Particle in a Box Simulation

About this experiment

The particle in a box is the simplest quantum system with quantised energy. An electron is trapped between two infinitely high walls, so its wavefunction must vanish at both ends and only standing waves that fit a whole number of half wavelengths are allowed. Pick a quantum number and watch the stationary state rotate in the complex plane while its probability density stays fixed, then mix two states and watch the electron slosh from wall to wall with a period set by their energy difference.

Learning goals

Solve the time-independent Schrödinger equation for an infinite square well and apply the boundary conditions
Explain why the energy is quantised and scales as n² and 1/L²
Distinguish the wavefunction from the probability density and compute the probability of finding the particle in a region
Explain why a stationary state has a time-independent probability density while a superposition does not
Relate level spacings to the wavelength of emitted or absorbed photons

How to use it

1In Single state, step n from 1 to 5. Count the nodes and note how the energy grows.
2Press Run time with n = 2. Watch Re Ψ and Im Ψ. Why does |Ψ|² not change at all?
3Drag a region over the left quarter of the box. Compare the probability for n = 1, 2 and a large n. What happens as n grows?
4Halve the box width. By what factor does every energy change? Check the photon wavelength for 2 → 1.
5Switch to Superposition with n₁ = 1 and n₂ = 2 and run time. Follow ⟨x⟩ and compare its period with the beat period readout.
6Try n₁ = 1 and n₂ = 3. Why does ⟨x⟩ stay in the middle even though |Ψ|² changes?

The model

ψ_{n} (x) = \frac{2}{L} sin \frac{nπ x}{L}, n = 1, 2, 3, \dots

Stationary states that vanish at both walls

E_{n} = \frac{n ^{2} h ^{2}}{8 m L ^{2}}

Allowed energies: 0.376 eV × n² for an electron in a 1 nm box

Ψ (x, t) = \frac{1}{2} (ψ_{n_{1}} e^{- i E_{n_{1}} t /ℏ} + ψ_{n_{2}} e^{- i E_{n_{2}} t /ℏ})

Equal superposition of two states; |Ψ|² oscillates with period h/|E₂ − E₁|

P (a < x < b) = \frac{b - a}{L} - \frac{sin ( 2 nπ b / L ) - sin ( 2 nπ a / L )}{2 nπ}

Probability of finding the particle between a and b in state n

Assumptions

The walls are infinitely high, so the wavefunction is exactly zero outside the box. A finite well lets the wavefunction leak a little into the walls and lowers every level.
The particle is an electron with mass 9.109 × 10⁻³¹ kg moving in one dimension, with no spin and no interactions.
Time runs much faster on screen: one oscillation or one beat lasts about four seconds, whatever its real length in femtoseconds.
Re Ψ and Im Ψ depend on the arbitrary choice of global phase; only |Ψ|² and differences of phase between states are physical.

Try this in class

Q1Why can there be no state with n = 0? What does that say about the lowest possible energy?
Q2For n = 2 the probability density is zero in the middle of the box. How can the electron get from one half to the other?
Q3A conjugated dye molecule is modelled as a 1.2 nm box. Estimate the wavelength it absorbs for the 3 → 4 transition. Is it visible?
Q4Show that the probability of finding the particle in the left half is exactly 1/2 for every stationary state.
Q5At very large n, the probability density looks almost uniform. How does that connect quantum and classical physics?

Frequently asked questions

Why is the energy quantised?

The wavefunction must be zero at both walls, so only sine waves with a whole number of half wavelengths across the box are allowed. Each allowed wavelength gives a definite momentum and therefore a definite kinetic energy.

Is the particle really moving in a stationary state?

Its probability density does not change with time, and its mean momentum is zero. Each stationary state is an equal mix of waves moving left and right. Motion you can see, such as sloshing, only appears in superpositions of different energies.

Where is this model used?

As a first estimate for electrons in conjugated molecules such as dyes and carotene, in quantum wells in semiconductor lasers and LEDs, and in nanocrystals called quantum dots, whose colour changes with their size.