Transmission Plot
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About this simulation

Physics

For a 1D symmetric rectangular barrier V(x) = U for −a < x < a (full width 2a, height U), the plane-wave transmission probability T(E) depends on mass m and energy E. Below the barrier (E < U), transmission occurs via quantum tunnelling; above (E > U), T oscillates with barrier width. Conservation of probability gives R + T = 1, where R is the reflection probability.

Formulas

E < U (tunneling — derived in Lecture 9):

\[ T = \frac{4k^2\eta^2}{4k^2\eta^2 + (k^2+\eta^2)^2\sinh^2(2\eta a)} \]

E > U (above barrier — obtained by substituting η → iκ):

\[ T = \frac{4k^2\kappa^2}{4k^2\kappa^2 + (k^2-\kappa^2)^2\sin^2(2\kappa a)} \]
\[ \eta = \sqrt{\frac{2m(U-E)}{\hbar^2}}\,,\quad \kappa = \sqrt{\frac{2m(E-U)}{\hbar^2}}\,,\quad E = \frac{\hbar^2 k^2}{2m} \]

The simulation sets ℏ = 1 in the computation. The E > U formula extends the lecture derivation, which covers the tunneling case explicitly.

Simulation visuals

The main plot shows T (and optionally R = 1 − T) vs energy or wavenumber; hover for readouts. A moving marker scans the curve (optional). The barrier diagram below the plot is a schematic visualization of the incident, transmitted, and barrier-interior waves.

Controls

View mode: T vs E or T vs k. Parameters: half-width a, barrier height U, mass m, scanner speed. Display: classical prediction, region shading, resonance markers (T = 1), animated scanner. Top bar: Reset, Start/Pause (animation). Theme: dark/light toggle. Current values panel shows T, E, k, and η or related quantity for the active point.

Transmission T
Classical limit

Guided Inquiry

1 · Barrier and beam

Here’s our setup. A steady beam of particles of energy E meets a rectangular barrier of height U and full width 2a. Instead of one energy at a time, the curve shows you the transmitted fraction T for every energy at once. You can change the barrier’s height and half-width, and the particle’s mass, too.

2 · The classical barrier

Let’s start with the classical picture. Classically, a particle either has enough energy to clear the barrier or it doesn’t. So at a fixed energy below the top, how do you think the transmitted fraction depends on the barrier’s width 2a?

3 · A lighter particle

Now let’s change the particle. We’ll lower its mass from 1.00 to 0.25 electron masses and leave the barrier height and width alone. At an energy half the barrier height, what do you think the transmitted fraction does?

4 · Perfect transmission

Now look above the barrier top. There the curve dips below 1 and comes back, reaching exactly 1 at the marked energies. What do you think makes the barrier perfectly transparent right there?

5 · Below the top

One last question. Every energy that makes the barrier perfectly transparent lies above its top. Could there also be such an energy below the top, for some choice of height, width and mass? What’s your hunch?

Listen
View mode
Parameters (ℏ = 1)
Barrier Half-Width a 1.00 Å
Barrier Height U 5.00 eV
Particle Mass m 1.00 me
Animated scanner speed 1.00×
Current Values
T—
R = 1 − T—
E (eV)—
k (Å⁻¹)—
η (Å⁻¹)—
Display
//refined following the curriculum feedback.