One-Dimensional Scattering and Tunneling

One-dimensional scattering asks how a prescribed incoming flux is redistributed among outgoing channels by a spatially localized potential. Tunneling is one regime of that problem: a finite classically forbidden region supports evanescent amplitudes that can connect propagating waves on its two sides.

This chapter owns exact one-dimensional matching problems and their physical interpretation. The general $S$-matrix, partial waves, Born approximation, analytic structure, and multichannel scattering belong in Approximation and Semiclassical Methods. Smooth-barrier estimates belong on Barrier Penetration and Tunneling. The pages here provide the canonical solvable laboratory those later methods must reproduce.

One-Dimensional Scattering Revisited is the graduate bridge from these exact models to flux-normalized channel matrices, transfer-matrix poles, resonances, and phase-sensitive WKB comparisons.

The scattering question

Consider

$$\hat H = -\frac{\hbar^2}{2m} \frac{d^2}{dx^2} +V(x),$$

with asymptotic constants

$$V(x)\to V_L \quad\text{as }x\to-\infty, \qquad V(x)\to V_R \quad\text{as }x\to+\infty.$$

For energy above both asymptotic potentials, define

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

A state incident from the left has asymptotic form

$$\psi(x) \sim e^{ik_Lx}+r e^{-ik_Lx} \quad (x\to-\infty),$$

and

$$\psi(x) \sim t e^{ik_Rx} \quad (x\to+\infty).$$

The complex amplitudes $r$ and $t$ contain phase as well as magnitude. Probabilities come from the one-dimensional current

$$j = \frac{\hbar}{m} \operatorname{Im} \left( \psi^*\frac{d\psi}{dx} \right),$$

so the physical coefficients are

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

For a real, time-independent potential with one open channel on each side,

$$R+T=1.$$

This conservation law is a check on the solution, not the definition of $R$ and $T$. Reflection and Transmission Coefficients owns the current derivation, flux normalization, unequal-velocity factor, and failure modes when absorption or additional channels are present.

One interface: reflection without a classical turning point

At a finite step, both $\psi$ and $\psi'$ are continuous. For

$$V(x) = \begin{cases} 0, & x\lt0,\\ V_0, & x\gt0, \end{cases}$$

and $E\gt V_0$, the left and right wavenumbers are

$$k = \frac{\sqrt{2mE}}{\hbar}, \qquad q = \frac{\sqrt{2m(E-V_0)}}{\hbar}.$$

Matching gives

$$r = \frac{k-q}{k+q}, \qquad t = \frac{2k}{k+q}.$$

Even above the step, $R$ is generally nonzero because the wavelength changes abruptly. For $0\lt E\lt V_0$, the right-side wave is evanescent. It penetrates over the length $1/\kappa$, where

$$\kappa = \frac{\sqrt{2m(V_0-E)}}{\hbar},$$

but a semi-infinite forbidden region carries no current to $+\infty$. Thus $T=0$ for the sub-threshold step. Potential Step owns both cases and the classical comparison.

Two interfaces: interference and tunneling

A finite rectangular barrier adds a second matching surface:

$$V(x) = \begin{cases} 0, & x\lt0,\\ V_0, & 0\lt x\lt a,\\ 0, & x\gt a. \end{cases}$$

For $0\lt E\lt V_0$, both exponentials are needed inside the finite forbidden region. The exact transmission probability is

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

When $\kappa a\gg1$,

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

The dimensionless opacity $\kappa a$ is the decisive scale. Width, mass, and the energy deficit $V_0-E$ enter the exponent, so modest parameter changes can alter transmission by many orders of magnitude.

For $E\gt V_0$, the interior wavenumber is $q=\sqrt{2m(E-V_0)}/\hbar$, and

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

There is still reflection, except at phase-matched energies $qa=n\pi$, where $T=1$. Finite Potential Barrier owns the complete two-interface calculation, including the finite barrier-top limit and above-barrier transparency. Rectangular Barrier Tunneling isolates the forbidden-energy regime and its exponential approximation.

Four related phenomena

The same forbidden-region mathematics appears in physically distinct settings.

Phenomenon	State description	Observable
Sub-threshold step	propagating wave matched to a semi-infinite evanescent tail	$R=1$, no asymptotic transmitted flux
Finite-barrier tunneling	incoming and outgoing scattering channels joined through an evanescent region	current ratio $T$
Double-well tunneling	nearly degenerate bound configurations mixed through a barrier	level splitting and coherent transfer time
Resonant transmission	a leaky standing wave between barriers	a peak in $T(E)$ with finite width

Quantum Tunneling owns the conceptual distinctions, the warning against “energy borrowing,” the relation to bound-state splitting, and the caution that no single universal tunneling time follows from $T$ alone.

