At a glance
This sim launches a quantum wave packet, with the mass of an electron, at a rectangular barrier and reports how much of it ends up on each side. It is the simplest model of an electron meeting a thin insulating layer or a gap between two surfaces.
What you'll learn
- How the barrier's height U and thickness, and the packet's energy E, decide the share on each side.
- What Re ψ, Im ψ and |ψ|² each show, and how they differ.
- How the textbook T relates to what a real packet does.
- What changes when E is below, at or above U.
The sim at t = 0, default settings: the barrier and the energy E (top), Re ψ with |ψ|² shaded (bottom).
What's on screen
- Barrier V(x) (red, top panel): height U, thickness 2a.
- Carrier E (dashed cyan): the packet's central energy.
- Re ψ (blue, bottom panel): the real part of the wave function.
- Im ψ (amber): the imaginary part.
- |ψ|² (purple): the probability density. The shaded band marks the barrier.
- Outcome bar: Reflection and Transmission estimates, "— %" until the packet has met the barrier.
- Wavepacket Info and Physics (sidebar): the packet's numbers, and the plane-wave T.
See itLaunch the default packet at 3× speed. Watch for: the packet reaching the barrier, then the Reflection and Transmission estimates replacing "— %" after about 7 s.
The controls
Press ⌖ on a row to see where that control sits in the sim.
| Control | What it changes | Range | |
|---|---|---|---|
| Barrier Height U | The height of the barrier. | 0.01–1.00 eV opens at 0.30 eV | |
| Barrier Half-Width a | Half the barrier's thickness: the barrier fills −a to +a, so it is 2a thick. | 0.10–2.00 nm opens at 0.50 nm | |
| Carrier E | The packet's central energy ħ²k₀²/2m (the dashed line). It also sets the packet's speed. | 0.02–0.80 eV opens at 0.20 eV | |
| Initial Width σ | How spread out the packet starts: |ψ|² has standard deviation σ. | 0.50–3.00 nm opens at 1.50 nm | |
| Initial Position x₀ | Where the packet's center starts, to the left of the barrier. | −10 to −3 nm opens at −6.0 nm | |
| Re ψ · Im ψ · |ψ|² (Display) | Show or hide each curve. | on / off all open on | |
| Animation Speed | Playback rate only. The physics is unchanged. | 0.1×–3.0× opens at 1.0× | |
| Outcome bar | Reflection and Transmission estimates, their sum, and a badge saying whether E is below, at or above U. | readout | |
| Wavepacket Info | ⟨H⟩₀, k₀, λ, group velocity, ΔE (energy spread) and k₀σ, plus one barrier line. | panel | |
| Physics | The plane-wave formula for the current regime, and its value of T. | panel | |
| Pause / Play | Freezes the packet, or lets it run. | top bar | |
| Reset | Restarts the packet from t = 0 with the sliders as they are. | top bar |
Fixed: electron mass, one dimension, no spin; the view spans −12 to +18 nm. Every slider except Animation Speed restarts the packet, and the sim restarts it by itself after each pass.
The packet and the barrier
The sim evolves ψ(x, t) with the time-dependent Schrödinger equation, in one dimension, for a particle of electron mass:
The barrier is V = U for |x| ≤ a and 0 elsewhere.
At t = 0 the packet is a Gaussian of width σ centered at x₀, moving right with wave number k₀:
The initial state, as the sim builds it.
- E = ħ²k₀²/2m is the packet's central energy (the dashed line). Its speed is vg = ħk₀/m.
- A packet confined to a width σ is a sum of many wave numbers, spread by 1/(2σ). Its energy spread is ΔE = ħvg/2σ: 58 meV at the defaults.
- A narrower packet has a wider ΔE. The sim flags k₀σ as "(low)" below 3: the packet is then too narrow for one energy to describe it well.
- The plane-wave formulas in the Physics panel are for a single energy. Section 07 returns to the difference.
See itNarrow the packet to σ = 0.50 nm. Watch for: ΔE in Wavepacket Info rising from 58 to about 175 meV, with k₀σ flagged "(low)".
Inside the barrier
Where E < U, the Schrödinger equation gives real exponentials instead of sines. The wave does not end at the wall: it decays.
κ = √(2m(U − E))/ħ; the decay length is 1/κ.
- At the defaults, 1/κ = 0.617 nm (Wavepacket Info prints it) and the barrier is 2a = 1.0 nm thick: 1.6 decay lengths.
- If any wave is left at the far face, V = 0 again, so it carries on as an ordinary traveling wave with the same k: the same wavelength and speed, but a smaller amplitude.
- The barrier removes amplitude, not energy.
The steady-state wave at the defaults: ripples on the left, decay in the barrier, a smaller steady wave on the right (|t|² = T = 13.1%).
See itRun the default packet and freeze it at t = 24 fs. Watch for: the purple |ψ|² reaching into the shaded band and thinning across it, with a low tail already beyond the far wall.
Thickness and height set T
Matching ψ and its slope at both faces gives the exact plane-wave T that the Physics panel evaluates:
Exact, for E < U.
