Level 3 · Undergraduate core teaching kit · First- and second-year university
Quantum mechanics I–II
Use the learner record during the live investigation, then use the instructor guide to facilitate comparison, address misconceptions, and assess evidence-bounded reasoning.
Learner lab record
Discrete spectrum and basis-state audit
How do Hamiltonian parameters and boundary conditions determine allowed energies and measurement probabilities?
Setup
Use the quantum-spectrum laboratory. Keep the model family fixed while changing one Hamiltonian parameter, then compare normalized state weights and adjacent energy gaps.
Predict first
- 1. Predict whether every energy shifts by the same factor under the chosen scale change.
- 2. Predict what normalized probabilities must sum to.
| Variable | Role | Unit |
|---|---|---|
| Hamiltonian scale or confinement parameter | independent | declared model unit |
| Quantum number | state index | integer |
| Allowed energies and gaps | dependent | eV or model energy |
| State probabilities | dependent | % |
Observation columns
Analyze
- 1. Which boundary condition produces discreteness?
- 2. How do amplitude and probability differ?
- 3. Which trend belongs to this Hamiltonian rather than all quantum systems?
- 4. What result would reveal failed normalization?
Conclusion frame
Changing ___ from ___ to ___ changed the n=___ energy from ___ to ___ while total probability remained ___; this follows from ___.
Instructor guide · 55–70 minutes
Teach the investigation, not the interface
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.
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
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.
Extension
Add a weak perturbation, predict first-order shifts, and compare them with the laboratory's numerical spectrum.