University Quantum Mechanics II · topic UQMII-06.05
Degenerate levels and the good states
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.
Degenerate levels and the good states: predict a finite change
The good states of a degenerate level are the combinations of its states that diagonalise the perturbation inside the level; the first-order shifts are the eigenvalues of that small matrix, Wᵢⱼ = ⟨i|H′|j⟩. Use the supplied topic model Y_out = 0.5*(1 - d/(2*sqrt(d*d/4 + w*w))), where Y_out is LOWER STATE: P(m = +1) (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x. d is Splitting δ from an axial field, measured in meV. Keep w (Coupling |W| = V₀/2) = 0.3 meV; phi0 (Orientation φ₀ of the hills) = 30 ° fixed. For this exercise use only 0.2125 meV ≤ d ≤ 1.7875 meV 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 = 0.559 meV and 0.7795 meV. 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.
LOWER STATE: P(m = +1) against d
Predictions from the supplied Degenerate levels and the good states 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): LOWER STATE: P(m = +1)
Horizontal axis (x): d: Splitting δ from an axial field / meV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| d: Splitting δ from an axial field / meV | LOWER STATE: P(m = +1) |
|---|---|
| 0.2125 | 0.33307642887770283 |
| 0.60625 | 0.14461949685805386 |
| 1 | 0.07125353714372784 |
| 1.3937499999999998 | 0.04074765624656507 |
| 1.7874999999999999 | 0.025990868264848865 |
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 = 0.559 meV, substitute into Y_out = 0.5*(1 - d/(2*sqrt(d*d/4 + w*w))): Y₀ = 0.159167.
- At d = 0.7795 meV, the same fixed-parameter model gives Y₁ = 0.103783.
- Signed change: ΔY_out = Y₁ − Y₀ = -0.0553844; the input change is Δd = 0.2205 meV.
- Average sensitivity is ΔY_out/Δd = -0.251176 1 per meV. 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 = 0.159167, Y₁ = 0.103783; ΔY_out = -0.0553844; average sensitivity = -0.251176 1 per meV.
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.
Degenerate levels and the good states: infer an input with uncertainty
The good states of a degenerate level are the combinations of its states that diagonalise the perturbation inside the level; the first-order shifts are the eigenvalues of that small matrix, Wᵢⱼ = ⟨i|H′|j⟩. Use the supplied topic model Y_out = 0.5*(1 - d/(2*sqrt(d*d/4 + w*w))), where Y_out is LOWER STATE: P(m = +1) (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x. d is Splitting δ from an axial field, measured in meV. Keep w (Coupling |W| = V₀/2) = 0.3 meV; phi0 (Orientation φ₀ of the hills) = 30 ° fixed. For this exercise use only 0.2125 meV ≤ d ≤ 1.7875 meV 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 = 0.054834779, with a bounded reading uncertainty ±0.0043266. 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 0.0001575 meV, 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.
LOWER STATE: P(m = +1) against d
Predictions from the supplied Degenerate levels and the good states 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): LOWER STATE: P(m = +1)
Horizontal axis (x): d: Splitting δ from an axial field / meV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| d: Splitting δ from an axial field / meV | LOWER STATE: P(m = +1) |
|---|---|
| 0.2125 | 0.33307642887770283 |
| 0.60625 | 0.14461949685805386 |
| 1 | 0.07125353714372784 |
| 1.3937499999999998 | 0.04074765624656507 |
| 1.7874999999999999 | 0.025990868264848865 |
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(0.2125) = 0.333076 and Y_out(1.7875) = 0.0259909. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives d_line = 0.2125 + (0.0548348 − 0.333076)(1.7875 − 0.2125)/(0.0259909 − 0.333076) = 1.63956 meV. 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 = 1.17325 meV; substituting back gives Y_out = 0.054834779.
- The interpolation error in the input is |d_line − d| = 0.466314 meV. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 0.050508179 and 0.059161379. Sort the roots even if the response decreases: 1.12113 meV ≤ d ≤ 1.23153 meV.
- The lower and upper input deviations are 0.0521246 meV and 0.0582845 meV. 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 ≈ 1.17325 meV; compatible interval [1.12113, 1.23153] meV. Endpoint interpolation gives 1.63956 meV.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.