Potential wells and tunneling

Take a particle of mass $m$ in $0<x<L$ with zero potential inside and ideal infinite walls outside. The wavefunction vanishes at both endpoints. Inside, the stationary Schrödinger equation has sinusoidal solutions, and fitting the boundary conditions requires an integer number of half-wavelengths. The allowed wave numbers are $k_n=n\pi/L$ with positive integer $n$. There is no nonzero normalized $n=0$ state satisfying both boundaries.

Using $E=\hbar^2k^2/(2m)$ gives energies proportional to $n^2/L^2$. The ground state therefore has positive kinetic energy even when the potential inside is zero. This is not energy borrowed from somewhere: it is the lowest permitted eigenvalue for this Hamiltonian and its boundaries. Infinite walls are an idealization, but the inverse-square size dependence is an instructive confinement scale. Finite wells modify both energies and wavefunction tails.

In a region of constant potential $V_0>E$, the stationary equation becomes $\psi''=\kappa^2\psi$, with real positive $\kappa$. Its solutions are growing and decaying exponentials instead of sines and cosines. In a semi-infinite forbidden region, normalizability selects the decaying branch. Across a finite barrier, both exponential terms can be needed to match the solution at both interfaces. Discarding one too early can spoil the boundary conditions.

For a finite step in the potential with constant mass and no singular delta term, the wavefunction and its first derivative are continuous. Matching these conditions determines reflected and transmitted amplitudes. A bound-state tail is not by itself a transmitted outgoing particle flux. Tunneling transmission is defined for a scattering setup with an incoming wave and an outgoing channel; it compares probability currents rather than merely comparing wavefunction heights at two points.

For a time-independent barrier, the incoming and transmitted components have the same total energy. The classical statement that a particle cannot occupy the barrier region is not the wave equation's rule. Quantum transmission follows from a nonzero solution matched across the full region. Calling it temporary energy borrowing obscures the actual calculation and incorrectly suggests that energy conservation is suspended during traversal.

The exponential factor makes width and mass powerful controls. Doubling a thick barrier's width multiplies the leading transmission by another factor $e^{-2\kappa a}$. Increasing the mass increases $\kappa$ and suppresses penetration for the same energy deficit. The approximation becomes unreliable near a barrier top or in a thin barrier; resonant structures also require a fuller calculation. A sensible estimate always states which regime makes its simplified expression useful.