University Advanced Quantum Mechanics · topic UAQM-07.04
The coupled radial equations for G(r) and F(r)
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.
The coupled radial equations for G(r) and F(r): predict a finite change
The coupled radial equations are the two real first-order equations, one from each block row of the Dirac equation, obeyed by the radial functions G(r) and F(r) of an electron's upper and lower components in any central potential. Use the supplied topic model Y_out = (p/(1+sqrt(1+p^2))), where Y_out is F/G AMPLITUDE cp/(E + mc²) (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x. p is Momentum p, measured in mc. Keep st (State: κ = −2, −1, +1, +2) = 2 fixed. For this exercise use only 0.469687 mc ≤ p ≤ 3.58031 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 = 1.15402 mc and 1.58951 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.
F/G AMPLITUDE cp/(E + mc²) against p
Predictions from the supplied The coupled radial equations for G(r) and F(r) 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): F/G AMPLITUDE cp/(E + mc²)
Horizontal axis (x): p: Momentum p / mc
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| p: Momentum p / mc | F/G AMPLITUDE cp/(E + mc²) |
|---|---|
| 0.4696875 | 0.22314953731926265 |
| 1.24734375 | 0.4799861607722101 |
| 2.025 | 0.6214599166488232 |
| 2.80265625 | 0.7049438851187437 |
| 3.5803125 | 0.758968054915741 |
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 = 1.15402 mc, substitute into Y_out = (p/(1+sqrt(1+p^2))): Y₀ = 0.456675.
- At p = 1.58951 mc, the same fixed-parameter model gives Y₁ = 0.552315.
- Signed change: ΔY_out = Y₁ − Y₀ = 0.0956394; the input change is Δp = 0.435488 mc.
- Average sensitivity is ΔY_out/Δp = 0.219615 1 per mc. 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 = 0.456675, Y₁ = 0.552315; ΔY_out = 0.0956394; average sensitivity = 0.219615 1 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.
The coupled radial equations for G(r) and F(r): infer an input with uncertainty
The coupled radial equations are the two real first-order equations, one from each block row of the Dirac equation, obeyed by the radial functions G(r) and F(r) of an electron's upper and lower components in any central potential. Use the supplied topic model Y_out = (p/(1+sqrt(1+p^2))), where Y_out is F/G AMPLITUDE cp/(E + mc²) (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x. p is Momentum p, measured in mc. Keep st (State: κ = −2, −1, +1, +2) = 2 fixed. For this exercise use only 0.469687 mc ≤ p ≤ 3.58031 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 = 0.66312351, with a bounded reading uncertainty ±0.014377. 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 0.000311062 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.
F/G AMPLITUDE cp/(E + mc²) against p
Predictions from the supplied The coupled radial equations for G(r) and F(r) 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): F/G AMPLITUDE cp/(E + mc²)
Horizontal axis (x): p: Momentum p / mc
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| p: Momentum p / mc | F/G AMPLITUDE cp/(E + mc²) |
|---|---|
| 0.4696875 | 0.22314953731926265 |
| 1.24734375 | 0.4799861607722101 |
| 2.025 | 0.6214599166488232 |
| 2.80265625 | 0.7049438851187437 |
| 3.5803125 | 0.758968054915741 |
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(0.469687) = 0.22315 and Y_out(3.58031) = 0.758968. The target lies between them. On the selected branch the response is strictly increasing.
- Endpoint interpolation gives p_line = 0.469687 + (0.663124 − 0.22315)(3.58031 − 0.469687)/(0.758968 − 0.22315) = 3.0239 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 = 2.36717 mc; substituting back gives Y_out = 0.66312351.
- The interpolation error in the input is |p_line − p| = 0.656731 mc. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 0.64874651 and 0.67750051. Sort the roots even if the response decreases: 2.24043 mc ≤ p ≤ 2.50466 mc.
- The lower and upper input deviations are 0.126743 mc and 0.137486 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 ≈ 2.36717 mc; compatible interval [2.24043, 2.50466] mc. Endpoint interpolation gives 3.0239 mc.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.