Q009 · Shape, prepare and measure

Quantum Well Spectroscopy

How much do the energy levels rise when a quantum well is made narrower? For an infinitely deep well the energies go as one over the width squared, so halving the width makes each level four times higher. Every bound level rises as the well narrows, and the gap between the lowest two widens.

Drag the walls, prepare bound levels and collect an energy spectrum. Design a gap, then estimate it from separate energy records of the two lowest prepared states.

Model lowest energy gap $\text{—}$

Selected-level tail probability $\text{—}$

First, make a prediction

Will a narrower well have a larger energy gap?

Drag either wall. Compare finite and infinite walls, then measure the lowest two states. Can you design a gap of 0.70 within five percent?

Make a prediction, then gather evidence.

Well width

w

4

Exterior potential

V_0

8

Mass ratio

\mu

1

PreparationPotential shape

Changing the selected level keeps labeled records. Changing the potential, basis, detector noise or seed clears them.

State details, numerics & sampling

Selected level, from zero

n

0

Energy readout noise

\sigma_E

0.08

Sine basis size

N_b

64

Samples per acquisition

N

400

Reproducible seed

s

2614

All quantities are dimensionless; the lesson defines physical units. Raising the basis size tests truncation, not detector accuracy. The record limit is 30,000.

The Lab solves the one-dimensional Schrödinger equation in a sine basis. A finite well sits inside a numerical outer box, and box states above the threshold are not bound levels. Energy readouts are ideal projective measurements with Gaussian detector noise added; they are not an optical absorption spectrum.

Recorded evidence & model checks

Simulated independent preparations

Measure a gap from two sets of records

Blue bars are recorded energies. The separate lower ruler marks model levels. The gap estimate subtracts sample means from the two known preparations, with one standard error; it never reads the model gap. The mixture has no hidden level labels in its records.

Measured lowest gap

\text{—}

Model bound states / displayed states

\text{—}

Selected-state edge probability

\text{—}

Basis and boundary checks

\text{—}

Finite and curved wells use a numerical box extending eight units beyond each wall. Weak bound states can shift or be missed. Only below-threshold states qualify, and only the lowest eight are available for preparation. The infinite-well box is physical. A matrix residual alone does not establish continuum accuracy.

Sources: MIT 8.04 · 11 · Feynman III · 8

Start with the essentials

Focus question: Can you design an energy gap and then measure it?
One-sentence intuition: Computed states, model references and measured outcomes play different roles, so keep them apart.

Core mathematical model

Dimensionless units

X=x/\ell,\quad \mu=m/m_*,\quad \mathcal E=Em_*\ell^2/\hbar^2,\quad \tau=\hbar t/(m_*\ell^2)

Pick a reference mass and length. The controls, axes and CSV all use these units. In the double well, the mass ratio is fixed at one.

Sine-basis approximation

u_j(X)=\sqrt{2/L}\sin\!\left[\frac{j\pi(X+L/2)}{L}\right],\quad H_{jk}=\frac{j^2\pi^2}{2\mu L^2}\delta_{jk}+\int u_jVu_k\,dX

Every retained sine mode takes part in the evolution, and the potential integrals are done analytically. The displayed matrix residual checks the eigensolver; it does not measure basis or boundary error.

Finite or curved well

V_{\rm square}(X)=\begin{cases}0,&|X|<w/2\\V_0,&|X|\ge w/2\end{cases},\qquad V_{\rm curved}(X)=\min\!\left[V_0,\frac{4V_0X^2}{w^2}\right]

Energy is measured from the bottom of the well. Only eigenvalues below the outside potential count as bound. The numerical box extends eight units beyond each wall, and states near the threshold need a check of the box size.

Exact levels of an infinite well

\mathcal E_n=\frac{(n+1)^2\pi^2}{2\mu w^2},\quad n=0,1,\ldots

Here the hard walls are physical, and the sine basis is exact for the low levels shown. A wider well lowers the energies, and so does a heavier particle. Index zero is the ground state.

Matching conditions for a finite well

k\tan(kw/2)=\kappa\quad\text{or}\quad-k\cot(kw/2)=\kappa,\qquad k=\sqrt{2\mu\mathcal E},\ \kappa=\sqrt{2\mu(V_0-\mathcal E)}

Requiring the wavefunction and its derivative to be continuous gives an even and an odd branch. These continuum equations give an independent check of the numerical spectrum.

Recorded energies

Y_j=\mathcal E_n+\eta_j,\quad \eta_j\sim\mathcal N(0,\sigma_E^2),\qquad \widehat{\Delta E}=\overline Y_1-\overline Y_0

When you prepare a selected state, its level is known. An equal incoherent mixture samples the four lowest available states, but the records do not say which level each event came from. The readout noise comes from an ideal detector model, not from a natural linewidth.

Uncertainty of the gap estimate

\mathrm{SE}(\widehat{\Delta E})=\sqrt{\frac{s_0^2}{N_0}+\frac{s_1^2}{N_1}}

Each known preparation needs at least two records. This standard error leaves out numerical and calibration errors. The design target is a gap of 0.70, within five percent.

Common difficulties

The numerical box is not the whole line

Typical misconceptionEvery discrete computed energy is a bound state.

Better mental modelA finite well binds only the states below the outside threshold. Check the basis and the box size separately.

More events do not fix the model

Typical misconceptionSampling noise is the only source of error.

Better mental modelA finite basis, artificial boundaries and missing detector physics each need their own check.

Run the experiment

01
Shape the well
Drag the walls and compare finite, infinite and curved wells.
What to observe: The level spacings and the wavefunction tails respond in different ways.
02
Measure the gap
Measure the lowest pair, then collect a mixed-state spectrum.
What to observe: Because you know which state was prepared, you can estimate the gap without hidden labels.
03
Check the basis
Select a high state and compare with a larger basis.
What to observe: Sharp boundaries can need many more modes.