At a glance
A particle in a finite well has a wave that oscillates inside the well and extends past its walls. This sim builds that wave for a well of depth V₀ and half-width a, and shows what the energies, the tails and the number of bound states depend on.
What you'll learn
- How the wave inside the well joins the wave outside, and what that demands of the energy.
- How to read a bound state's energy E and its tail length δ off the η–k panel.
- How far the wave reaches past the walls, and what sets that distance.
- How the number of bound states depends on V₀ and a, through a single number z₀.
The well: V = 0 outside, −V₀ for |x| < a. The orange curve sketches a wave of the kind the sim draws at an energy E.
What's on screen
- Grey step (left plot): the potential V(x). The blue-tinted region is the well, between the dashed lines at x = ±a.
- Coloured curves: each state's ψ, drawn on its own level line (E0, E1, …) across the well.
- η–k plot (right): the matching curve (orange even, violet odd) and the blue circle. Each crossing is a dot in its state's colour.
- Bound States panel: N, and each state's E (eV) and δ (nm).
See itSwitch the sim to the odd family, with the same well.
Watch for: the orange curve in the η–k plot replaced by a violet one, and the wavefunctions on the left changing shape.
The controls
Press ⌖ on a row to see where that control sits in the sim.
| Control | What it changes | Range | |
|---|---|---|---|
| V₀ (Well Depth) | The depth of the well, in eV. | 1–32 eV opens at 10 | |
| a (Half-width) | Half the width of the well, in nm. The well runs from −a to +a. | 0.1–0.6 nm opens at 0.5 | |
| Parity | Which family of states the sim shows: even (tan) or odd (−cot). It sets the matching curve, the wavefunctions and the Bound States list. | Even / Odd opens on Even | |
| Wavefunction display | Draws each state as the wave ψ itself, Re(ψ), or as its probability density |ψ|². | Re(ψ) / |ψ|² opens on Re(ψ) | |
| Wavefunction visibility | Hides the curves and the level lines, and marks the well's width 2a and depth V₀ instead. | on / off opens on | |
| Bound States | N for the selected family, then each state's E and δ. N counts one family at a time. | panel | |
| Maximize (each plot) | Enlarges that plot to fill the stage. | on / off | |
| Reset | Returns V₀ and a to their opening values. | top bar |
Fixed: ħ²/2m = 0.0381 eV·nm² (the free-electron value). V = 0 outside the well and −V₀ inside, so every bound energy is negative.
A bound wave dies away outside
Take an energy E between −V₀ and 0, with zero at infinity. The wave has a different form in each region:
k and η are real and positive. For odd states the outside wave changes sign across x = 0.
- Inside, E is above V, so ψ oscillates.
- Outside, E is below V. The equation allows e−η|x| and e+η|x|, and only the decaying one can be normalised, ∫|ψ|² dx = 1.
- At x = ±a the inside wave must join the outside wave smoothly. At most energies the join also needs the growing exponential, so there is no bound state there.
The sim's defaults, V₀ = 10 eV and a = 0.5 nm, using the sim's own construction of ψ. Left: E = −2.86 eV, an allowed energy. Right: E = −3.50 eV, which is not.
Nothing discrete is imposed by hand: requiring a normalisable wave selects the energies.
See itThe sim hides its curves for an instant, then draws them again.
Watch for: every curve returning on one of its own level lines, E0, E1, …, and nowhere else.
Matching at the walls picks the energies
Requiring ψ and its slope to be continuous at x = a leaves one condition on k and η:
The matching condition for the even and odd families.
The two are not independent, because both come from the same E:
A circle in (ka, ηa), radius z₀.
- The η–k plot draws both: the matching curve and the circle. Its axes are k and η in nm⁻¹, so the circle's radius is z₀/a = √(V₀/0.0381) nm⁻¹, which is 16.2 nm⁻¹ at the defaults.
- Each crossing of curve and circle is one bound state. The sim numbers them 0, 1, 2, …
- z₀ = a√(V₀/0.0381), with a in nm and V₀ in eV, is 8.10 at the defaults. Only V₀ and a enter it.
See itSame well, other family: the curve changes and the circle does not.
Watch for: the blue circle staying the same size while the violet −cot branches cross it at new places.
Reading energy and tail from a crossing
A crossing gives a pair (k, η), and both the energy and the tail length follow from it:
The Bound States panel prints E in eV and δ in nm for each crossing.
