Piecewise-Constant Potentials and Bound States

Potentials that are constant in pieces — wells, steps, and barriers — are the simplest setting in which the qualitative features of quantum mechanics appear: quantized bound-state energies, tunneling, and scattering resonances. In each region the time-independent Schrödinger equation has constant-coefficient solutions (exponentials or sinusoids); physics enters through the matching conditions at the boundaries.

1. Matching Conditions

Wherever $V(x)$ is finite, both $\psi$ and ${\psi }^{\prime }$ are continuous. At a $\delta$-function potential ${\psi }^{\prime }$ jumps; where $V\to \infty$ (hard wall) $\psi$ must vanish. Bound states additionally require $\psi \to 0$ as $|x|\to \infty$ (normalizability), which is what selects a discrete set of allowed energies.

2. Infinite Square Well

For $V=0$ on $[0,L]$ and $V=\infty$ outside, $\psi$ vanishes at both walls. The solutions are standing waves

$${\psi }_n(x)=\sqrt{\frac{2}{L}}\sin \frac{n\pi x}{L},E_n=\frac{{\hslash }^2{\pi }^2{n}^2}{2m{L}^2},n=1,2,\dots$$

Key lessons: energies are quantized by the boundary conditions, the spectrum grows as $n^2$, there is a nonzero ground-state energy $E_1>0$ (a consequence of confinement + uncertainty), and ${\psi }_n$ has $n-1$ interior nodes and definite parity about the center.

3. Finite Square Well

For a well of depth $V_0$ and width $2a$, bound states ($-V_0<E<0$) are oscillatory inside and exponentially decaying outside, $\psi \sim e^{-\kappa |x|}$ with $\kappa =\sqrt{-2mE}/\hslash$. Matching $\psi ,{\psi }^{\prime }$ at the edges yields transcendental quantization conditions (separately for even and odd parity):

$$k\tan (ka)=\kappa (\text{even}),k\cot (ka)=-\kappa (\text{odd}),k=\sqrt{2m(E+V_0)}/\hslash .$$

There are finitely many bound states, and — in 1D — at least one always exists, however shallow the well. The wavefunction leaks into the classically forbidden region ($|x|>a$), an intrinsically quantum effect.

4. Step Potential and the Barrier: Tunneling

For a particle of energy $E$ incident on a step or barrier, one solves for transmitted and reflected amplitudes and forms the transmission $T$ and reflection $R$ coefficients from the probability current, with $R+T=1$.

For a rectangular barrier of height $V_0>E$ and width $a$, the wavefunction is exponentially damped inside but nonzero on the far side — the particle tunnels:

$$T\approx \frac{1}{1+\frac{V_0^2{\sinh }^2(\kappa a)}{4E(V_0-E)}}\stackrel{\kappa a\gg 1}{\to }T\approx 16\frac{E(V_0-E)}{V_0^2}{e}^{-2\kappa a},\kappa =\frac{\sqrt{2m(V_0-E)}}{\hslash }.$$

The exponential sensitivity $e^{-2\kappa a}$ to width and mass underlies $\alpha$-decay, the scanning tunneling microscope, and Josephson junctions. The smooth-barrier generalization is the WKB tunneling formula. Above the barrier ($E>V_0$) $T$ oscillates, reaching unity at resonances where the barrier width is a half-integer number of internal wavelengths.

5. The Delta-Function Potential

For $V(x)=-\lambda \delta (x)$ ($\lambda >0$), integrating the Schrödinger equation across $x=0$ gives the jump condition ${\psi }^{\prime }(0^{+})-{\psi }^{\prime }(0^{-})=-\frac{2m\lambda }{{\hslash }^2}\psi (0)$. There is exactly one bound state,

$$\psi (x)=\sqrt{\kappa }{e}^{-\kappa |x|},\kappa =\frac{m\lambda }{{\hslash }^2},E=-\frac{m{\lambda }^2}{2{\hslash }^2},$$

a compact model for a short-range attractive potential and a useful building block (e.g. the Kronig–Penney model of a 1D crystal, obtained by periodically repeating it).

See also

• Wave mechanics — the PDE, current, and matching.
• WKB approximation — tunneling through smooth barriers.
• Scattering theory — the 3D generalization of $R,T$.