University Advanced Quantum Mechanics · topic UAQM-08.09
Midterm consolidation: symmetries, spin scattering and relativistic one-particle theory
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.
Midterm consolidation: symmetry, scattering, Dirac theory: predict a finite change
Relativistic one-particle theory is the quantum mechanics of a single particle obeying a Lorentz-covariant wave equation, Klein-Gordon or Dirac, in a prescribed external field: exact for its free and Coulomb solutions, but unable to change the number of particles. Use the supplied topic model Y_out = 0.511*(1/sqrt(1-b^2)-1), where Y_out is ELECTRON KINETIC ENERGY (MeV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x. b is Electron speed β = v/c, dimensionless. Keep eta (Sommerfeld parameter η) = 2; s (Spin s of each identical particle) = 0.5 fixed. For this exercise use only 0.104125 ≤ b ≤ 0.875875 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 b = 0.27391 and 0.381955. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δb across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
ELECTRON KINETIC ENERGY against b
Predictions from the supplied Midterm consolidation: symmetry, scattering, Dirac theory 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): ELECTRON KINETIC ENERGY / MeV
Horizontal axis (x): b: Electron speed β = v/c
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| b: Electron speed β = v/c | ELECTRON KINETIC ENERGY / MeV |
|---|---|
| 0.104125 | 0.0027928658442349483 |
| 0.2970625 | 0.024158086462507444 |
| 0.49 | 0.075195766818946 |
| 0.6829375 | 0.1885430826219897 |
| 0.875875 | 0.547983812560736 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of b separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At b = 0.27391, substitute into Y_out = 0.511*(1/sqrt(1-b^2)-1): Y₀ = 0.0203202 MeV.
- At b = 0.381955, the same fixed-parameter model gives Y₁ = 0.041922 MeV.
- Signed change: ΔY_out = Y₁ − Y₀ = 0.0216019 MeV; the input change is Δb = 0.108045.
- Average sensitivity is ΔY_out/Δb = 0.199934 MeV 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 = 0.0203202 MeV, Y₁ = 0.041922 MeV; ΔY_out = 0.0216019 MeV; average sensitivity = 0.199934 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.
Midterm consolidation: symmetry, scattering, Dirac theory: infer an input with uncertainty
Relativistic one-particle theory is the quantum mechanics of a single particle obeying a Lorentz-covariant wave equation, Klein-Gordon or Dirac, in a prescribed external field: exact for its free and Coulomb solutions, but unable to change the number of particles. Use the supplied topic model Y_out = 0.511*(1/sqrt(1-b^2)-1), where Y_out is ELECTRON KINETIC ENERGY (MeV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x. b is Electron speed β = v/c, dimensionless. Keep eta (Sommerfeld parameter η) = 2; s (Spin s of each identical particle) = 0.5 fixed. For this exercise use only 0.104125 ≤ b ≤ 0.875875 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 = 0.11351958 MeV, with a bounded reading uncertainty ±0.016609 MeV. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown b by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000077175, 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.
ELECTRON KINETIC ENERGY against b
Predictions from the supplied Midterm consolidation: symmetry, scattering, Dirac theory 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): ELECTRON KINETIC ENERGY / MeV
Horizontal axis (x): b: Electron speed β = v/c
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| b: Electron speed β = v/c | ELECTRON KINETIC ENERGY / MeV |
|---|---|
| 0.104125 | 0.0027928658442349483 |
| 0.2970625 | 0.024158086462507444 |
| 0.49 | 0.075195766818946 |
| 0.6829375 | 0.1885430826219897 |
| 0.875875 | 0.547983812560736 |
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(b) = 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(0.104125) = 0.00279287 MeV and Y_out(0.875875) = 0.547984 MeV. The target lies between them. On the selected branch the response is strictly increasing.
- Endpoint interpolation gives b_line = 0.104125 + (0.11352 − 0.00279287)(0.875875 − 0.104125)/(0.547984 − 0.00279287) = 0.260865. 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 b = 0.574892; substituting back gives Y_out = 0.11351958 MeV.
- The interpolation error in the input is |b_line − b| = 0.314027. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 0.09691058 MeV and 0.13012858 MeV. Sort the roots even if the response decreases: 0.541681 ≤ b ≤ 0.603937.
- The lower and upper input deviations are 0.0332115 and 0.0290446. 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 b ≈ 0.574892; compatible interval [0.541681, 0.603937]. Endpoint interpolation gives 0.260865.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.