Quantum Particle in a 1D Box

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Guided Inquiry

1 · The box

Here’s our particle, trapped between two hard walls. The top box draws its wave function ψ(x); below it you’ll see |ψ(x)|², the probability density for where the particle can be found — each dot beneath marks one position measurement. The quantum number n counts the half-waves that fit between the walls.

2 · The classical particle

Let’s start with classical physics. Picture a ball bouncing elastically between the walls at constant speed. Over many snapshots, where do you think you’d find it?

3 · Raise n to 3

Now let’s set n = 3 — three half-waves will fit instead of one. After many measurements, how many separate humps do you think the histogram will show?

4 · Count the forbidden points

See those empty points? They’re nodes — places the particle is never found. n is yours to change, so feel free to try a few. For quantum number n, how many nodes do you think lie inside the box?

5 · The price of confinement

Now let’s think about energy. Going from n = 1 to n = 3 squeezed three half-waves into the same box. Compared with E₁, what do you think the energy at n = 3 is?

6 · Can it rest?

One last question. Every state you’ve seen has n ≥ 1. Could a confined particle ever have zero energy — just sitting still somewhere in the box? What do you think?

Listen

Parameters

Energy Level n 1

Box Length L 1.00

Override ω with animation speed

Animation ω (0–10ω₀) 1.00ω₀

Physical Mode: ω = n² × ω₀ = 1 × ω₀

Display

Measurement rate 10 particles/s

Measurements: 0

State properties

Time (phase): 0.00

Energy E: 1.00 E₁

Oscillation ω: 1.00 ω₀

Position ⟨x⟩: 0.50 L

Uncertainty Δx: 0.18 L

⟨p⟩: 0.00 ℏ/L

Eigenstate ψ₁(x)

Wave: √(2/L) sin(πx/L)

λ: 2L/1 = 2.00

The Physics

The infinite square well · what this sim shows, the physics behind it, and every control

At a glance

A particle trapped between two impenetrable walls is described by a standing wave, and only some waves are allowed. This sim shows those allowed states, the probability density for the particle's position, and the position results a measurement would give.

What you'll learn

How the number of nodes and the energy of a state depend on n.
What |ψ|² tells you, and how a histogram of position results builds it up.
How the picture changes as n gets large.
Which parts of the display change with time, and which do not.

The sim shows the wave above, the probability density below, and each measured position as a green dot. Drawn for n = 1.

What's on screen

Wave ψ (top plot, blue): the real part of the wave function. Red dots mark where it crosses zero inside the box.
Probability density |ψ|² (bottom plot, pink): probability per unit length, so the area under it is 1.
Green dot and fading trail: simulated position results, one at a time. Green bars (when the histogram is on): the results counted into 50 bins.
Energy levels (left of the top plot): n = 1 to 10, with the current n highlighted.

See itStart from the opening state, n = 1, with the histogram on. Watch for: a green dot jumping to a new position inside the box, and the Measurements count rising.

The controls

Press ⌖ on a row to see where that control sits in the sim.

Control	What it changes	Range
Energy Level n	Which state is shown. Changing n clears the histogram and sets the measurement rate back to 10 /s.	1–10 opens at 1
Box Length L	Readout only. Lengths in the sim are in units of L.	1.00 fixed
Override ω	Off: the wave oscillates at the frequency its energy sets. On: you set the frequency with the slider below. This changes only the viewing speed.	on / off opens off
Animation ω	The oscillation frequency while Override is on. The slider is disabled when Override is off.	0–10 ω₀ opens at 1 ω₀
Measurement rate	How many simulated position results are drawn each second.	1–25 /s opens at 10 /s
Show / Hide Histogram	Shows or hides the green bars. Results are counted either way.	on / off opens off
Clear Histogram	Empties the bins and the Measurements count.	button
State properties	Time, energy E in units of E₁, frequency ω, ⟨x⟩ in L, Δx in L and ⟨p⟩ in ħ/L. The sim prints ⟨x⟩ = 0.50 L and ⟨p⟩ = 0 as fixed values, exact for every n.	panel
Eigenstate	The wave function and wavelength of the current state.	panel
Pause / Start	Stops or resumes the animation. Pausing also stops the measurements.	top bar
Reset	Returns the sim to its opening state: n = 1, Override off, rate 10 /s, histogram off and empty.	top bar

Fixed: ħ = m = L = 1 in the sim's units. The walls are infinitely high.

Only some waves fit between the walls

The walls are infinitely high, so the particle cannot be outside the box and ψ must be exactly zero at x = 0 and x = L. Inside, V = 0, and the condition ψ(0) = 0 leaves a sine wave.

Schrödinger's equation inside the box, and the solutions the walls allow.

Only sines that are zero at both walls survive: those that fit a whole number n of half-wavelengths. n counts the humps.
A state with quantum number n has n − 1 nodes: points inside the box where ψ = 0. These are the red dots in the sim.
n = 0 would give ψ = 0 everywhere: no particle. The list starts at n = 1.

