Level 3 · Undergraduate core teaching kit · First- and second-year university
Statistical mechanics and ensembles
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
Two-level ensemble population and heat-capacity map
How do temperature, energy gap, and degeneracy turn microscopic possibilities into macroscopic averages?
Setup
Use the canonical ensemble laboratory. Hold degeneracies fixed while sweeping temperature, then change the excited-state multiplicity at one temperature.
Predict first
- 1. Predict the excited population as temperature approaches zero.
- 2. Predict how added excited-state degeneracy competes with a fixed energy penalty.
| Variable | Role | Unit |
|---|---|---|
| Temperature | independent | K or declared scale |
| Energy gap | independent | eV or model energy |
| State degeneracies | controlled, then independent | count |
| Population, mean energy, entropy, heat capacity | dependent | %, energy, entropy, energy/K |
Observation columns
Analyze
- 1. Where is population competition strongest?
- 2. Why does high degeneracy matter even at fixed energy?
- 3. Which run shows a heat-capacity peak?
- 4. What physical assumption justifies the canonical ensemble?
Conclusion frame
At gap ___, raising temperature from ___ to ___ changed excited population from ___ to ___; degeneracy altered the balance because ___.
Instructor guide · 50–65 minutes
Teach the investigation, not the interface
Learning target: Learners derive qualitative ensemble behavior from Boltzmann weights and distinguish microstate multiplicity from energy alone.
Prepare
- • Review exponential weights and normalization.
- • Define degeneracy as a count of distinct states.
- • Sketch the low- and high-temperature limits.
Facilitation moves
- • Ask for limiting behavior before calculation.
- • Keep probability per state separate from total level population.
- • Connect the heat-capacity peak to changing occupation.
Accessibility and participation
- • Pair exponential notation with population percentages.
- • Use a token model for degeneracy.
- • Offer a table-first route before interpreting curves.
Evidence of learning
- • Correct low/high-temperature limits
- • A degeneracy-controlled comparison
- • A population-based heat-capacity explanation
Misconception checks
The lowest-energy level is always occupied with 100% probability.
At finite temperature, excited states receive Boltzmann weight; degeneracy can amplify their total population.
Entropy is merely disorder or confusion.
Here it is a quantitative property of the normalized ensemble distribution and accessible multiplicity.
Extension
Add a third level and determine whether one effective two-level approximation remains accurate over a chosen temperature range.