Q009 · Shape / prepare / measure

Quantum Well Spectroscopy

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

One-dimensional sine-basis Schrödinger model. A finite well uses a numerical outer box; above-threshold box states are not bound levels. Energy readouts are ideal projective measurements with added Gaussian detector noise, 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 and measure a gap?
One-sentence intuition: Numerical states, model references and measured outcomes have different roles.

Core mathematical model

Units

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

Choose a reference mass and length. These units apply to controls, axes and CSV. The double-well mass ratio is fixed at one.

Spatial 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

All retained sine modes evolve. Potential integrals are analytic; the displayed matrix residual measures the eigensolver, not 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]

The energy origin is the well bottom. Only eigenvalues below the exterior potential qualify as bound. The outer numerical box extends eight units beyond each wall; near-threshold states require a domain check.

Infinite-well benchmark

\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 displayed low levels. Widening the well lowers energies; increasing mass also lowers them. Index zero denotes the ground state.

Independent finite-well matching

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)}

Continuity of the wavefunction and its derivative gives even and odd branches. These continuum equations independently test the numerical spectrum.

Recorded energy

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

The prepared level is known in a selected-state run. An equal incoherent mixture samples the lowest four available states but records no hidden level identity. Readout noise is an explicit ideal detector model, not 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}}

At least two records from each known preparation are required. This standard error excludes numerical and calibration errors. The design goal is a gap of 0.70 within five percent.

Common difficulties

A numerical box is not the whole line

Typical misconceptionEvery discrete computed energy is a bound state.

Better mental modelA finite well only binds states below the exterior threshold; compare basis and domain separately.

More events do not fix the model

Typical misconceptionSampling noise is the only source of error.

Better mental modelFinite basis, artificial boundaries and omitted detector physics need separate checks.

Run the experiment

01
Shape
Drag the walls and compare finite, infinite and curved wells.
What to observe: Level spacings and tails respond differently.
02
Measure
Measure the lowest pair and collect a mixed-state spectrum.
What to observe: Known preparations permit a gap estimate without hidden labels.
03
Refine
Select a high state and compare a larger basis.
What to observe: Sharp boundaries can need substantially more modes.