---
title: Barrier Penetration and Quantum Tunneling
module: Wave Mechanics in One Dimension
moduleNumber: 3
lessonNumber: 6
order: 306
summary: >
  Unbound states scatter rather than bind. A particle meeting a step is partly reflected
  even when it has more than enough energy to pass, and a particle meeting a barrier taller
  than its energy has a nonzero chance of appearing on the far side. Matching the wave
  function across the boundaries gives the reflection and transmission coefficients and the
  exponential tunneling probability that explains alpha decay, the scanning tunneling
  microscope, and the ammonia clock.
topics: [Wave Mechanics in One Dimension]
draft: false
sources:
  - book: Tipler & Llewellyn
    ref: "Ch. 6 — The Schrödinger Equation; §6-6 Reflection and Transmission of Waves, Alpha Decay, NH₃ Atomic Clock"
---

The [square-well problems](/quantum-mechanics/wave-mechanics-1d/particle-in-infinite-and-finite-square-wells) were
bound states: the potential exceeds the energy far away, so the wave function decays and the
energy is quantized. This lesson takes the opposite regime, **unbound states**, where
$E > V(x)$ as $x \to \pm\infty$ in at least one direction. There the wave function
oscillates without decaying and any energy is allowed, so the spectrum is continuous.[^tl-66]

A single plane wave $\psi \sim e^{ikx}$ is not normalizable over all space. The physical
fix is to describe a _beam_: normalize to the particle density $\rho$, so that
$\int_a^b|\psi|^2\,\d x = \rho(b-a) = N$ counts the $N$ particles in the interval. Scattering
problems then ask what fraction of an incident beam is reflected and what fraction
transmitted.

## The step potential

Consider a beam moving right into a potential that jumps at the origin:

$$
V(x) =
\begin{cases}
0, & x < 0, \\
V_0, & x > 0.
\end{cases}
$$

Classically the outcome is a rule about energy. If $E > V_0$ the particle continues,
slowing to speed $v = \sqrt{2(E-V_0)/m}$; if $E < V_0$ it is turned around and reflected. A
ball rolling at a step in the ground either climbs and continues or rolls back, with no
middle case.[^tl-step]

Quantum mechanically, the Schrödinger equation is solved on each side and the pieces joined.
For $E > V_0$ both regions are classically allowed, with

$$
k_1 = \frac{\sqrt{2mE}}{\hbar}\ (x<0),
\qquad
k_2 = \frac{\sqrt{2m(E-V_0)}}{\hbar}\ (x>0),
$$

and general solutions

$$
\psi_{\mathrm{I}}(x) = A e^{ik_1 x} + B e^{-ik_1 x},
\qquad
\psi_{\mathrm{II}}(x) = C e^{ik_2 x}.
$$

The term $Ae^{ik_1 x}$ is the incident beam, $Be^{-ik_1 x}$ the reflected beam, and
$Ce^{ik_2 x}$ the transmitted beam; there is no left-moving wave in region II, so its
coefficient is zero. Requiring $\psi$ and $\d\psi/\d x$ continuous at $x = 0$ gives

$$
A + B = C, \qquad k_1 A - k_1 B = k_2 C,
$$

which solve to

$$
B = \frac{k_1 - k_2}{k_1 + k_2}\,A,
\qquad
C = \frac{2k_1}{k_1 + k_2}\,A.
$$

### Reflection and transmission coefficients

The observable rates are the squared amplitudes, weighted by the wave numbers because the
transmitted particles move at a different speed:

$$
R = \frac{|B|^2}{|A|^2} = \Big(\frac{k_1 - k_2}{k_1 + k_2}\Big)^2,
\qquad
T = \frac{k_2|C|^2}{k_1|A|^2} = \frac{4k_1 k_2}{(k_1 + k_2)^2},
$$

and they conserve particles,

$$
R + T = 1.
$$

> **Definition (Reflection and transmission coefficients).** $R$ is the fraction of an
> incident beam reflected, $T$ the fraction transmitted. They satisfy $R + T = 1$. For a
> step with $E > V_0$, $R = \big(\tfrac{k_1-k_2}{k_1+k_2}\big)^2$ depends only on the size
> of the wave-number change, not its direction.

