At a glance
This sim shows what the wave of a quantum particle does at the walls of a finite potential well, and how many bound states the well holds. Real walls have finite height, and this is the simplest model that includes that.
What you'll learn
- How the wave behaves at the walls and beyond them.
- How the state number n sets the energy and the probability found outside the well.
- How the depth V₀ and the width 2a set the number of bound states.
- How the Ψ curves and the |ψ|² curve differ for a state of definite energy.
The model: depth V₀, width 2a. The sim measures energy from the rim, where V = 0.
What's on screen
- Blue line (upper panel): the potential V(x), with dashed walls at x = ±a.
- Yellow dashed line: the selected state's energy, labelled n and E.
- Coloured curve (lower panel): Re Ψ, pink for n = 1. The purple curve is Im Ψ.
- Green curve and fill: the probability density |ψ|², on its own scale. |x| > a is shaded.
- Sidebar: Energy, Probability outside well, and the list of bound states.
See itOpen the default well on its lowest state, n = 1. Watch for: the blue well in the upper panel with the yellow dashed energy line inside it, and the pink Ψ curve with the green |ψ|² fill below.
The controls
Press ⌖ on a row to see where that control sits in the sim.
| Control | What it changes | Range | |
|---|---|---|---|
| V₀ | The depth of the well. The floor sits at −V₀ and the rim at 0. | 5–50 eV opens at 20.0 | |
| 2a | The width of the well. | 0.20–1.50 nm opens at 0.50 | |
| Bound-state selector ◀ ▶ | Chooses which bound state is drawn. The ↑ and ↓ keys do the same. | n = 1 up to the last level opens at n = 1 | |
| Energy levels list | One row per bound state with its energy. Click a row to select that state. | list | |
| ψ(x) and |ψ|² | ψ(x) shows or hides the two Ψ curves, coloured and purple. |ψ|² shows or hides the green curve and fill. | on / off both open on | |
| Animation speed | How fast the Ψ curves turn. It changes the display only, not the physics. | 0.10–1.00× opens at 0.25× | |
| Readouts | Energy: the selected state's E, measured from the rim. Probability outside well: the share of the total probability at |x| > a, in percent. | panel | |
| Play / Pause | Starts or stops the turning of the Ψ curves. The space bar does the same. | top bar | |
| Reset | Returns V₀, 2a, n and the speed to their opening values and starts the animation. | top bar |
Fixed: one electron mass, so ħ²/2m = 0.0381 eV·nm². The sim prints numbers without unit labels; they are eV and nm. Energies are measured from the rim, so bound states have negative E.
Oscillating inside, decaying outside
The potential is V = −V₀ for |x| < a and V = 0 for |x| > a. A bound state has an energy between the floor and the rim:
The well, and the energy window a bound state must sit in.
The time-independent Schrödinger equation then has a different form on each side of the wall, set by the sign of E − V:
Inside, E > V. Outside, E < V.
- Inside, ψ is a sine or cosine with wavenumber k.
- Outside, the solution that grows with |x| cannot be normalised, so only the decaying exponential remains. It never reaches zero.
- At the wall, V jumps by a finite amount, so ψ and ψ′ are continuous there. The wave does not stop at x = ±a. E is the same on both sides; only the sign of E − V changes.
See itLook at the default well on n = 1, where the dashed lines mark the walls. Watch for: the pink curve and the green fill continuing past the dashed walls, bending smoothly into the shaded region instead of stopping.
How far the tail reaches
Outside the walls, ψ falls as e−κ(|x|−a), with κ = √(2m|E|)/ħ. The rim is at E = 0, so |E| is how far the level lies below it. The decay length 1/κ is the distance over which ψ falls by a factor e.
The share outside is the area under |ψ|² beyond the walls, as a fraction of the total area, which is 1.
- At the default well (V₀ = 20 eV, 2a = 0.50 nm), n = 1 has E = −18.913 eV, 1/κ = 0.045 nm and 0.83% outside. n = 4 has E = −3.763 eV, 1/κ = 0.101 nm and 23.30% outside.
- Closer to the rim means smaller κ, a longer tail and a larger share outside. As |E| goes to zero, 1/κ grows without limit.
The tail past the wall at the default well, for n = 1 and n = 4. Dotted lines mark 1/κ.
See itTake the default well on n = 1, then on n = 4. Watch for: the Outside well readout going from 0.83% to 23.30% as the green fill and the Ψ curves beyond the dashed walls grow.
The number of levels depends on one number
Matching ψ and ψ′ at x = a gives one condition for even states and one for odd states. Here z = ka and z₀ = a√(V₀/(ħ²/2m)):
The two quantisation conditions.
Each crossing of a branch with the circle κa = √(z₀² − z²) is one bound state. Here z₀ = 5.73, so there are four.
