# Instructor guide: Discrete spectrum and basis-state audit

Course: Quantum mechanics I–II

Suggested time: 55–70 minutes

## Learning target

Learners connect a specified Hamiltonian and boundary conditions to a normalized discrete spectrum without universalizing one model.

## Prepare

- Review eigenvalue equations and normalization.
- State the model Hamiltonian and boundary conditions.
- Prepare one classical-continuum comparison.

## Facilitation moves

- Ask what operator defines energy.
- Separate amplitudes from squared magnitudes.
- Require model-specific language for every spectrum trend.

## Misconception checks

- **Every quantum system has equally spaced levels.** Equal spacing is specific to the harmonic oscillator; other Hamiltonians produce different spectra.
- **A probability distribution means the system secretly occupies every measured outcome classically.** The state supplies amplitudes and outcome statistics; ontological interpretations go beyond that shared operational rule.

## Accessibility and participation

- Provide energy tables alongside level diagrams.
- Read ket and operator notation aloud in plain language.
- Use patterns and state labels in addition to color.

## Evidence of learning

- A normalized probability audit
- A parameter-controlled spectral comparison
- A stated Hamiltonian and boundary condition

## Extension

Add a weak perturbation, predict first-order shifts, and compare them with the laboratory's numerical spectrum.

## Evidence boundary

Assess the learner's reasoning only within the declared model and recorded observations. Do not upgrade a simulation result into a claim about an unmodeled physical system.