See itSet n = 3. Watch for: two red node dots on the wave, three humps in the pink curve, and λ = 2L/3 in the card.

The energy ladder: E_n = n²E₁

Fitting n half-waves into the box fixes the wavelength. The de Broglie relation, p = h/λ, turns that into a momentum.

Inside the box V = 0, so the energy is all kinetic:

The levels are not evenly spaced: the gap above level n is (2n + 1)E₁, so 3E₁, 5E₁, 7E₁, and so on.
The lowest energy is E₁ = π²ħ²/2mL², not zero. Confinement forces the particle to carry kinetic energy.

This slider moves the sim too.

See itSet n = 4. Watch for: the highlighted level four steps up the ladder, with the Energy readout at 16 E₁.

Where the particle is found

|ψ|² is a probability density: probability per unit length. The probability of a result between a and b is the area under it, and the total area is 1.

Each green dot is one simulated position result, drawn at random from |ψ|². The histogram counts the dots into 50 bins, scaled to the same height as the pink curve, so the bars approach the curve as results accumulate.
The nodes are places where |ψ|² = 0, so no result lands there. At n = 2 that includes the centre, yet the average position is ⟨x⟩ = L/2: an average need not be a place where results occur.
⟨p⟩ = 0 for every n. A standing wave is an equal mix of right- and left-moving waves, but the kinetic energy is not zero.

See itSet n = 2 and measure at 25 /s with the histogram on. Watch for: green bars growing in two humps, with almost nothing in the middle.

Large n looks classical

A classical particle bouncing between the walls at constant speed spends equal time everywhere: its density is flat, 1/L. At n = 1 the quantum density is far from flat: it peaks at the centre and is zero at the walls.

As n grows, the humps get narrower. Their spacing is L/n, so averaged over a few humps the density approaches 1/L.
Δx follows: 0.18 L at n = 1, 0.27 L at n = 2, 0.288 L at n = 10, approaching the flat-density value L/√12 = 0.289 L.
The nodes never go away. They become too close together to resolve. A 1 g bead moving at 1 mm/s in a 10 cm box has n ≈ 3 × 10²⁶.

See itSet n = 10 and measure at 25 /s with the histogram on. Watch for: ten humps in the pink curve, and green dots spread across the whole box instead of crowding the middle.

What moves on screen, and what does not

The top plot draws the real part of Ψ, ψ_n(x) cos(ω_nt). The shape is fixed; its height rises, falls and changes sign. The nodes stay where they are.
|Ψ|² has no t in it, so the pink curve does not move, and neither does the distribution the green dots follow. A state like this is called stationary.
ω_n = E_n/ħ, so ω_n ∝ n². The sim writes it as ω = n²ω₀, with ω₀ = 1 as its on-screen time unit.
Override ω sets the frequency by hand. That changes only how fast you watch the wave oscillate. The energy and |ψ|² do not change.

See itSet n = 2 with Override off. Watch for: the blue wave rising and falling in place while the pink curve stays still, and ω reading 4 ω₀.

Where it matters

Dye molecules. Treat the electrons along a 1 nm conjugated chain as free electrons in a box. Two per level, six electrons fill n = 1–3, so the first jump is 3 → 4: 7E₁ ≈ 2.6 eV, about 470 nm. This is a first-order estimate.
Quantum dots. E₁ ∝ 1/L²: a smaller dot has a larger gap and emits bluer light. This 1D box gives the trend only; real dots confine in three dimensions, with an effective mass.
Confinement energy. Squeezing a particle raises its lowest energy. An electron in a 0.5 nm box has E₁ ≈ 1.5 eV.

Go deeper: a photon carries a jump, not E₁

Absorption or emission moves the electron between levels, so the photon energy is a difference such as E₂ − E₁ = 3E₁.
In a 0.5 nm box that is about 4.5 eV (275 nm); in a 1 nm box, about 1.1 eV (1100 nm).

Summary

In one line. The walls force ψ to zero at both ends, so only n half-waves fit. That gives E_n = n²E₁, a density |ψ_n|² with n − 1 nodes, and a flat classical average at large n.

Key points

ψ_n(x) = √(2/L) sin(nπx/L), with n = 1, 2, 3, …
E_n = n²E₁, and E₁ > 0: the particle is never at rest.
|ψ|² is a probability density. The dots and the histogram sample it.
In a stationary state |Ψ|² does not depend on time; only the phase of Ψ does.

Common mix-ups

"The lowest state is n = 0, where the particle sits still." n = 0 gives ψ = 0 everywhere. The lowest state is n = 1, with energy E₁.
"Triple n, triple the energy." Momentum triples, and energy goes as p², so it rises nine times.
"At a node the particle moves through quickly." At a node |ψ|² = 0: a position result is never found there. The sim does not draw a trajectory.

Things to try on your own

Set n = 4. Count the red nodes and read the energy, then compare with n − 1 and n²E₁.
Set n = 2 with the histogram on and let it run past a few hundred results. Compare the two middle bins with the tallest ones.
Set n = 3, turn Override on and lower ω. Check that the energy readout does not change.