Quantum States & the Wavefunction

A particle in a box has places it is never found, a lowest energy it cannot give up, and a probability that only moves when two levels share the state.

26 min3 checkpoints6 simulation tasks

Put a particle in a box and ask where it is. A ball bouncing between the walls is equally likely to be anywhere. The quantum answer is a curve.

Predict

An electron in a 1 nm box is in the n = 2 state. How often is it found in the middle fifth of the box, from 0.4 to 0.6 nm?

Where it is found

Try it

The middle fifth holds 4.9%. Keep the strip a fifth of the box wide and move it to where the electron is found most often.

GoalA strip a fifth of the box wide that holds over 30%.

ψ(x), the amplitude|ψ(x)|², the probability densityallowed energiesnodes and the walls

Marked region, from

In units of the well width. The probability of finding it here is integrated live.

Marked region, to

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The places it is never found

ψ must be zero at both walls: an infinite wall is one the particle cannot be in. Inside, the $n$ -th state also crosses zero $n - 1$ times. These nodes sit at $x = L/n, 2L/n, \dots$ , and they do not move.

Read the graph

Ground state, left quarter of the box marked. A bouncing ball would be there a quarter of the time. How often is the electron?

ψ(x), the amplitude|ψ(x)|², the probability densityallowed energiesnodes and the walls

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Probability in the marked region

Run the simulation first, then read the value off it.

Try it

Now n = 3. Find a strip a tenth of the box wide where the electron is almost never found: under 2%.

GoalA strip a tenth of the box wide that holds under 2%.

ψ(x), the amplitude|ψ(x)|², the probability densityallowed energiesnodes and the walls

Marked region, from

In units of the well width. The probability of finding it here is integrated live.

Marked region, to

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Why the box has a lowest energy

Only certain wavelengths fit: ψ must be zero at both walls, so a whole number of half-wavelengths fits across the box. With de Broglie's $p = h/\lambda$ , that single constraint gives every allowed energy:

E_n = \frac{n^2h^2}{8mL^2}

Predict

In a 1 nm box an electron's gap from n = 1 to n = 2 is 1.128 eV. Halve the width to 0.5 nm. What happens to that gap?

Try it

In a 1 nm box, the photon that lifts the electron to n = 2 is infrared, 1099 nm. Squeeze the box until that photon is visible light.

GoalThe photon that lifts the electron to n = 2 is visible light, 400 to 700 nm.

ψ(x), the amplitude|ψ(x)|², the probability densityallowed energiesnodes and the walls

Well width L

The box is drawn to scale: squeeze it and watch every energy climb.

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Practice

An electron is trapped in a box nm wide. What is its lowest energy, $E_1$ ?

Ground state E₁ = h²/8mL²

A stationary state is stationary

Predict

Run A: one electron in a 1 nm box, pure n = 2. Run B: n = 1 and n = 2 mixed. In which does the left-half probability change with time?

Try it

Levels 1 and 2 together, left half marked. Run it and watch the probability slosh into the left half.

GoalOver 85% of the probability in the left half.

ψ(x), the amplitude|ψ(x)|², the probability densityallowed energiesnodes and the walls

probability in the marked region

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Quantum states add: $\Psi = a\psi_1 + b\psi_2$ is a state too. Each piece carries its own time factor, and because $E_1 \neq E_2$ they drift out of step. The density gains a third term, and it is the whole of the motion:

|\Psi|^2 = a^2|\psi_1|^2 + b^2|\psi_2|^2 + 2ab\,\psi_1\psi_2\cos\!\left(\frac{(E_2-E_1)t}{\hbar}\right)

Try it

The mixture of 1 and 2 beats once every 3.666 fs. Choose a different second level so the density beats about twice as fast: once every 1 to 2 fs.

GoalA beat that repeats every 1 to 2 fs.

ψ(x), the amplitude|ψ(x)|², the probability densityallowed energiesnodes and the walls

Second level in the mixture

Set it equal to n and there is no mixture, just the one state.

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What comes next

Energy came in levels because a wave had to fit between two walls: no new postulate, only de Broglie's $\lambda = h/p$ and a boundary condition. And "where is it" became a question with a distribution, computed by squaring something that is not itself observable. Real walls are not infinite, so a real wavefunction leaks a little way into them, and a particle can occasionally be found on the far side of a barrier it lacks the energy to climb.

Ideas this lesson took on

Each of these is a common intuition the lesson put to the test. If one still feels right, the section that tests it is worth another look.

“The particle is equally likely to be found anywhere in the box”
“|ψ|² is the shape of the particle, smeared out”
“Halving the width of the box halves the energy”
“A particle in a box is always moving, so its probability must move too”

Finished with this lesson?

Progress is stored on this device only. There is no account and nothing is sent anywhere.

Research behind this lesson

McKagan et al. 2008: Examining student descriptions of atoms on final exams across several versions of a transformed modern-physics curriculum, the authors found that where the curriculum does not explicitly connect the different models of the atom, many students keep a Bohr-like planetary picture instead of the Schrodinger one.
Singh 2001: A test given to 89 students at six universities, with follow-up interviews, at the end of a full year of upper-level quantum mechanics. Stationary states, eigenstates and the time dependence of expectation values were the hardest ideas, the difficulties were common across institutions and teaching approaches, and they traced to an inability to discriminate between related concepts and a tendency to over-generalise.
White & Gunstone 1992: The Predict–Observe–Explain cycle: committing to a prediction before observing makes a conflicting result impossible to ignore, which is what forces a belief to be revised rather than merely supplemented.

Quantum States & the Wavefunction

Introduction

An electron in a 1 nm box is in the n = 2 state. How often is it found in the middle fifth of the box, from 0.4 to 0.6 nm?

Where it is found#

A Particle in a Box simulation

The places it is never found#

A Particle in a Box simulation

A Particle in a Box simulation

Why the box has a lowest energy#

In a 1 nm box an electron's gap from n = 1 to n = 2 is 1.128 eV. Halve the width to 0.5 nm. What happens to that gap?

A Particle in a Box simulation

A stationary state is stationary#

Run A: one electron in a 1 nm box, pure n = 2. Run B: n = 1 and n = 2 mixed. In which does the left-half probability change with time?

A Particle in a Box simulation

A Particle in a Box simulation

What comes next#

Ideas this lesson took on

Research behind this lesson

A Particle in a Box simulation

A Particle in a Box simulation

A Particle in a Box simulation

A Particle in a Box simulation

A Particle in a Box simulation

A Particle in a Box simulation

Where it is found

The places it is never found

Why the box has a lowest energy

A stationary state is stationary

What comes next