University Quantum Mechanics II · topic UQMII-11.07
Bohr-Sommerfeld quantisation and the quantum bouncer
Connect the ideas, make your working visible, and test the result. Use the hints when you need a starting point; open the full solution after your attempt.
Further practice
Work through a fresh case before tackling the extension.
Bohr-Sommerfeld quantisation and the quantum bouncer: predict a finite change
Bohr-Sommerfeld quantisation fixes bound-state energies by requiring the phase ∫p dx/ħ accumulated between the ends of the classical motion to equal (n − 1)π plus the phase lost at each end: π/4 at a smooth turning point, π/2 at a hard wall. Use the supplied topic model Y_out = (2/3)*(en/0.60179)^1.5/pi, where Y_out is PHASE ∫p dz / πħ (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. en is Trial energy E, measured in peV. Keep ht (Absorber height h) = 20 μm fixed. For this exercise use only 1.4675 peV ≤ en ≤ 4.9325 peV on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. Calculate Y_out at en = 2.2298 peV and 2.7149 peV. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δen across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
PHASE ∫p dz / πħ against en
Predictions from the supplied Bohr-Sommerfeld quantisation and the quantum bouncer model, with all other inputs fixed as stated in the question. 5 calibration points support a trend comparison; they are not measured data or a straight-line law.
Vertical axis (y): PHASE ∫p dz / πħ
Horizontal axis (x): en: Trial energy E / peV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| en: Trial energy E / peV | PHASE ∫p dz / πħ |
|---|---|
| 1.4675 | 0.8080878611976668 |
| 2.33375 | 1.6205891038018245 |
| 3.2 | 2.6020544819524294 |
| 4.06625 | 3.7271985849346034 |
| 4.9325 | 4.979572490273787 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of en separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At en = 2.2298 peV, substitute into Y_out = (2/3)*(en/0.60179)^1.5/pi: Y₀ = 1.51353.
- At en = 2.7149 peV, the same fixed-parameter model gives Y₁ = 2.0334.
- Signed change: ΔY_out = Y₁ − Y₀ = 0.51987; the input change is Δen = 0.4851 peV.
- Average sensitivity is ΔY_out/Δen = 1.07168 1 per peV. Its positive sign means that this response increases over the specified interval.
- This is a secant slope between two settings. It equals a point derivative only for a locally linear relation or in a justified small-interval limit.
Answer Y₀ = 1.51353, Y₁ = 2.0334; ΔY_out = 0.51987; average sensitivity = 1.07168 1 per peV.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
Extra challenging questions
Work through the linked stages and explain what each result means.
Bohr-Sommerfeld quantisation and the quantum bouncer: infer an input with uncertainty
Bohr-Sommerfeld quantisation fixes bound-state energies by requiring the phase ∫p dx/ħ accumulated between the ends of the classical motion to equal (n − 1)π plus the phase lost at each end: π/4 at a smooth turning point, π/2 at a hard wall. Use the supplied topic model Y_out = (2/3)*(en/0.60179)^1.5/pi, where Y_out is PHASE ∫p dz / πħ (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. en is Trial energy E, measured in peV. Keep ht (Absorber height h) = 20 μm fixed. For this exercise use only 1.4675 peV ≤ en ≤ 4.9325 peV on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 3.0805282, with a bounded reading uncertainty ±0.28486. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown en by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0003465 peV, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.
PHASE ∫p dz / πħ against en
Predictions from the supplied Bohr-Sommerfeld quantisation and the quantum bouncer model, with all other inputs fixed as stated in the question. 5 calibration points support a trend comparison; they are not measured data or a straight-line law.
Vertical axis (y): PHASE ∫p dz / πħ
Horizontal axis (x): en: Trial energy E / peV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| en: Trial energy E / peV | PHASE ∫p dz / πħ |
|---|---|
| 1.4675 | 0.8080878611976668 |
| 2.33375 | 1.6205891038018245 |
| 3.2 | 2.6020544819524294 |
| 4.06625 | 3.7271985849346034 |
| 4.9325 | 4.979572490273787 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
The model's endpoint predictions bracket the reading. Solve Y_out(en) = Y_measured and then solve again at Y_measured − u and Y_measured + u; a nonlinear inverse can give unequal input deviations.
Reveal the worked solution
- Endpoint predictions are Y_out(1.4675) = 0.808088 and Y_out(4.9325) = 4.97957. The target lies between them. On the selected branch the response is strictly increasing.
- Endpoint interpolation gives en_line = 1.4675 + (3.08053 − 0.808088)(4.9325 − 1.4675)/(4.97957 − 0.808088) = 3.35508 peV. This estimates the inverse of the endpoint chord, rather than assuming that chord is the physical model.
- Use the actual equation, not the chord, and bracket its root. Evaluating the midpoint and retaining the sign-changing half-interval gives en = 3.58115 peV; substituting back gives Y_out = 3.0805282.
- The interpolation error in the input is |en_line − en| = 0.226071 peV. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 2.7956682 and 3.3653882. Sort the roots even if the response decreases: 3.35683 peV ≤ en ≤ 3.79865 peV.
- The lower and upper input deviations are 0.224319 peV and 0.217499 peV. They differ because the inverse response is nonlinear; a symmetric error bar would only be a local approximation. This interval includes reading uncertainty only, not model error or uncertainty in fixed inputs.
Answer en ≈ 3.58115 peV; compatible interval [3.35683, 3.79865] peV. Endpoint interpolation gives 3.35508 peV.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.