Transfer matrices and layered potentials

In a constant-potential region, collect right- and left-moving amplitudes into

$$\mathbf v_j = \begin{pmatrix} A_j\\ B_j \end{pmatrix}.$$

Interface matrices impose matching, while propagation matrices add phase or exponential factors. A layered structure becomes one ordered product,

$$\mathbf v_R = M(E)\mathbf v_L.$$

This turns repeated matching into reusable linear algebra and exposes interference in multiple barriers. It also carries a numerical warning: evanescent propagation introduces factors such as $e^{\kappa a}$, so a mathematically correct transfer matrix can be badly conditioned for opaque or long stacks. Scattering-matrix or stabilized recursive methods are then preferable.

Transfer Matrix Method owns the matrix convention, interface derivation, extraction of $r$ and $t$, matrix ordering, and stability checks.

Resonances and quasi-bound states

Two barriers can temporarily confine amplitude in the region between them. Finite leakage turns an ideal bound level into a quasi-bound resonance. Near an isolated resonance, a useful line shape is

$$T(E) \approx \frac{\Gamma_L\Gamma_R} {(E-E_{\mathrm{res}})^2+\Gamma^2/4}, \qquad \Gamma=\Gamma_L+\Gamma_R.$$

The scale

$$\tau \sim \frac{\hbar}{\Gamma}$$

connects a narrow energy width with a long-lived intermediate state. Symmetric leakage can produce unit peak transmission even when each barrier is individually opaque. Resonant Transmission owns the double-barrier picture, reflection phases, peak-height condition, width–lifetime relation, and bridge to scattering poles.

A point interaction

The delta potential

$$V(x)=\lambda\delta(x)$$

is free everywhere except at one point. The wavefunction is continuous, but its derivative obeys

$$\psi'(0^+)-\psi'(0^-) = \frac{2m\lambda}{\hbar^2} \psi(0).$$

With $\gamma=m\lambda/\hbar^2$ and $E=\hbar^2k^2/(2m)$,

$$R = \frac{\gamma^2}{k^2+\gamma^2}, \qquad T = \frac{k^2}{k^2+\gamma^2}.$$

Attractive and repulsive interactions of equal strength magnitude have the same probabilities but different scattering phases. Only the attractive case has a normalizable bound-state pole. Scattering from a Delta Potential owns the amplitudes, narrow-barrier limit, parity channels, and pole interpretation.

From stationary waves to localized particles

Stationary scattering states are generalized eigenstates, not normalizable particle states. A physical incoming packet is assembled from them with spectral amplitude $a(k)$. After the collision and sufficient spatial separation, the outgoing probabilities are

$$P_{\mathrm R} = \int_0^\infty \lvert a(k)\rvert^2R(k)\,dk,$$

and

$$P_{\mathrm T} = \int_0^\infty \lvert a(k)\rvert^2T(k)\,dk.$$

For a narrow packet centered at $k_0$, these reduce to $R(k_0)$ and $T(k_0)$ only when the coefficients vary slowly across the packet bandwidth. Thresholds and narrow resonances can invalidate that approximation. The phases of $r(k)$ and $t(k)$ also reshape and shift outgoing packets, but such shifts do not by themselves define a unique traversal time.

Wave Packets and Scattering owns the asymptotic packet construction, late-time norm accounting, spectral averaging, phase delays, and numerical checks.

Applications and scope

Alpha decay, scanning tunneling microscopy, Josephson junctions, molecular inversion, and tunnel diodes all reuse tunneling language, but they do not share one complete microscopic model. Nuclear structure, electronic densities of states, superconducting phase coherence, molecular rotations, and semiconductor bands belong in their specialist homes.

Tunneling Applications: First Encounters maps the transferable barrier intuition and states what each elementary model omits. Its role is orientation, not a substitute for nuclear, molecular, superconducting, or transport theory.

Reading route

1. Potential Step
2. Reflection and Transmission Coefficients
3. Finite Potential Barrier
4. Rectangular Barrier Tunneling
5. Quantum Tunneling
6. Wave Packets and Scattering
7. Transfer Matrix Method
8. Resonant Transmission
9. Scattering from a Delta Potential
10. Tunneling Applications: First Encounters
11. One-Dimensional Scattering Revisited

Page map

Page	Central question
Potential Step	How can one finite interface reflect an above-threshold wave?
Finite Potential Barrier	What do two interfaces add to exact scattering?
Rectangular Barrier Tunneling	How does forbidden-region opacity control transmission?
Transfer Matrix Method	How are many interfaces composed systematically and stably?
Reflection and Transmission Coefficients	Why are scattering probabilities current ratios?
Resonant Transmission	How do quasi-bound states create narrow transparency windows?
Quantum Tunneling	Which phenomena count as tunneling, and what common stories are misleading?
Scattering from a Delta Potential	How does a point interaction encode scattering and a bound-state pole?
Wave Packets and Scattering	How do stationary amplitudes become late-time particle probabilities?
Tunneling Applications: First Encounters	Which part of elementary tunneling survives in real applications?
One-Dimensional Scattering Revisited	How do these models fit into $S$-matrix, pole, resonance, and semiclassical language?