Two consequences have no classical analog.[^tl-stepR] Even with $E > V_0$, $R \ne 0$: some
particles reflect off a step they have ample energy to cross, exactly like the partial
reflection of light at the boundary between two transparent media. And $R$ depends on
$|k_1 - k_2|$ only, so a step _down_ reflects as strongly as a step _up_ of the same size.

$$
% caption: A beam crossing a downward step keeps its energy but changes wavelength; the
% incident region carries a short wave, the far region a longer one, and part of the beam
% reflects even though every particle has enough energy to pass.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % potential step outline
  \draw[black, thick] (-3.4,1.3) -- (0,1.3) -- (0,0) -- (3.6,0);
  \node[black!70, font=\scriptsize] at (-2.9,1.55) {$V_0$};
  \node[black!70, font=\scriptsize] at (3.3,0.28) {$V = 0$};
  \draw[black, dashed] (0,-1.4) -- (0,1.9);
  \node[black, font=\scriptsize, anchor=south] at (-1.7,-1.9) {Region I};
  \node[black, font=\scriptsize, anchor=south] at (1.9,-1.9) {Region II};
  % left short wavelength wave (baseline y = -0.7)
  \begin{scope}[yshift=-0.7cm]
    \draw[acc, very thick] (-3.4,0) sin (-3.05,0.5) cos (-2.7,0) sin (-2.35,-0.5) cos (-2.0,0)
      sin (-1.65,0.5) cos (-1.3,0) sin (-0.95,-0.5) cos (-0.6,0) sin (-0.25,0.5) cos (0,0.3);
  \end{scope}
  % right long wavelength wave (baseline y = -0.7)
  \begin{scope}[yshift=-0.7cm]
    \draw[acc, very thick] (0,0.3) sin (0.6,0.65) cos (1.2,0) sin (1.8,-0.65) cos (2.4,0)
      sin (3.0,0.65) cos (3.6,0);
  \end{scope}
  \node[acc, font=\scriptsize, anchor=south] at (-1.7,-0.05) {short wave};
  \node[acc, font=\scriptsize, anchor=south] at (1.7,-0.05) {long wave};
\end{tikzpicture}
$$

### Penetration when $E < V_0$

For $E < V_0$ the far region is classically forbidden. Then $k_2$ becomes imaginary and the
transmitted solution is a real decaying exponential,

$$
\psi_{\mathrm{II}}(x) = C e^{-\alpha x},
\qquad
\alpha = \frac{\sqrt{2m(V_0 - E)}}{\hbar}.
$$

Now $B$ and $A$ have equal modulus, so $R = 1$ and $T = 0$: every particle is eventually
reflected, as classical physics predicts. But the reflection is not instantaneous at
$x = 0$. The wave penetrates a short distance into the barrier, with density

$$
|\psi_{\mathrm{II}}|^2 = |C|^2 e^{-2\alpha x},
$$

falling off over a length $\sim 1/\alpha$, before the beam is fully turned back. This is
the optical analog of total internal reflection, where the field leaks a fraction of a
wavelength beyond the reflecting surface. The penetration does not violate energy
conservation: localizing a particle in the tail of width $\Delta x \sim 1/\alpha$ costs a
momentum uncertainty that supplies exactly enough kinetic energy to keep any measured value
non-negative, the same argument used for the
[finite well](/quantum-mechanics/wave-mechanics-1d/particle-in-infinite-and-finite-square-wells).

