The Potential Barrier

Setting up the problem

The previous lesson studied what happens when a particle moving in a region of zero potential encounters an abrupt step to a higher potential. Now we extend that idea: instead of a step that goes on forever, the elevated potential exists only over a finite interval and then returns to its original value. This configuration is the rectangular potential barrier.

Concretely, consider a particle of mass $m$ and total energy $E$ moving to the right along the $x$ -axis. The potential energy is

V(x) = \begin{cases} 0 & x < 0 \\ V_0 & 0 \leq x \leq a \\ 0 & x > a, \end{cases}

where $V_0 > 0$ is the barrier height and $a$ is the barrier width. The barrier divides all of space into three regions, which are conventionally labelled I, II, and III.

The three regions

Region I ( $x < 0$ ) — the particle travels freely toward the barrier. There is both an incident wave (moving right) and a reflected wave (moving left):

\psi_\text{I}(x) = A e^{ikx} + B e^{-ikx}, \qquad k = \sqrt{\frac{2mE}{\hbar^2}}.

The wave number $k$ is real and positive because $E > 0$ in the free region.

Region II ( $0 \leq x \leq a$ ) — the character of the solution depends on the sign of $E - V_0$ .

If $E > V_0$ (the particle has more energy than the barrier), the wave number inside the barrier is $k' = \sqrt{2m(E-V_0)}/\hbar$ , which is again real, and the solution is a superposition of oscillating exponentials $e^{\pm ik'x}$ .
If $E < V_0$ (the classical case where the particle is energetically forbidden from the barrier), the quantity $E - V_0$ is negative, so we define $\kappa = \sqrt{2m(V_0-E)}/\hbar > 0$ and the general solution inside the barrier takes the form
$\psi_\text{II}(x) = C e^{\kappa x} + D e^{-\kappa x}.$
Both terms are real exponentials rather than oscillating waves. The coefficient $\kappa$ is called the decay constant (or attenuation constant) of the evanescent wave.

Region III ( $x > a$ ) — the potential has returned to zero. The boundary conditions at $x = a$ will determine the transmitted amplitude. Because there is no barrier to the right, there is no source of left-moving waves in this region, so the transmitted wave is purely

\psi_\text{III}(x) = F e^{ikx},

with the same free-particle wave number $k$ as in region I (energy is conserved).

Joining the pieces

A physically acceptable wavefunction must be continuous and have a continuous first derivative everywhere — including at the two boundaries $x = 0$ and $x = a$ . Applying these four matching conditions (continuity of $\psi$ and $d\psi/dx$ at each boundary) generates four equations relating the five constants $A$ , $B$ , $C$ , $D$ , and $F$ . One overall scale is arbitrary (it is fixed by normalization), so the system is exactly determined and can be solved to find the ratios $B/A$ and $F/A$ .

Transmission and reflection coefficients

Once $F/A$ is known, the transmission coefficient $T$ is defined as the ratio of the probability current carried by the transmitted wave to that carried by the incident wave. Because both region I and region III have the same wave number $k$ , the probability currents are proportional to $|F|^2$ and $|A|^2$ respectively, giving

T = \frac{|F|^2}{|A|^2}, \qquad R = \frac{|B|^2}{|A|^2}.

By conservation of probability (particles are neither created nor destroyed) we have $T + R = 1$ for all values of $E$ and $V_0$ .

Classical intuition — and where it fails

Classical mechanics says unambiguously: if $E < V_0$ the particle cannot cross the barrier and $T = 0$ ; if $E > V_0$ the particle always crosses and $T = 1$ . Quantum mechanics disagrees on both counts.

When $E > V_0$ : $T$ is generally less than 1 because the abrupt change in potential produces a reflected wave, exactly as an abrupt change in the index of refraction reflects light. At special energies where $k' a = n\pi$ (integer $n$ ), the reflected waves cancel by interference and $T$ reaches 1 — a phenomenon called resonance transmission.
When $E < V_0$ : $T$ is greater than zero because the evanescent wave in region II has a nonzero value at $x = a$ and feeds amplitude into region III. This non-zero transmission for a classically forbidden energy is quantum tunneling, the topic of the next lesson.

Summary

| Quantity | Region I | Region II ( $E < V_0$ ) | Region III | |---|---|---|---| | Potential | $0$ | $V_0$ | $0$ | | Solution type | Oscillating | Exponential | Oscillating | | Wave number / decay constant | $k = \sqrt{2mE}/\hbar$ | $\kappa = \sqrt{2m(V_0-E)}/\hbar$ | $k$ |

The rectangular barrier is the simplest model in which both reflection and transmission depend non-trivially on energy. It captures the essential quantum physics — evanescent waves, tunneling, and resonance — in a form where the algebra is fully tractable from the time-independent Schrödinger equation.