Across the thickness 2a the amplitude falls by about e−2κa, so the probability falls by about e−4κa:
The thick-barrier form. At the defaults (2κa = 1.6) it gives 13.9%, against the exact 13.10%.
κ is inside the exponent, so T is steep in both a and U − E:
| Change | Physics panel T |
|---|---|
| defaults: U = 0.30 eV, E = 0.20 eV, a = 0.50 nm | 13.10% |
| E raised to 0.25 eV | 21.77% |
| E lowered to 0.10 eV | 3.58% |
| U raised to 0.50 eV | 1.39% |
| a doubled to 1.00 nm | 0.54% |
E
U
a
These are the sim's real sliders: dragging one here restarts the packet.
See itRun a = 0.50 nm, then a = 1.00 nm. Watch for: the Physics panel's T and the Transmission estimate both dropping sharply when a is doubled.
Above the barrier, some still reflects
For E > U the wave oscillates inside the barrier, with a different wave number k₂ = √(2m(E − U))/ħ. A change in wave number reflects part of a wave, so the two faces still reflect.
Exact, for E > U.
- At E = 0.60 eV, U = 0.30 eV, a = 0.50 nm the plane-wave formula gives T = 98.66%, so R = 1.34%.
- The two reflections interfere. T = 1 when 2k₂a = nπ, a whole number of half-wavelengths of the inner wave. For these values the first is a = π/2k₂ = 0.56 nm: set a = 0.56 and the Physics panel reads 100.00%.
- At E = U the sim uses T = 1/(1 + k₁²a²). For U = E = 0.30 eV and a = 0.50 nm, that is 33.7%.
See itSend the packet in at E = 0.60 eV. Watch for: a small reflected packet moving back to the left, and a Reflection estimate above zero.
A packet's outcome differs
The Physics panel gives T for one energy. A packet is a spread of energies, and each component meets the barrier with its own T. The packet's transmitted share is the average:
|φ(k)|² is the packet's distribution of wave numbers.
- T rises steeply with E, so the higher-energy components gain more than the lower-energy ones lose. The average sits above the single-energy value: about 16% against 13.10% at the defaults.
- A wider ΔE widens the gap. A narrower σ means a wider ΔE.
- The transmitted part is made of the higher-energy components, so it leaves slightly faster than the incoming packet: about 0.29 against 0.265 nm/fs at the defaults.
- The bar is an "estimate": it adds back the probability absorbed at the edges of the grid, and appears only after the packet has met the barrier.
| Initial width | ΔE | Transmission estimate |
|---|---|---|
| σ = 0.50 nm | 175 meV | 27.9% |
| σ = 1.50 nm (default) | 58 meV | 16.0% |
| σ = 3.00 nm, x₀ = −10 nm | 29 meV | 14.1% |
U = 0.30 eV, E = 0.20 eV, a = 0.50 nm throughout. The Physics panel's T is 13.10% in every row.
See itRun σ = 1.50 nm, then σ = 0.50 nm. Watch for: the Transmission estimate staying above the Physics panel's 13.10% both times, and rising when σ is narrower.
Where it matters
- Scanning tunneling microscope. The current through the vacuum gap between a sharp tip and a surface depends steeply on the gap. In this rectangular model with a 4 eV barrier, 0.1 nm more gap cuts T by a factor of about 8, which is why single atoms show up as bumps.
- Alpha decay. Gamow explained it as an alpha particle tunneling through the barrier around the nucleus. Because T is so steep, half-lives range from under a microsecond to billions of years.
- Flash memory. Charge is moved through a thin insulating oxide by tunneling, and the same effect is a leak in very thin transistor insulators.
T against half-width a: this sim's barrier (U − E = 0.10 eV) and a 4 eV vacuum gap.
Summary
In one line. A wave meeting a barrier is partly reflected and partly transmitted; for E < U the transmitted share falls roughly as e−4κa.
Key points
- Where E < U the wave decays as e−κx; what reaches the far face keeps its original k.
- T is steep in the thickness 2a and in U − E, through κ = √(2m(U − E))/ħ.
- For E > U some wave still reflects; T = 1 only when 2k₂a = nπ.
- A packet's outcome is T averaged over its energy spread.
Common mix-ups
- "The particle borrows energy to climb the barrier." The transmitted wave has the same k, hence the same energy. The barrier removes amplitude, not energy.
- "With E above U, everything gets through." Waves reflect wherever k changes, even above the barrier.
- "The bar should match the Physics panel exactly." The panel is for one energy; the packet has a spread.
Things to try on your own
- At U = 0.30 eV and E = 0.20 eV, find the smallest a (to 0.01 nm) where the Physics panel's T falls below 1%.
- At E = 0.60 eV, sweep a and note where the Physics panel reads 100.00%. Compare with a = nπ/2k₂.
- At the defaults, set σ to 0.50, 1.50 and 3.00 nm (with x₀ = −10 nm for the last) and record the Transmission estimate each time.