$$
% caption: Reflection and transmission at a step versus energy in units of the step height;
% below the step everything reflects, and above it the transmission rises toward one while a
% residual reflection persists.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (5.6,0) node[right, font=\scriptsize, text=black] {$\frac{E}{V_0}$};
  \draw[->, black] (0,0) -- (0,3.4) node[above, font=\scriptsize, text=black] {$R,\,T$};
  \draw[black, dashed] (0,3.0) -- (5.4,3.0);
  \node[black, font=\scriptsize, anchor=east] at (-0.05,3.0) {$1$};
  % step top at E/V0 = 1 -> x = 1.8
  \draw[black, dashed] (1.8,0) -- (1.8,3.2);
  \node[black, font=\scriptsize, anchor=north] at (1.8,-0.1) {$1$};
  % R = 1 for E<V0, then falls
  \draw[black, very thick, dashed] (0,3.0) -- (1.8,3.0)
    .. controls (2.6,2.1) and (3.6,0.9) .. (5.4,0.55);
  \node[black, font=\scriptsize, anchor=west] at (4.2,0.9) {$R$};
  % T = 0 for E<V0, then rises
  \draw[acc, very thick] (0,0) -- (1.8,0)
    .. controls (2.6,0.9) and (3.6,2.1) .. (5.4,2.45);
  \node[acc, font=\scriptsize, anchor=west] at (4.2,2.15) {$T$};
\end{tikzpicture}
$$

## The rectangular barrier

Place a wall of finite width $a$ in the beam's path:

$$
V(x) =
\begin{cases}
V_0, & 0 < x < a, \\
0, & \text{otherwise}.
\end{cases}
$$

Take $E < V_0$. Classically the beam is entirely reflected. Quantum mechanically the wave
decays inside the barrier as $e^{-\alpha x}$, but if the barrier is thin the wave has not
died out at $x = a$, and it must join a right-moving oscillatory wave beyond. That surviving
amplitude is a transmitted beam: some particles cross a barrier they classically cannot
enter. This is **tunneling**.[^tl-barrier]

$$
% caption: A wave incident on a barrier taller than its energy decays exponentially inside
% and emerges with reduced amplitude on the far side; the surviving oscillation is the
% transmitted, tunneled beam.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % barrier outline
  \draw[black, thick] (-3.2,0) -- (0,0) -- (0,2.0) -- (1.7,2.0) -- (1.7,0) -- (4.4,0);
  \node[black!70, font=\scriptsize] at (0.85,2.25) {$V_0$};
  \node[black, font=\scriptsize] at (0.85,0.9) {barrier width $a$};
  \draw[black, dashed] (0,-1.6) -- (0,2.2);
  \draw[black, dashed] (1.7,-1.6) -- (1.7,2.2);
  \node[black, font=\scriptsize] at (-1.6,-1.9) {Region I};
  \node[black, font=\scriptsize] at (0.85,-1.9) {II};
  \node[black, font=\scriptsize] at (3.0,-1.9) {Region III};
  % incident wave (larger amplitude), baseline y=-0.8
  \begin{scope}[yshift=-0.8cm]
    \draw[acc, very thick] (-3.2,0) sin (-2.85,0.55) cos (-2.5,0) sin (-2.15,-0.55) cos (-1.8,0)
      sin (-1.45,0.55) cos (-1.1,0) sin (-0.75,-0.55) cos (-0.4,0) sin (-0.05,0.4) cos (0,0.45);
    % exponential decay inside barrier
    \draw[acc, very thick] (0,0.45) .. controls (0.6,0.28) and (1.2,0.16) .. (1.7,0.12);
    % small transmitted wave
    \draw[acc, very thick] (1.7,0.12) sin (2.05,0.2) cos (2.4,0) sin (2.75,-0.2) cos (3.1,0)
      sin (3.45,0.2) cos (3.8,0) sin (4.15,-0.2) cos (4.4,0);
  \end{scope}
  \node[acc, font=\scriptsize, anchor=south] at (-1.7,0.1) {incident};
  \node[acc, font=\scriptsize, anchor=south] at (3.1,0.05) {transmitted};
\end{tikzpicture}
$$

