Barrier Penetration and Quantum Tunneling

The square-well problems 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.¹

A single plane wave $ψ \sim e^{ik x}$ is not normalizable over all space. The physical fix is to describe a beam: normalize to the particle density $ρ$ , so that $\int_{a}^{b} ∣ ψ ∣^{2} d x = ρ (b - a) = N$ 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) = {0, V_{0}, x < 0, x > 0.

Classically the outcome is a rule about energy. If $E > V_{0}$ the particle continues, slowing to speed $v = 2 (E - V_{0}) / m$ ; 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.²

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{2 m E}{ℏ} (x < 0), k_{2} = \frac{2 m ( E - V _{0} )}{ℏ} (x > 0),

and general solutions

ψ_{I} (x) = A e^{i k_{1} x} + B e^{- i k_{1} x}, ψ_{II} (x) = C e^{i k_{2} x} .

The term $A e^{i k_{1} x}$ is the incident beam, $B e^{- i k_{1} x}$ the reflected beam, and $C e^{i k_{2} x}$ the transmitted beam; there is no left-moving wave in region II, so its coefficient is zero. Requiring $ψ$ and $d ψ / d x$ continuous at $x = 0$ gives

A + B = C, k_{1} A - k_{1} B = k_{2} C,

which solve to

B = \frac{k _{1} - k _{2}}{k _{1} + k _{2}} A, C = \frac{2 k _{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}} = (\frac{k _{1} - k _{2}}{k _{1} + k _{2}})^{2}, T = \frac{k _{2} ∣ C ∣ ^{2}}{k _{1} ∣ A ∣ ^{2}} = \frac{4 k _{1} k _{2}}{( k _{1} + k _{2} ) ^{2}},

and they conserve particles,

R + T = 1.

Two consequences have no classical analog.³ Even with $E > V_{0}$ , $R \neq = 0$ : 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.

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.

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,

ψ_{II} (x) = C e^{- α x}, α = \frac{2 m ( V _{0} - E )}{ℏ} .

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

∣ ψ_{II} ∣^{2} = ∣ C ∣^{2} e^{- 2 α x},

falling off over a length $\sim 1/ α$ , 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 $Δ x \sim 1/ α$ 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.

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.

The rectangular barrier

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

V (x) = {V_{0}, 0, 0 < x < a, otherwise .

Take $E < V_{0}$ . Classically the beam is entirely reflected. Quantum mechanically the wave decays inside the barrier as $e^{- α x}$ , 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.⁴

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.

Matching $ψ$ and $ψ^{'}$ at both walls gives the transmission coefficient exactly:

T = [1 + \frac{sinh ^{2} ( α a )}{4 \frac{E}{V _{0}} ( 1 - \frac{E}{V _{0}} )}]^{- 1} .

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

T \approx 16 \frac{E}{V _{0}} (1 - \frac{E}{V _{0}}) e^{- 2 α a}, α = \frac{2 m ( V _{0} - E )}{ℏ} .

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.

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.

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 $V (r) = k Z z e^{2} / r$ that repels it outside the nuclear radius $R$ .⁵ The alpha particle, with energy $E$ well below the barrier top, rattles against the wall $N \approx v /2 R$ times per second and tunnels out with probability $T$ per attempt, giving a decay rate $decay rate = \frac{1}{τ} \approx \frac{v}{2 R} e^{- 2 2 m (V_{0} - E) a /ℏ} .$
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 $1 0^{- 6} s$ to $1 0^{10}$ years. This extreme sensitivity, derived from tunneling, is the resolution of the puzzle that radioactive decay rates vary over forty orders of magnitude.

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.

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 $0.5 nm$ in the gap alters the current by a factor of $1 0^{4}$ .
Ammonia clock. In $N H_{3}$ 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 $f = 2.3786 \times 1 0^{10} Hz$ , 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 $e^{- 2 α a}$ : 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.

Tipler & Llewellyn, Modern Physics, §6-6 — unbound states, beam normalization to particle density, and the reflection/transmission framework for scattering from potentials. ↩
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 $R + T = 1$ . ↩
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 $E < V_{0}$ . ↩
Tipler & Llewellyn, Modern Physics, §6-6 — the rectangular barrier, the exact transmission coefficient with $sinh^{2} (α a)$ , and the thick-barrier limit $T \approx 16 \frac{E}{V _{0}} (1 - \frac{E}{V _{0}}) e^{- 2 α a}$ . ↩
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. ↩