University Quantum Mechanics II · topic UQMII-14.09
The Born-Oppenheimer approximation for molecules
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.
The Born-Oppenheimer approximation for molecules: predict a finite change
The Born-Oppenheimer approximation writes a molecular state as one electronic eigenstate φₙ(r; R), found with the nuclei clamped at R, times a nuclear wavefunction χ(R) that moves in the electronic energy Eₙ(R) as its potential. Use the supplied topic model Y_out = 3401.4/mu, where Y_out is ROTATIONAL CONSTANT B (meV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. mu is Nuclear reduced mass μ/mₑ, dimensionless. Keep no other adjustable input fixed. For this exercise use only 100 ≤ mu ≤ 180, at the specified test settings separated by 1. 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 mu = 118 and 119. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δmu across this interval. State whether the response increases or decreases; explain why a finite difference between these settings does not require interpolation between them.
ROTATIONAL CONSTANT B against mu
Predictions from the supplied The Born-Oppenheimer approximation for molecules 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): ROTATIONAL CONSTANT B / meV
Horizontal axis (x): mu: Nuclear reduced mass μ/mₑ
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| mu: Nuclear reduced mass μ/mₑ | ROTATIONAL CONSTANT B / meV |
|---|---|
| 100 | 34.014 |
| 120 | 28.345000000000002 |
| 140 | 24.295714285714286 |
| 160 | 21.25875 |
| 180 | 18.89666666666667 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of mu separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At mu = 118, substitute into Y_out = 3401.4/mu: Y₀ = 28.8254 meV.
- At mu = 119, the same fixed-parameter model gives Y₁ = 28.5832 meV.
- Signed change: ΔY_out = Y₁ − Y₀ = -0.24223 meV; the input change is Δmu = 1.
- Average sensitivity is ΔY_out/Δmu = -0.24223 meV per unit input. Its negative sign means that this response decreases over the specified interval.
- This finite difference compares two specified test settings without interpolating between them. A point derivative is not required. Any quantum numbers or particle counts must also obey their physical admissibility conditions.
Answer Y₀ = 28.8254 meV, Y₁ = 28.5832 meV; ΔY_out = -0.24223 meV; average sensitivity = -0.24223 meV per unit input.
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.
The Born-Oppenheimer approximation for molecules: infer an input with uncertainty
The Born-Oppenheimer approximation writes a molecular state as one electronic eigenstate φₙ(r; R), found with the nuclei clamped at R, times a nuclear wavefunction χ(R) that moves in the electronic energy Eₙ(R) as its potential. Use the supplied topic model Y_out = 3401.4/mu, where Y_out is ROTATIONAL CONSTANT B (meV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. mu is Nuclear reduced mass μ/mₑ, dimensionless. Keep no other adjustable input fixed. For this exercise use only 100 ≤ mu ≤ 180, at the specified test settings separated by 1. 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 = 22.828188 meV, with a bounded reading uncertainty ±1.2094 meV. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. Determine every allowed value of mu consistent with that bound. Compare with a straight-line interpolation between the two endpoint predictions, and explain why a fractional setting may be inadmissible. Do not assume that the supplied calibration points prove monotonicity outside this branch.
ROTATIONAL CONSTANT B against mu
Predictions from the supplied The Born-Oppenheimer approximation for molecules 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): ROTATIONAL CONSTANT B / meV
Horizontal axis (x): mu: Nuclear reduced mass μ/mₑ
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| mu: Nuclear reduced mass μ/mₑ | ROTATIONAL CONSTANT B / meV |
|---|---|
| 100 | 34.014 |
| 120 | 28.345000000000002 |
| 140 | 24.295714285714286 |
| 160 | 21.25875 |
| 180 | 18.89666666666667 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Enumerate the permitted settings in the stated range, evaluate Y_out at each, and retain every setting satisfying |Y_out − Y_measured| ≤ u. The plotted predictions are supplied calibration evidence.
Reveal the worked solution
- Endpoint predictions are Y_out(100) = 34.014 meV and Y_out(180) = 18.8967 meV. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives mu_line = 100 + (22.8282 − 34.014)(180 − 100)/(18.8967 − 34.014) = 159.195. This estimates the inverse of the endpoint chord, rather than assuming that chord is the physical model.
- Evaluate the model at each permitted setting. The central reading corresponds to mu = 149; only the allowed settings 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157 lie inside the stated reading bound.
- The interpolation error in the input is |mu_line − mu| = 10.1946.
- The compatible allowed values span 142 to 157. Values between specified test settings are outside this exercise's input domain even if an interpolation passes through them.
- The quoted output bound restricts which specified settings remain compatible. It does not change physical restrictions on quantum numbers or counts, and the chosen grid alone does not establish physical quantisation.
Answer Compatible mu: 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157. Endpoint interpolation predicts 159.195; check admissibility against the discrete settings.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.