Matching $\psi$ and $\psi'$ at both walls gives the transmission coefficient exactly:

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

When the barrier is thick or tall, $\alpha a \gg 1$, the hyperbolic sine is dominated by its
growing exponential and $T$ simplifies to

$$
T \approx 16\,\frac{E}{V_0}\Big(1 - \frac{E}{V_0}\Big)\,e^{-2\alpha a},
\qquad
\alpha = \frac{\sqrt{2m(V_0 - E)}}{\hbar}.
$$

> **Theorem (Tunneling probability).** A particle of energy $E$ incident on a rectangular
> barrier of height $V_0 > E$ and width $a$ transmits with probability
> $T \approx 16\,\tfrac{E}{V_0}\!\big(1-\tfrac{E}{V_0}\big)e^{-2\alpha a}$ for $\alpha a \gg 1$.
> The probability falls exponentially with the barrier width and with
> $\sqrt{V_0 - E}$, so tunneling is significant only for thin, low barriers.

The exponential dependence is what makes tunneling a sharp, sensitive effect. Doubling the
barrier width squares the (small) transmission; a change in width of one atomic diameter can
change $T$ by orders of magnitude.

$$
% caption: Transmission drops exponentially as the barrier widens, so a thin barrier lets a
% measurable fraction through while a barrier a few penetration depths wide is effectively
% opaque.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (5.6,0) node[right, font=\scriptsize, text=black] {barrier width $a$};
  \draw[->, black] (0,0) -- (0,3.4) node[above, font=\scriptsize, text=black] {$T$};
  \node[black, font=\scriptsize, anchor=east] at (-0.05,3.0) {$1$};
  \draw[black, dashed] (0,3.0) -- (0.2,3.0);
  % exponential decay curve
  \draw[acc, very thick] (0,3.0) .. controls (0.7,1.5) and (1.5,0.7) .. (2.6,0.35)
    .. controls (3.6,0.16) and (4.6,0.06) .. (5.4,0.04);
  \node[acc, font=\scriptsize, anchor=south west] at (2.7,0.35) {exponential decay};
\end{tikzpicture}
$$

## Physical realizations

Tunneling is not a curiosity; the exponential in $T$ appears throughout nuclear, atomic, and
device physics.

- **Alpha decay.** Gamow, Condon, and Gurney modeled the nucleus as a well holding an alpha
  particle, bounded by the Coulomb barrier $V(r) = kZze^2/r$ that repels it outside the
  nuclear radius $R$.[^tl-alpha] The alpha particle, with energy $E$ well below the barrier
  top, rattles against the wall $N \approx v/2R$ times per second and tunnels out with
  probability $T$ per attempt, giving a decay rate

  $$
  \text{decay rate} = \frac{1}{\tau} \approx \frac{v}{2R}\,e^{-2\sqrt{2m(V_0 - E)}\,a/\hbar}.
  $$

  A small rise in $E$ lowers both the barrier height $V - E$ and its thickness, and because
  the rate depends _exponentially_ on both, a modest change in alpha energy (roughly 4 to 7
  MeV across natural emitters) spans lifetimes from $10^{-6}\ \text{s}$ to $10^{10}$ years.
  This extreme sensitivity, derived from tunneling, is the resolution of the puzzle that
  [radioactive decay](/nuclear-physics/radioactive-decay/decay-law-modes) rates vary over
  forty orders of magnitude.