Common mistakes

Mistake	Correction
Setting $T=\lvert t\rvert^2$ without checking asymptotic velocities	define transmission from current and include $k_R/k_L$ when needed
Calling every evanescent tail tunneling transmission	require a far-side propagating channel for transmitted flux
Discarding a growing exponential inside a finite barrier	retain both interior solutions until both interfaces are matched
Assuming $E\gt V_0$ forbids reflection	a wavelength mismatch can reflect an above-barrier wave
Saying tunneling borrows energy	stationary scattering and bound-state splitting conserve energy
Treating a quasi-bound resonance as a normalizable bound state	identify leakage, width $\Gamma$, and the outgoing channels
Comparing a broad packet with one value $T(k_0)$	average $T(k)$ over the packet spectrum
Trusting long products of opaque-barrier transfer matrices	monitor conditioning and use a stable scattering formulation
Reading a packet’s plotted peak height as probability	integrate $\lvert\psi\rvert^2$ after outgoing packets separate
Applying the rectangular-barrier formula directly to every device	state the missing dimensional, many-body, and materials physics

Exercises

1. The velocity factor

A left-incident state has asymptotic wavenumbers $k_L$ and $k_R$ and amplitudes $r$ and $t$. Derive the current-based expressions for $R$ and $T$.

Solution

For a right-moving plane wave $A e^{ikx}$,

$$j = \frac{\hbar k}{m} \lvert A\rvert^2.$$

The reflected wave moves left, so its current is negative. With incident amplitude one,

$$j_{\mathrm{inc}} = \frac{\hbar k_L}{m}, \qquad j_{\mathrm{ref}} = -\frac{\hbar k_L}{m} \lvert r\rvert^2,$$

and

$$j_{\mathrm{trans}} = \frac{\hbar k_R}{m} \lvert t\rvert^2.$$

Therefore

$$R = \frac{\lvert j_{\mathrm{ref}}\rvert} {j_{\mathrm{inc}}} = \lvert r\rvert^2,$$

and

$$T = \frac{j_{\mathrm{trans}}} {j_{\mathrm{inc}}} = \frac{k_R}{k_L} \lvert t\rvert^2.$$

2. Penetration versus transmission

Explain why a sub-threshold semi-infinite step has a nonzero wavefunction in the forbidden region but $T=0$, whereas a finite barrier can have $T\gt0$.

Solution

The semi-infinite step supports only a decaying evanescent solution on its forbidden side. A single real exponential carries zero current, and there is no second interface at which it could reconnect to a propagating outgoing channel. Hence the tail is nonzero near the boundary but $T=0$ asymptotically.

A finite barrier ends at a second interface. Its interior evanescent solution can match there onto a propagating wave in the far-side allowed region. That outgoing wave carries current, so the current ratio $T$ can be positive.

3. Opacity scaling

In the opaque-barrier regime, compare the leading transmission exponent after (a) doubling the barrier width and (b) multiplying the particle mass by four, with $V_0-E$ fixed.

Solution

The leading factor is

$$T \propto e^{-2\kappa a}, \qquad \kappa = \frac{\sqrt{2m(V_0-E)}}{\hbar}.$$

Doubling $a$ changes the exponent from $-2\kappa a$ to $-4\kappa a$.

Multiplying $m$ by four doubles $\kappa$, so it produces the same exponent $-4\kappa a$. The prefactor may respond differently, but both changes double the leading opacity.

4. A packet crossing a narrow resonance

A transmission resonance has unit peak height but width much smaller than the incoming packet’s energy spread. Must the packet transmit with probability near one? Explain.

Solution

No. The packet probability is the spectral average

$$P_{\mathrm T} = \int \lvert a(k)\rvert^2T(k)\,dk.$$

Only the narrow fraction of the packet spectrum inside the resonance sees $T\approx1$; components outside it may be strongly reflected. A unit value at one energy does not imply unit transmission for a broad packet. Near a narrow resonance, the outgoing packet can also be spectrally filtered and delayed.

References

• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• E. Merzbacher, Quantum Mechanics, 3rd ed., Wiley, 1998.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.
• J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover, 2006.
• S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge University Press, 1995.