# Learner lab record: Discrete spectrum and basis-state audit

Course: Quantum mechanics I–II

Name: ____________________  Date: ____________________  Group: ____________________

## Investigation question

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.

## Variables

| 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 | % |

## Predict before changing controls

1. Predict whether every energy shifts by the same factor under the chosen scale change.

   ________________________________________________________________

2. Predict what normalized probabilities must sum to.

   ________________________________________________________________

## Observation table

| model parameter | state n | energy | adjacent gap | amplitude | probability |
| --- | --- | --- | --- | --- | --- |
|   |   |   |   |   |   |
|   |   |   |   |   |   |
|   |   |   |   |   |   |
|   |   |   |   |   |   |
|   |   |   |   |   |   |
|   |   |   |   |   |   |

## 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?

   ________________________________________________________________

## Evidence-bounded conclusion

Changing ___ from ___ to ___ changed the n=___ energy from ___ to ___ while total probability remained ___; this follows from ___.

________________________________________________________________