$$
% caption: In alpha decay the particle sits in the nuclear well at energy E, below the peak
% of the Coulomb barrier; its wave function decays through the barrier and re-emerges as a
% free outgoing wave at the radius where the potential drops back to E.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,-1.4) -- (0,3.4) node[above, font=\scriptsize, text=black] {$V(r)$};
  \draw[->, black] (0,0) -- (6.2,0) node[right, font=\scriptsize, text=black] {$r$};
  % square well: deep box for r<R
  \draw[black, thick] (0.55,3.0) -- (0.55,-1.2) -- (1.3,-1.2) -- (1.3,3.0);
  \node[black, font=\scriptsize, anchor=south] at (0.9,3.0) {well};
  % Coulomb tail decreasing 1/r from barrier top
  \draw[black, thick] (1.3,3.0) .. controls (2.2,1.7) and (3.4,1.1) .. (6.0,0.55);
  % energy level E inside and outside
  \draw[black, thick, dashed] (0.55,1.2) -- (5.6,1.2);
  \node[black, font=\scriptsize, anchor=west] at (5.4,1.45) {$E$};
  \node[black, font=\scriptsize, anchor=south] at (1.3,-0.7) {$R$};
  % emergence radius where Coulomb = E
  \draw[black, dashed] (3.05,0) -- (3.05,1.2);
  \node[black, font=\scriptsize, anchor=north] at (3.05,-0.05) {$r_1$};
  % wavefunction: oscillate in well, decay in barrier, oscillate outside
  \draw[acc, very thick] (0.6,2.0) sin (0.75,2.35) cos (0.9,2.0) sin (1.05,1.65) cos (1.2,2.0);
  \draw[acc, very thick] (1.3,2.0) .. controls (2.0,1.55) and (2.6,1.4) .. (3.05,1.35);
  \draw[acc, very thick] (3.05,1.35) sin (3.4,1.55) cos (3.75,1.35) sin (4.1,1.15) cos (4.45,1.35)
    sin (4.8,1.55) cos (5.15,1.35);
\end{tikzpicture}
$$

- **Scanning tunneling microscope.** The gap between a sharp conducting tip and a specimen is
  a barrier for surface electrons. A small bias drives a tunneling current whose exponential
  dependence on the gap width is so steep that holding the current constant while scanning
  traces the surface to atomic resolution; a change of $0.5\ \text{nm}$ in the gap alters the
  current by a factor of $10^4$.
- **Ammonia clock.** In $\mathrm{NH_3}$ the nitrogen atom sits in a double-well potential,
  one minimum above and one below the plane of the three hydrogens, separated by a central
  barrier. The nitrogen tunnels back and forth through the barrier at $f = 2.3786\times10^{10}\ \text{Hz}$,
  a frequency stable enough to have served as the standard in the first atomic clocks.
- **Tunnel diode and Josephson junction.** Electron tunneling across a thin insulating gap
  underlies fast electronic devices whose current responds to the barrier on the same
  exponential curve.

The common thread is the exponential penetration factor $e^{-2\alpha a}$: a small tail of the
wave function reaching past a classically impassable region, turned into a measurable current,
a decay rate, or a clock frequency.

[^tl-66]: **Tipler & Llewellyn**, _Modern Physics_, §6-6 — unbound states, beam normalization to particle density, and the reflection/transmission framework for scattering from potentials.
[^tl-step]: **Tipler & Llewellyn**, _Modern Physics_, §6-6 — the step potential with $E > V_0$: continuity conditions, amplitudes $B$ and $C$, and the coefficients $R$ and $T$ with $R + T = 1$.
[^tl-stepR]: **Tipler & Llewellyn**, _Modern Physics_, §6-6 — partial reflection despite $E > V_0$ and the dependence of $R$ on $|k_1 - k_2|$ only (a step down reflects like a step up); penetration and total reflection for $E < V_0$.
[^tl-barrier]: **Tipler & Llewellyn**, _Modern Physics_, §6-6 — the rectangular barrier, the exact transmission coefficient with $\sinh^2(\alpha a)$, and the thick-barrier limit $T \approx 16\,\tfrac{E}{V_0}(1-\tfrac{E}{V_0})e^{-2\alpha a}$.
[^tl-alpha]: **Tipler & Llewellyn**, _Modern Physics_, §6-6, Alpha Decay — the Gamow-Condon-Gurney tunneling model of alpha decay and the exponential energy dependence of the decay rate; the STM, ammonia clock, and tunnel diode as further applications.