University Advanced Quantum Mechanics · topic UAQM-04.01
Dirac's requirements and the algebra of alpha and beta
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.
Dirac's requirements and the algebra of α and β: predict a finite change
Dirac's requirements are that the Hamiltonian H = cα⋅p + βmc² be Hermitian and square to p²c² + m²c⁴; they hold exactly when α₁, α₂, α₃ and β are Hermitian, square to 1 and anticommute in pairs, which no ordinary numbers can do. Use the supplied topic model Y_out = 1.022*m*sqrt(max(1-s^2,0)), where Y_out is SOLID GAP AT ITS MINIMUM (MeV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x; max(a,b) selects the larger value. s is Overlap s, where αβ + βα = 2s, dimensionless. Keep m (Mass m in electron masses) = 1 mₑ; p0 (Probe momentum p₀) = 1 MeV/c fixed. For this exercise use only 0.10625 ≤ s ≤ 0.89375 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 s = 0.2795 and 0.38975. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δs across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
SOLID GAP AT ITS MINIMUM against s
Predictions from the supplied Dirac's requirements and the algebra of α and β 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): SOLID GAP AT ITS MINIMUM / MeV
Horizontal axis (x): s: Overlap s, where αβ + βα = 2s
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| s: Overlap s, where αβ + βα = 2s | SOLID GAP AT ITS MINIMUM / MeV |
|---|---|
| 0.10625 | 1.0162149156766742 |
| 0.303125 | 0.9739157214363764 |
| 0.5 | 0.8850779626676962 |
| 0.6968749999999999 | 0.7329708435271745 |
| 0.8937499999999999 | 0.45843386092625193 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of s separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At s = 0.2795, substitute into Y_out = 1.022*m*sqrt(max(1-s^2,0)): Y₀ = 0.981269 MeV.
- At s = 0.38975, the same fixed-parameter model gives Y₁ = 0.941181 MeV.
- Signed change: ΔY_out = Y₁ − Y₀ = -0.0400879 MeV; the input change is Δs = 0.11025.
- Average sensitivity is ΔY_out/Δs = -0.363609 MeV 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 = 0.981269 MeV, Y₁ = 0.941181 MeV; ΔY_out = -0.0400879 MeV; average sensitivity = -0.363609 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.
Dirac's requirements and the algebra of α and β: infer an input with uncertainty
Dirac's requirements are that the Hamiltonian H = cα⋅p + βmc² be Hermitian and square to p²c² + m²c⁴; they hold exactly when α₁, α₂, α₃ and β are Hermitian, square to 1 and anticommute in pairs, which no ordinary numbers can do. Use the supplied topic model Y_out = 1.022*m*sqrt(max(1-s^2,0)), where Y_out is SOLID GAP AT ITS MINIMUM (MeV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x; max(a,b) selects the larger value. s is Overlap s, where αβ + βα = 2s, dimensionless. Keep m (Mass m in electron masses) = 1 mₑ; p0 (Probe momentum p₀) = 1 MeV/c fixed. For this exercise use only 0.10625 ≤ s ≤ 0.89375 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 = 0.82767559 MeV, with a bounded reading uncertainty ±0.028281 MeV. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown s by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00007875, 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.
SOLID GAP AT ITS MINIMUM against s
Predictions from the supplied Dirac's requirements and the algebra of α and β 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): SOLID GAP AT ITS MINIMUM / MeV
Horizontal axis (x): s: Overlap s, where αβ + βα = 2s
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| s: Overlap s, where αβ + βα = 2s | SOLID GAP AT ITS MINIMUM / MeV |
|---|---|
| 0.10625 | 1.0162149156766742 |
| 0.303125 | 0.9739157214363764 |
| 0.5 | 0.8850779626676962 |
| 0.6968749999999999 | 0.7329708435271745 |
| 0.8937499999999999 | 0.45843386092625193 |
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(s) = 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(0.10625) = 1.01621 MeV and Y_out(0.89375) = 0.458434 MeV. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives s_line = 0.10625 + (0.827676 − 1.01621)(0.89375 − 0.10625)/(0.458434 − 1.01621) = 0.372438. 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 s = 0.586625; substituting back gives Y_out = 0.82767559 MeV.
- The interpolation error in the input is |s_line − s| = 0.214187. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 0.79939459 MeV and 0.85595659 MeV. Sort the roots even if the response decreases: 0.54639 ≤ s ≤ 0.623044.
- The lower and upper input deviations are 0.0402351 and 0.0364194. 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 s ≈ 0.586625; compatible interval [0.54639, 0.623044]. Endpoint interpolation gives 0.372438.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.