University Physics V · topic UPV-11.01
The central-field Hamiltonian and a complete set of observables
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 Hamiltonian and its commuting set: predict a finite change
A central-field Hamiltonian is H = p²/2μ + V(r) with V depending only on the distance |r|; because it commutes with L² and L_z, its eigenvectors carry the labels |n l m⟩ and the three-dimensional eigenproblem splits into one half-line radial problem for each l. Use the supplied topic model Y_out = -13.606/max(a,0.05)^2, where Y_out is SPURIOUS E −ħ²/2μa² (eV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; max(a,b) selects the larger value. a is Decay length a of e^(−r/a), measured in a₀. Keep k (Wavenumber k of sin(kr)) = 1.5 1/a₀ fixed. For this exercise use only 0.586875 a₀ ≤ a ≤ 2.71312 a₀ 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 a = 1.05465 a₀ and 1.35232 a₀. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δa across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
SPURIOUS E −ħ²/2μa² against a
Predictions from the supplied The central-field Hamiltonian and its commuting set 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): SPURIOUS E −ħ²/2μa² / eV
Horizontal axis (x): a: Decay length a of e^(−r/a) / a₀
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| a: Decay length a of e^(−r/a) / a₀ | SPURIOUS E −ħ²/2μa² / eV |
|---|---|
| 0.586875 | -39.50383397922925 |
| 1.1184375 | -10.876947353867415 |
| 1.65 | -4.99761248852158 |
| 2.1815625 | -2.8588748706482474 |
| 2.713125 | -1.8483782957810917 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of a separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At a = 1.05465 a₀, substitute into Y_out = -13.606/max(a,0.05)^2: Y₀ = -12.2325 eV.
- At a = 1.35232 a₀, the same fixed-parameter model gives Y₁ = -7.43992 eV.
- Signed change: ΔY_out = Y₁ − Y₀ = 4.79254 eV; the input change is Δa = 0.297675 a₀.
- Average sensitivity is ΔY_out/Δa = 16.0999 eV per a₀. 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 = -12.2325 eV, Y₁ = -7.43992 eV; ΔY_out = 4.79254 eV; average sensitivity = 16.0999 eV per a₀.
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 Hamiltonian and its commuting set: infer an input with uncertainty
A central-field Hamiltonian is H = p²/2μ + V(r) with V depending only on the distance |r|; because it commutes with L² and L_z, its eigenvectors carry the labels |n l m⟩ and the three-dimensional eigenproblem splits into one half-line radial problem for each l. Use the supplied topic model Y_out = -13.606/max(a,0.05)^2, where Y_out is SPURIOUS E −ħ²/2μa² (eV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; max(a,b) selects the larger value. a is Decay length a of e^(−r/a), measured in a₀. Keep k (Wavenumber k of sin(kr)) = 1.5 1/a₀ fixed. For this exercise use only 0.586875 a₀ ≤ a ≤ 2.71312 a₀ 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 = -3.8337213 eV, with a bounded reading uncertainty ±0.2978 eV. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown a by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000212625 a₀, 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.
SPURIOUS E −ħ²/2μa² against a
Predictions from the supplied The central-field Hamiltonian and its commuting set 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): SPURIOUS E −ħ²/2μa² / eV
Horizontal axis (x): a: Decay length a of e^(−r/a) / a₀
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| a: Decay length a of e^(−r/a) / a₀ | SPURIOUS E −ħ²/2μa² / eV |
|---|---|
| 0.586875 | -39.50383397922925 |
| 1.1184375 | -10.876947353867415 |
| 1.65 | -4.99761248852158 |
| 2.1815625 | -2.8588748706482474 |
| 2.713125 | -1.8483782957810917 |
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(a) = 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(0.586875) = -39.5038 eV and Y_out(2.71312) = -1.84838 eV. The target lies between them. On the selected branch the response is strictly increasing.
- Endpoint interpolation gives a_line = 0.586875 + (-3.83372 − -39.5038)(2.71312 − 0.586875)/(-1.84838 − -39.5038) = 2.60102 a₀. 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 a = 1.88389 a₀; substituting back gives Y_out = -3.8337213 eV.
- The interpolation error in the input is |a_line − a| = 0.717133 a₀. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = -4.1315213 eV and -3.5359213 eV. Sort the roots even if the response decreases: 1.81472 a₀ ≤ a ≤ 1.96162 a₀.
- The lower and upper input deviations are 0.069165 a₀ and 0.0777282 a₀. 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 a ≈ 1.88389 a₀; compatible interval [1.81472, 1.96162] a₀. Endpoint interpolation gives 2.60102 a₀.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.