University Quantum Mechanics II · topic UQMII-07.07
The central-field approximation, penetration and quantum defects
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 central field, penetration and quantum defects: predict a finite change
The quantum defect δₗ is the amount by which an alkali level's effective quantum number falls short of its principal quantum number, E = −13.606 eV/(n − δₗ)², one number per l measuring how far that orbital penetrates the core. Use the supplied topic model Y_out = -13.606/(3-d)^2, where Y_out is TRIAL E AT n = 3 (eV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. d is Trial quantum defect δ, dimensionless. Keep l (Orbital quantum number l (s, p, d)) = 0 fixed. For this exercise use only 0.154062 ≤ d ≤ 1.29594 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 d = 0.405275 and 0.565137. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δd across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
TRIAL E AT n = 3 against d
Predictions from the supplied The central field, penetration and quantum defects 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): TRIAL E AT n = 3 / eV
Horizontal axis (x): d: Trial quantum defect δ
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| d: Trial quantum defect δ | TRIAL E AT n = 3 / eV |
|---|---|
| 0.1540625 | -1.6798857654761004 |
| 0.43953125 | -2.075350754116887 |
| 0.725 | -2.628861248641469 |
| 1.01046875 | -3.4373910071239506 |
| 1.2959375 | -4.685537606276788 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of d separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At d = 0.405275, substitute into Y_out = -13.606/(3-d)^2: Y₀ = -2.02091 eV.
- At d = 0.565137, the same fixed-parameter model gives Y₁ = -2.29499 eV.
- Signed change: ΔY_out = Y₁ − Y₀ = -0.27408 eV; the input change is Δd = 0.159862.
- Average sensitivity is ΔY_out/Δd = -1.71448 eV per unit input. Its negative sign means that this response decreases 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 = -2.02091 eV, Y₁ = -2.29499 eV; ΔY_out = -0.27408 eV; average sensitivity = -1.71448 eV 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 central field, penetration and quantum defects: infer an input with uncertainty
The quantum defect δₗ is the amount by which an alkali level's effective quantum number falls short of its principal quantum number, E = −13.606 eV/(n − δₗ)², one number per l measuring how far that orbital penetrates the core. Use the supplied topic model Y_out = -13.606/(3-d)^2, where Y_out is TRIAL E AT n = 3 (eV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. d is Trial quantum defect δ, dimensionless. Keep l (Orbital quantum number l (s, p, d)) = 0 fixed. For this exercise use only 0.154062 ≤ d ≤ 1.29594 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 = -2.9450895 eV, with a bounded reading uncertainty ±0.18978 eV. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown d by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000114188, 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.
TRIAL E AT n = 3 against d
Predictions from the supplied The central field, penetration and quantum defects 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): TRIAL E AT n = 3 / eV
Horizontal axis (x): d: Trial quantum defect δ
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| d: Trial quantum defect δ | TRIAL E AT n = 3 / eV |
|---|---|
| 0.1540625 | -1.6798857654761004 |
| 0.43953125 | -2.075350754116887 |
| 0.725 | -2.628861248641469 |
| 1.01046875 | -3.4373910071239506 |
| 1.2959375 | -4.685537606276788 |
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(d) = 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(0.154062) = -1.67989 eV and Y_out(1.29594) = -4.68554 eV. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives d_line = 0.154062 + (-2.94509 − -1.67989)(1.29594 − 0.154062)/(-4.68554 − -1.67989) = 0.634725. 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 d = 0.850606; substituting back gives Y_out = -2.9450895 eV.
- The interpolation error in the input is |d_line − d| = 0.215881. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = -3.1348695 eV and -2.7553095 eV. Sort the roots even if the response decreases: 0.777816 ≤ d ≤ 0.916682.
- The lower and upper input deviations are 0.0727903 and 0.0660761. 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 d ≈ 0.850606; compatible interval [0.777816, 0.916682]. Endpoint interpolation gives 0.634725.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.