University Advanced Quantum Mechanics · topic UAQM-08.05
Charge conjugation: psi^c = i gamma^2 psi* and the positron wavefunction
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.
Charge conjugation and the positron wavefunction: predict a finite change
Charge conjugation is the antilinear map ψ → ψ^c = iγ²ψ*, complex conjugation followed by the matrix iγ², which turns every solution of the Dirac equation for charge q into a solution for charge −q with the same mass. Use the supplied topic model Y_out = -p/sqrt(1+p^2), where Y_out is VELOCITY v_z, BOTH ROWS (c). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x. p is Electron momentum p_z, measured in mc. Keep th (Rest-frame spin angle θ from +z) = 60 ° fixed. For this exercise use only 0.31875 mc ≤ p ≤ 2.68125 mc 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 p = 0.8385 mc and 1.16925 mc. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δp across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
VELOCITY v_z, BOTH ROWS against p
Predictions from the supplied Charge conjugation and the positron wavefunction 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): VELOCITY v_z, BOTH ROWS / c
Horizontal axis (x): p: Electron momentum p_z / mc
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| p: Electron momentum p_z / mc | VELOCITY v_z, BOTH ROWS / c |
|---|---|
| 0.31875 | -0.3036951958106041 |
| 0.9093749999999999 | -0.6727878620767288 |
| 1.5 | -0.8320502943378437 |
| 2.0906249999999997 | -0.90211140049248 |
| 2.68125 | -0.9369559659498268 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of p separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At p = 0.8385 mc, substitute into Y_out = -p/sqrt(1+p^2): Y₀ = -0.642518 c.
- At p = 1.16925 mc, the same fixed-parameter model gives Y₁ = -0.759967 c.
- Signed change: ΔY_out = Y₁ − Y₀ = -0.117449 c; the input change is Δp = 0.33075 mc.
- Average sensitivity is ΔY_out/Δp = -0.3551 c per mc. 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 = -0.642518 c, Y₁ = -0.759967 c; ΔY_out = -0.117449 c; average sensitivity = -0.3551 c per mc.
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.
Charge conjugation and the positron wavefunction: infer an input with uncertainty
Charge conjugation is the antilinear map ψ → ψ^c = iγ²ψ*, complex conjugation followed by the matrix iγ², which turns every solution of the Dirac equation for charge q into a solution for charge −q with the same mass. Use the supplied topic model Y_out = -p/sqrt(1+p^2), where Y_out is VELOCITY v_z, BOTH ROWS (c). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x. p is Electron momentum p_z, measured in mc. Keep th (Rest-frame spin angle θ from +z) = 60 ° fixed. For this exercise use only 0.31875 mc ≤ p ≤ 2.68125 mc 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 = -0.86944149 c, with a bounded reading uncertainty ±0.010127 c. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown p by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00023625 mc, 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.
VELOCITY v_z, BOTH ROWS against p
Predictions from the supplied Charge conjugation and the positron wavefunction 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): VELOCITY v_z, BOTH ROWS / c
Horizontal axis (x): p: Electron momentum p_z / mc
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| p: Electron momentum p_z / mc | VELOCITY v_z, BOTH ROWS / c |
|---|---|
| 0.31875 | -0.3036951958106041 |
| 0.9093749999999999 | -0.6727878620767288 |
| 1.5 | -0.8320502943378437 |
| 2.0906249999999997 | -0.90211140049248 |
| 2.68125 | -0.9369559659498268 |
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(p) = 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(0.31875) = -0.303695 c and Y_out(2.68125) = -0.936956 c. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives p_line = 0.31875 + (-0.869441 − -0.303695)(2.68125 − 0.31875)/(-0.936956 − -0.303695) = 2.42937 mc. 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 p = 1.75988 mc; substituting back gives Y_out = -0.86944149 c.
- The interpolation error in the input is |p_line − p| = 0.669499 mc. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = -0.87956849 c and -0.85931449 c. Sort the roots even if the response decreases: 1.68016 mc ≤ p ≤ 1.84872 mc.
- The lower and upper input deviations are 0.0797136 mc and 0.0888423 mc. 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 p ≈ 1.75988 mc; compatible interval [1.68016, 1.84872] mc. Endpoint interpolation gives 2.42937 mc.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.