Quantum Tunneling and the Particle in a Box

The idea: a particle is a wave

In quantum mechanics an electron is described by a wave function ψ(x, t), a complex number at every point. It is not a wave of anything you can touch: its size tells you where the electron is likely to be found. Shut the electron in a box with walls it cannot pass, and ψ must be zero at both walls, just like a guitar string held at both ends. Only standing waves fit, so only certain energies are allowed.

Why the energy is quantized

A box of width L holds a whole number n of half-wavelengths, as on a string: λₙ = 2L/n. The wave is

ψₙ(x) = √(2/L)·sin(nπx/L),   n = 1, 2, 3, …

De Broglie's rule gives the momentum p = h/λ = nh/(2L), and the energy of a free particle is p²/2m, so

Eₙ = n²·h²/(8mL²) = n²·E₁

For an electron in a 1 nm box E₁ = 0.376 eV, and the next levels are 1.50, 3.38 and 6.02 eV. Nothing in between is allowed. There is also no level with E = 0: confining a particle always costs energy (the zero-point energy). A box half as wide has levels four times higher, because E ∝ 1/L². (Demos 1 and 2; the energy-level panel draws the ladder.)

Probability density |ψ|²

The chance of finding the electron near x is |ψ(x)|², the probability density; over the whole box it adds up to 1. ψₙ has n − 1 nodes inside the box, places where |ψ|² = 0 and the electron is never found. In level n = 1 it is most likely in the middle; in n = 2 it is never in the middle.

In one level, ψ(x, t) = ψₙ(x)·e^(−iEₙt/ħ): the whole wave turns in the complex plane at the rate Eₙ/ħ. The real part, drawn in blue, swings up and down like a string. But |e^(−iEₙt/ħ)|² = 1, so |ψ|² never changes. That is why a level is called a stationary state (Demo 1).

Superpositions move

An electron can be in two levels at once: ψ = (ψ₁e^(−iE₁t/ħ) + ψ₂e^(−iE₂t/ħ))/√2. The two parts turn at different rates, so where they add up and where they cancel changes in time. |ψ|² sloshes from wall to wall with the period h/(E₂ − E₁) = 3.67 fs, and the average position ⟨x⟩ goes from 0.32 nm to 0.68 nm and back (Demo 3). This is the quantum picture of an atom giving off light: the charge sloshes at the frequency (E₂ − E₁)/h and the photon carries the energy E₂ − E₁ (here 1.13 eV, near-infrared).

Tunneling through a barrier

Now take a free wave packet (a lump of wave, see Group Velocity) with energy E and send it at a barrier of height V₀ > E. A classical ball would bounce back every time. The wave does not stop at the edge: inside the barrier the Schrödinger equation gives a decaying wave instead of an oscillating one,

ψ ∝ e^(−κx),   κ = √(2m(V₀ − E))/ħ

If the barrier is thin, some of the wave is still left at the far side and goes on as a smaller packet. The packet splits: a part R is reflected and a part T is transmitted, R + T = 1. That is tunneling. The particle is not split; T is the chance that it is found on the far side (Demos 4 to 6). While the packet hits the barrier, the incoming and reflected waves interfere and |ψ|² gets ripples half a wavelength apart, as in Standing Waves.

Exponential fall-off with width

The amplitude drops by e^(−κa) across a barrier of width a, so the probability drops by about e^(−2κa). For a plane wave the exact answer is

T = 1 / (1 + V₀²·sinh²(κa) / (4E(V₀ − E)))

and with E = V₀/2, as on the page, this is T = 1/cosh²(κa). In the page's units (ħ = m = 1, E = 0.5, V₀ = 1) κ = 1. The packet is solved numerically (Crank–Nicolson) and agrees with the plane-wave formula:

width a	plane wave 1/cosh²(a)	packet T (page)	ratio to the row above
0.5	0.786	0.785
1	0.420	0.420	0.53
2	0.0707	0.0727	0.17
3	0.0099	0.0107	0.15
4	0.0013	0.0016	0.15

Each extra unit of width multiplies T by roughly e^(−2) ≈ 0.14: on a log scale the points lie on a straight line (Demo 7). The packet holds a small spread of energies, and its faster parts tunnel more, so for wide barriers it gets through a little more than the plane wave.

This steepness is used everywhere. In a scanning tunneling microscope an electron tunnels across the vacuum gap from a sharp tip to a surface; with a work function of about 4 eV, κ ≈ 10 nm⁻¹, so moving the tip 0.1 nm changes the current about 8 times, enough to see single atoms. Alpha decay is an alpha particle tunneling out of the nucleus, and flash memory stores bits by tunneling electrons through a thin oxide.

See also

The box levels are the same counting as the harmonics of a guitar string in Standing Waves. That electrons are waves at all was shown by electron diffraction, the same fringes as light in Diffraction.

That light, too, is both a wave and a particle is shown in The Photoelectric Effect, where photons of energy hf knock electrons out of a metal.

What the page leaves out

The page stays in one dimension with ideal shapes: infinitely high walls, a rectangular barrier and no spin. A real well has finite walls, so ψ leaks a little into them and there are only a few levels. Energies above the barrier (where a packet can still partly reflect), smooth barriers (the WKB formula T ≈ e^(−2∫κ dx)), measurement and what "collapse" means, and how fast a particle tunnels are left out. The barrier scene uses units with ħ = m = 1 rather than electron-volts and nanometres.

References

OpenStax University Physics Volume 3, 7.4 The Quantum Particle in a Box

OpenStax University Physics Volume 3, 7.7 Quantum Tunneling of Particles through Potential Barriers

Wikipedia — Particle in a box

Wikipedia — Crank–Nicolson method