University Quantum Mechanics II · topic UQMII-02.05
Truncation: n = n_r + l + 1 and the Bohr energies derived
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.
Truncation: n = nᵣ + l + 1 and the Bohr energies derived: predict a finite change
Truncation is the requirement that the radial series v(ρ) = Σ cⱼρʲ end at a finite power ρ^(nᵣ), which happens only when ρ₀ = 2(nᵣ + l + 1) = 2n, and so only at the hydrogen energies Eₙ = −13.6 eV/n². Use the supplied topic model Y_out = -13.598/nu^2, where Y_out is TRIAL ENERGY −13.598 eV/ν² (eV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. nu is Trial ν = ρ₀/2, so E = −13.598 eV/ν², dimensionless. Keep l (Orbital quantum number l) = 1 fixed. For this exercise use only 0.925 ≤ nu ≤ 4.075 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 nu = 1.618 and 2.059. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δnu across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
TRIAL ENERGY −13.598 eV/ν² against nu
Predictions from the supplied Truncation: n = nᵣ + l + 1 and the Bohr energies derived 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 ENERGY −13.598 eV/ν² / eV
Horizontal axis (x): nu: Trial ν = ρ₀/2, so E = −13.598 eV/ν²
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| nu: Trial ν = ρ₀/2, so E = −13.598 eV/ν² | TRIAL ENERGY −13.598 eV/ν² / eV |
|---|---|
| 0.925 | -15.892476260043827 |
| 1.7125 | -4.6367520912142375 |
| 2.5 | -2.1756800000000003 |
| 3.2874999999999996 | -1.25818213361477 |
| 4.074999999999999 | -0.8188791448680798 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of nu separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At nu = 1.618, substitute into Y_out = -13.598/nu^2: Y₀ = -5.19419 eV.
- At nu = 2.059, the same fixed-parameter model gives Y₁ = -3.20747 eV.
- Signed change: ΔY_out = Y₁ − Y₀ = 1.98672 eV; the input change is Δnu = 0.441.
- Average sensitivity is ΔY_out/Δnu = 4.50504 eV per unit input. 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 = -5.19419 eV, Y₁ = -3.20747 eV; ΔY_out = 1.98672 eV; average sensitivity = 4.50504 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.
Truncation: n = nᵣ + l + 1 and the Bohr energies derived: infer an input with uncertainty
Truncation is the requirement that the radial series v(ρ) = Σ cⱼρʲ end at a finite power ρ^(nᵣ), which happens only when ρ₀ = 2(nᵣ + l + 1) = 2n, and so only at the hydrogen energies Eₙ = −13.6 eV/n². Use the supplied topic model Y_out = -13.598/nu^2, where Y_out is TRIAL ENERGY −13.598 eV/ν² (eV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. nu is Trial ν = ρ₀/2, so E = −13.598 eV/ν², dimensionless. Keep l (Orbital quantum number l) = 1 fixed. For this exercise use only 0.925 ≤ nu ≤ 4.075 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 = -1.6782346 eV, with a bounded reading uncertainty ±0.1289 eV. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown nu by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000315, 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 ENERGY −13.598 eV/ν² against nu
Predictions from the supplied Truncation: n = nᵣ + l + 1 and the Bohr energies derived 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 ENERGY −13.598 eV/ν² / eV
Horizontal axis (x): nu: Trial ν = ρ₀/2, so E = −13.598 eV/ν²
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| nu: Trial ν = ρ₀/2, so E = −13.598 eV/ν² | TRIAL ENERGY −13.598 eV/ν² / eV |
|---|---|
| 0.925 | -15.892476260043827 |
| 1.7125 | -4.6367520912142375 |
| 2.5 | -2.1756800000000003 |
| 3.2874999999999996 | -1.25818213361477 |
| 4.074999999999999 | -0.8188791448680798 |
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(nu) = 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(0.925) = -15.8925 eV and Y_out(4.075) = -0.818879 eV. The target lies between them. On the selected branch the response is strictly increasing.
- Endpoint interpolation gives nu_line = 0.925 + (-1.67823 − -15.8925)(4.075 − 0.925)/(-0.818879 − -15.8925) = 3.89542. 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 nu = 2.8465; substituting back gives Y_out = -1.6782346 eV.
- The interpolation error in the input is |nu_line − nu| = 1.04892. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = -1.8071346 eV and -1.5493346 eV. Sort the roots even if the response decreases: 2.7431 ≤ nu ≤ 2.96254.
- The lower and upper input deviations are 0.103396 and 0.116045. 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 nu ≈ 2.8465; compatible interval [2.7431, 2.96254]. Endpoint interpolation gives 3.89542.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.