At a glance
A steady electron beam meets a barrier, and the sim draws the wave on both sides and reads off the transmitted fraction T and the reflected fraction R. The same mathematics governs the tunnel current of a scanning tunnelling microscope and alpha decay.
What you'll learn
- How the wave behaves inside and beyond a barrier taller than the beam energy E.
- How T depends on E, the height U and the width 2a.
- How a single step differs from a barrier with two edges.
- How T changes with E once E passes U.
What's on screen
- Potential V(x) (red): 0 outside, U between x = −a and x = +a.
- Beam energy E (dashed cyan), on the same axis.
- Re ψ (blue): the real part of the wave, for x from −10 to 14 nm.
- Im ψ (amber): the imaginary part of the wave.
- |ψ|² (purple): probability density, relative to the incoming beam.
- Transmission bar: T, R and T + R. The Guided Inquiry reveals it step by step.
- T(E) spectrum and ANALYTICAL panels: T against E, and the formula in use.
The controls
Press ⌖ on a row to see where that control sits in the sim.
| Control | What it changes | Range | |
|---|---|---|---|
| Square Barrier / Step Potential | Barrier: V = U for −a ≤ x ≤ a. Step: V = U for every x ≥ 0, with no far edge. | two options opens on Square Barrier | |
| Potential Height U | The height of the barrier or step, in eV. Moving it restarts the clock at t = 0. | 0.01–2.00 eV opens at 0.50 | |
| Barrier Half-Width a | Half the barrier's width, in nm, so the full width is 2a. Not used in Step mode. | 0.05–3.00 nm opens at 0.50 | |
| Energy E | The energy of each electron in the beam, in eV. Moving it restarts the clock. | 0.01–2.00 eV opens at 0.30 | |
| Real part, Imaginary part, Probability | Three tick boxes that show or hide the blue, amber and purple curves. The physics does not change. | all on at the start | |
| Animation Speed | How fast the clock runs. At 1× it advances 2 fs per second of real time. | 0.1×–5× opens at 1× | |
| Pause / Play | Freezes the clock, or runs it. | top bar | |
| Reset | Sets the clock back to t = 0. The sliders keep their values. | top bar | |
| Transmission bar | The transmitted fraction T, the reflected fraction R, their sum, and a badge naming the regime (E < U, E > U or E ≈ U). | readout | |
| T(E) SPECTRUM | T against E for the current U and a. The dot marks the current E and the dashed line marks U. | panel | |
| ANALYTICAL | The formula for T in the current regime, with k, η (κ for a step) or k₂ evaluated at your settings. | panel |
Fixed: the particle is an electron (ħ²/2m = 0.0381 eV·nm²). The beam comes from the left with incident amplitude 1.
The wave decays inside the barrier
Outside the barrier V = 0 and ψ is a travelling wave with wavenumber k. Inside, where U > E, the solutions are real exponentials with decay constant η:
At the opening values k = 2.81 nm⁻¹, so the wavelength 2π/k is 2.24 nm, and η = 2.29 nm⁻¹. The amplitude inside falls by a factor e over 1/η = 0.44 nm.
A plane wave cannot be normalised, so the plot shows |ψ|² relative to the incoming beam, and T and R are ratios of probability current. The beam is a stationary state, so its time dependence is one phase factor:
T falls steeply with barrier width
For E < U the transmitted fraction through a barrier of full width 2a is exactly:
Once 2ηa is larger than about 2, sinh(2ηa) ≈ e2ηa/2 and this reduces to:
At E = 0.30 eV and U = 0.50 eV, doubling the full width from 1.00 nm to 2.00 nm lowers T by a factor of 96, not 2. Raising U also lowers T: at 2a = 1.00 nm, U = 0.80 eV gives T = 0.27%, against 3.86% for 0.50 eV.
These sliders move the sim too.
A step transmits nothing below U
Replace the barrier by a step, V = U for x ≥ 0, with the same E < U. Beyond the edge ψ is still a decaying exponential (κ in the sim's analytical panel), so |ψ|² reaches a short way into the step. There is no second edge to turn that tail back into a travelling wave. A real exponential carries no probability current, so no current leaves.
Above the step one edge still reflects. At E = 0.60 eV and U = 0.50 eV, k₁ = 3.97 nm⁻¹ and k₂ = 1.62 nm⁻¹ give T = 82.3%.
Above the barrier, T oscillates with energy
For E > U the wave inside is a travelling wave with wavenumber k₂, so the sinh becomes a sin:
Each edge reflects part of the wave, and T = 1 exactly when the two reflected parts cancel. That happens when sin(2k₂a) = 0: a whole number of half-wavelengths of the inner wave fits across the full width 2a. Between those energies T dips, but it is above 0.94 for 0.88 eV ≤ E ≤ 2.00 eV at the opening U and a.
The peaks move down and crowd as the barrier widens
Setting 2k₂a = nπ gives the energies of perfect transmission:
En − U grows as n² and falls as 1/(2a)². Doubling a from 0.50 nm to 1.00 nm cuts every En − U by 4, so the peaks sit lower and closer together, and a third peak enters the energy range.
Where it matters
- Scanning tunnelling microscope. A metal tip sits about half a nanometre above a surface, and the tunnel current through the gap changes by roughly a factor of ten for each 0.1 nm change in gap. That steepness lets it detect height changes far smaller than an atom.
- Alpha decay. An alpha particle inside a heavy nucleus tunnels through a barrier it cannot climb. The exponential sensitivity to the barrier is why half-lives range from microseconds to billions of years.
- Flash memory. In common NAND flash, electrons tunnel through a thin oxide layer to charge or discharge a storage gate.
Go deeper: the same exponential in a vacuum gap
- With U − E = 4.5 eV, a typical metal work function, η = 10.9 nm⁻¹. Each extra 0.1 nm of gap lowers T by a factor of about 9.
- A real tip–surface barrier is not rectangular, and a current needs a sum over many electrons. The plane-wave figures give the scale of the effect, not a prediction.
Summary
Key points
- k = √(2mE)/ħ outside, η = √(2m(U − E))/ħ inside, and T + R = 1.
- For E < U and 2ηa ≳ 2, T ≈ 16E(U − E)/U² · e−4ηa.
- A step has T = 0 for E < U and T = 4k₁k₂/(k₁ + k₂)² above.
- A barrier with E > U has T = 1 at En = U + n²π²ħ²/(2m(2a)²).
Common mix-ups
- "The electron borrows energy to cross the barrier." E is the same everywhere. The wave simply has a decaying part inside.
- "Where |ψ|² is not zero, it is transmitted." In Step mode |ψ|² reaches past x = 0 while T = 0. T counts current that gets away.
- "Above U, everything is transmitted." T reaches 1 only at En; a step still reflects 17.7% at E = 0.60 eV.
Things to try on your own
- At E = 0.30 eV and U = 0.50 eV, find the a, in steps of 0.01 nm, where T first falls below 1%.
- At a = 0.50 nm, find the E where T first reaches 99.9%, and compare it with E₁ = 0.876 eV.