Quantum mechanics

The particle in a box and quantum tunneling

An infinite well has discrete energy levels; for a finite barrier the wavefunction decays but remains nonzero beyond it, giving a tunneling probability.

Potential energy determines where classical motion is allowed, but the Schrödinger equation permits a wavefunction to extend into classically forbidden regions. Two basic models are the infinite well, which illustrates quantization, and the finite barrier, which illustrates tunneling.

E_n=\frac{n^2\pi^2\hbar^2}{2mL^2},\quad\psi_n(x)=\sqrt{\frac2L}\sin\frac{n\pi x}{L};\qquad T\approx e^{-2\kappa a},\quad\kappa=\frac{\sqrt{2m(V_0-E)}}{\hbar}

Definition: One-dimensional infinite well

For $V=0$ on $0<x<L$ and $V=\infty$ outside, boundary conditions $\psi(0)=\psi(L)=0$ allow only wave numbers $k_n=n\pi/L$ ( $n=1,2,...$ ). The ground-state energy is nonzero: a confined particle cannot have both zero position spread and zero momentum.

Change level

n

to see standing-wave nodes and

E_n∝n²

; switch to a finite barrier to observe a wave packet transmitted with reduced amplitude.

Penetrating a finite barrier

For a rectangular barrier of height $V_0$ and width $a$ , with $E<V_0$ , the solution decays in the barrier as $e^{-\kappa x}$ . Exact transmission also depends on matching at both boundaries; $T\approx e^{-2\kappa a}$ is a WKB approximation for a sufficiently thick/high barrier, not an exact formula in every case.

Example: Ground-state energy in a box

An electron is confined in a 1D box of length $L=1.00$ nm. Find $E_1$ in eV.

Solution

$E_1=h^2/(8m_eL^2)=6.02\times10^{-20}$ J $=0.376$ eV.

For a particle with $E<V_0$ incident on a barrier of height $V_0$ , the Schrödinger equation gives an evanescent solution in the forbidden region. For a rectangular barrier of width $a$ , transmission decreases approximately as $T\propto e^{-2\kappa a}$ , where $\kappa=\sqrt{2m(V_0-E)}/\hbar$ . A wider barrier or heavier particle therefore makes tunneling less likely. For a light particle and a thin barrier, the probability remains finite. This underlies scanning tunneling microscopes and alpha decay; the ideal one-dimensional model must be distinguished from a real material's structure.

Quick check

In a 1D infinite well, how does level $n$ energy scale?

For $E<V_0$ , what is the qualitative form of the wavefunction inside a rectangular barrier?

References

David J. Griffiths, Darrell F. Schroeter (2018). Introduction to Quantum Mechanics