V₀
a
These sliders move the sim too.
- Zero is set at infinity, so every bound energy is negative, and −V₀ is the floor of the well.
- N in the panel counts one family at a time. The line above counts both.
- The tail length δ = 1/η = ħ/√(2m|E|) is how far the wave extends past a wall before it has fallen to 1/e of its value there.
See itRun your current well through the sim.
Watch for: a level line on the plot for each E in the Bound States panel, with the panel's N counting the even states.
The wave reaches past the walls
The wave in the region |x| > a.
- Past a wall ψ is small but not zero, so the probability density |ψ|² is not zero where V is greater than E. A classical particle there would need negative kinetic energy. The wave itself needs no trajectory, so none is implied.
- The reach δ = 1/η is set by |E|, how far E sits below zero. A state near E = 0 reaches furthest. At the defaults the lowest even state has δ = 0.06 nm, and the least-bound odd state has δ = 0.41 nm.
- The density |ψ|² is a probability per unit length. The probability of finding the particle outside is the area under it there.
See itShow |ψ|² in place of Re(ψ).
Watch for: each curve rising above its own level line past the dashed wall lines, instead of stopping at them.
How many states a well holds
Each time z₀ passes a multiple of π/2, the circle reaches one more branch of the matching curves, and one more state appears. The two families take turns:
Counts for each family, and in total (⌈ ⌉ means round up).
- The even curve starts at the origin, so any circle crosses it. A one-dimensional well always binds at least one state.
- States arrive one at a time, alternating even, odd, even, … Never in pairs.
- Only z₀ sets the count. At a = 0.5 nm the whole V₀ slider gives 2 to 10 states in total.
The count against V₀ at a = 0.5 nm. The marked point is the opening well.
See itThe weakest well the sim can make, V₀ = 1 eV and a = 0.1 nm.
Watch for: N = 1, with one level near the top of the well and a wave extending far past the walls.
See itTwo different wells with the same z₀ = 7.25, run one after the other.
Watch for: the same N both times, with every energy changed.
Where it matters
- Quantum-well lasers. A layer a few nanometres thick, between layers of a wider-gap semiconductor, confines electrons in a finite well. Its thickness helps set the energy levels, and so the colour of the light. In GaAs the electron's effective mass is about 0.067 mₑ, so ħ²/2m is about 15 times the sim's value and the same well holds fewer levels.
- The deuteron. The neutron–proton well is a three-dimensional well of a similar kind. It holds a single bound state about 2.2 MeV below threshold. The scales are MeV and femtometres, not eV and nanometres.
- Tunnelling. The tail e−η|x| is the same wave behaviour that carries a particle through a thin barrier. A scanning tunnelling microscope's current changes steeply with the gap for this reason.
Go deeper: the weakest well
The shallow-limit estimate, to first order in z₀².
- For z₀ ≪ 1 the one state sits just below E = 0 and its tail is long, δ = 1/η ≈ a/z₀².
- The estimate is rough where z₀ is not small. At V₀ = 1 eV and a = 0.1 nm, z₀ = 0.51: it gives 0.26 eV, and the sim's solver gives 0.196 eV.
Summary
In one line. A finite well binds the energies at which the wave inside joins a decaying tail outside. The number z₀ = a√(V₀/0.0381) sets how many there are.
Key points
- Bound states have −V₀ < E < 0: ψ oscillates inside and decays outside.
- A crossing of the matching curve with the circle k² + η² = V₀/0.0381 gives k, η, E and δ = 1/η.
- |ψ|² is not zero past the walls, out to about δ.
- States appear one at a time as z₀ grows by π/2. A one-dimensional well always has one.
Common mix-ups
- "N = 3 means the well has three states." N counts one family. The defaults have 3 even and 3 odd.
- "A shallow enough well binds nothing." In one dimension it always binds one. The weakest well here has one, at E = −0.196 eV.
- "E < 0 means below the floor of the well." Zero is at infinity. The floor is −V₀.
Things to try on your own
- Set V₀ = 8 eV, a = 0.5 nm, then V₀ = 32 eV, a = 0.25 nm. Compare N and the energies for both families.
- At a = 0.5 nm, raise V₀ from 1 eV until a second even state appears. The exact threshold is z₀ = π, V₀ = 1.504 eV.
- At the defaults, show |ψ|² and compare the tails of E0 and E2 with their δ values.