp. 8–10.

Tunneling can therefore transmit a scattering wave, split bound levels, or give a quasibound state a finite width. It is not itself a fourth kind of state.

The bounded bridge to compact-object waves

For a massless perturbation of a nonrotating, asymptotically flat black hole, separation often gives the schematic radial equation

$$\frac{d^2u}{dr_*^2}+[\omega^2-V_{\rm eff}(r)]u=0.$$

With time dependence $e^{-i\omega t}$, its quasinormal modes are ingoing at the future horizon, $u\sim e^{-i\omega r_*}$ as $r_*\to-\infty$, and outgoing at infinity, $u\sim e^{+i\omega r_*}$ as $r_*\to+\infty$. Its complex frequency is an outgoing resonance, not a normalizable bound-state energy. The terminology is geometric: both fluxes leave the exterior barrier region, one by falling through the future horizon and the other by escaping to infinity. These boundary conditions and the resulting lower-half-plane decay convention are stated in Berti, Cardoso, and Starinets 2009, §3.1, Eqs. (27) and (30), printed pp. 17–18. Rotating horizons replace the horizon wavenumber by a co-rotating combination, while de Sitter, anti-de Sitter, and massive-field problems impose different outer conditions.

A greybody problem instead prescribes incident data and compares Wronskian-normalized fluxes through the effective curvature barrier. At real frequency in a conservative scattering problem, flux conservation survives, but long-range tails modify the asymptotic states, rotation gives the horizon flux a signed co-rotating frequency factor, and coupled channels replace the scalar equation $R+T=1$ by a summed or matrix relation. The canonical reflection, transmission, and absorption page derives those extensions. This page supplies the boundary-condition grammar, not a derivation of black-hole spectra.

Fixed-particle quantum mechanics also cannot create or annihilate quanta. Continue to Flat-Spacetime Quantum Field Theory for Gravitation for fields, Fock space, propagators, and interactions.

Common pitfalls

Using amplitudes as probabilities. Transmission is a ratio of fluxes. A wavenumber or group-velocity factor remains when the two open channels differ.

Classifying a pole without its sheet. The same real energy can sit beside different analytic continuations. Boundary conditions and sheet data separate bound, virtual, threshold, and resonance behavior.

Treating an outgoing resonance as a normalizable state. Its spatial growth is the analytic counterpart of temporal decay. It lies outside the self-adjoint Hamiltonian’s Hilbert-space spectrum.

Calling every small transmission a resonance. Generic tunneling can be weak without any nearby pole. Resonant transmission requires the additional phase and trapping conditions that create a quasibound mode.

Exercises

A single elastic resonance

Near an isolated elastic partial-wave resonance in an open channel with $E_0\gt0$, neglect the slowly varying background phase and take the Breit–Wigner form reviewed in Tong 2017, §6.2.7, Eq. (6.53), printed pp. 223–225:

$$S_\ell(E) =\frac{E-E_0-i\Gamma/2}{E-E_0+i\Gamma/2}, \qquad \Gamma\gt0.$$

Show that $|S_\ell|=1$ for real $E$, locate its pole, determine the probability lifetime, and show that the partial cross-section saturates its unitarity bound at $E=E_0$.

Solution

For real $E$, the numerator is the complex conjugate of the denominator, so $|S_\ell|=1$. The denominator vanishes at

$$E_{\rm pole}=E_0-i\Gamma/2,$$

which lies on the decaying continued sheet. Its time-dependent probability contains $e^{-\Gamma t/\hbar}$, hence $\tau=\hbar/\Gamma$.

At $E=E_0$,

$$S_\ell(E_0)=-1=e^{2i\delta_\ell},$$

so $\delta_\ell=\pi/2$ modulo $\pi$. Therefore

$$\sigma_\ell(E_0) =\frac{4\pi}{k^2}(2\ell+1),$$

where $k=k(E_0)$ is real. This is the elastic unitarity bound. A nonzero background phase or an inelastic channel changes the observed line shape and must be included in a less idealized model.

References

• Berti, Emanuele, Vitor Cardoso, and Andrei O. Starinets. “Quasinormal Modes of Black Holes and Black Branes.” Classical and Quantum Gravity 26 (2009): 163001. doi:10.1088/0264-9381/26/16/163001. Open PDF.
• Tong, David. Quantum Mechanics, §2.3.2, Eqs. (2.39)–(2.40), printed pp. 48–49; §2.4.2, Eqs. (2.51)–(2.54), printed pp. 60–62. Cambridge lecture notes, undated; official PDF inspected August 30, 2026. Official PDF.
• Tong, David. Topics in Quantum Mechanics, §2.2.5, printed pp. 47–49; §6.1, printed pp. 189–205; §§6.2.2–6.2.4, printed pp. 209–215; §6.2.7, printed pp. 223–225. Cambridge lecture notes, 2017. Official PDF.
• Zworski, Maciej. “Mathematical Study of Scattering Resonances.” Bulletin of Mathematical Sciences 7 (2017): 1–85. doi:10.1007/s13373-017-0099-4. Open PDF.