The energies, and the number of levels.
V₀
2a
These sliders move the sim too.
- Counting. V₀ and a enter the count only through z₀. A new level appears each time z₀ passes a multiple of π/2, alternately even and odd. A one-dimensional well always holds at least one. At the default well, a = 0.25 nm and z₀ = 0.25·√(20/0.0381) = 5.73, so N = ⌈3.65⌉ = 4.
- Scaling. Double 2a and divide V₀ by 4: z₀, N and every zn stay the same, so every En becomes exactly one quarter. For n = 1, −18.913 eV becomes −4.728 eV.
See itMake the well as shallow and narrow as the sliders allow: V₀ = 5 eV, 2a = 0.20 nm. Watch for: the list of bound states shrinking to a single entry, n = 1 at about −2.57.
See itTake the default well, then V₀ = 5 eV with 2a = 1.00 nm. Watch for: the n = 1 Energy going from −18.913 to −4.728, a quarter of the first value, with four levels in the list both times.
A soft wall lowers every level
An infinitely deep box of the same width 2a has its ground state at E = ħ²π²/(2m(2a)²). For 2a = 0.50 nm that is 1.504 eV above the floor. The sim does not draw a box; this number is calculated here.
- The finite well is lower. Its ground state, −18.913 eV, lies 1.087 eV above the floor at −20 eV.
- Why. The wave continues past the walls, so inside the well it turns through less than the box's half wavelength: ka = z₁ < π/2, where the box has exactly π/2. A smaller k means less kinetic energy.
- Every level. zn lies between (n−1)π/2 and nπ/2, so each level lies below the box level of the same n. A deeper well approaches the box: at V₀ = 50 eV the ground state is 1.219 eV above the floor.
See itLook at the ground state of the default well. Watch for: the yellow dashed line just above the well's floor, with Energy reading −18.913, which is 1.087 above −20.
A stationary state keeps its density
A state of definite energy has one phase factor in time, and it cancels in the density:
Ψ(x,t) = ψ(x) e−iEt/ħ.
- Two curves turn. The coloured curve is Re Ψ and the purple curve is Im Ψ. They trade height as the phase advances.
- One curve stays. The green |ψ|² never moves. This is what "stationary" means.
- Animation speed is a display rate. The sim turns the phase at a rate set by max(|E|, 0.25) times the speed slider, not at the true |E|/ħ, which is about 2.9×10¹⁶ rad/s for n = 1 at the default well, with E measured from the rim. The sense of rotation on screen is also a display choice.
See itRun n = 4 at speed 1× with both curve toggles on. Watch for: the coloured and purple curves trading height while the green |ψ|² fill stays put.
Where it matters
- Semiconductor quantum wells. A thin GaAs layer between AlGaAs layers is a well for electrons, a few nm wide and a few tenths of an eV deep, and quantum-well lasers use it. The electron's effective mass there is about 0.067 mₑ, so this sim's numbers do not carry over, but the physics does.
- Scanning tunnelling microscopy. The same decay law makes the tunnelling current between tip and surface fall as e−2κd with the gap d. For a work function of 4 to 5 eV, κ is about 10 to 11 nm⁻¹, so each extra 0.1 nm of gap cuts the current by about 8 to 10 times.
Go deeper: two more cases
- The deuteron. Its s-wave radial problem is the odd sector of this well. With V₀ = 35 MeV and a = 2 fm, the binding comes out near 1.7 MeV, against 2.22 MeV measured, with about two-thirds of the probability outside the force range.
- Two energies at once. Take (ψ₁ + ψ₄)/√2 at the default well. The density then oscillates at (E₄ − E₁)/ħ, with a period near 0.27 fs. The sim shows single states only.
Summary
In one line. A bound state oscillates inside the well and decays outside it, so some probability lies beyond the walls, more for levels closer to the rim.
Key points
- Bound energies lie between −V₀ and 0. The tail decays over 1/κ = ħ/√(2m|E|).
- The number of levels is N = ⌈2z₀/π⌉, never less than 1.
- Every level lies below the infinite-box level of the same n and width.
Common mix-ups
- "With energy below the rim, the particle stays between the walls." A classical marble would. The quantum state has probability beyond the walls, and the sim prints it.
- "The most tightly bound state reaches furthest." The reverse: κ grows with |E|, so n = 1 has the shortest tail.
Things to try on your own
- At the default well, step through n = 1 to 4 and note Probability outside well for each, beside its distance below the rim.
- Keep 2a = 0.50 nm and lower V₀ until the list drops from four levels to three. Compare with 13.54 eV, where z₀ = 3π/2.
- Set V₀ = 50 eV, 2a = 0.50 nm. From Energy, find how far the ground state sits above the floor, and compare with the box value, 1.504 eV.