University Quantum Field Theory I · topic UQFTI-15.03
Form factors in a Coulomb field: the high-energy contact potential on s 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.
Form factors in a Coulomb field: the s-state contact term: predict a finite change
The high-energy contact potential is the δ³(r) term that the slope of F₁ adds to the Coulomb potential of a static nucleus; it shifts only states whose wavefunction is non-zero at the nucleus, which are the s states. Use the supplied topic model Y_out = 10^lmu, where Y_out is PHOTON MASS μ_γ (eV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. lmu is Photon mass, log₁₀(μ_γ / 1 eV), dimensionless. Keep z (Nuclear charge Z) = 1 fixed. For this exercise use only -2.25625 ≤ lmu ≤ 3.25625 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 lmu = -1.0435 and -0.27175. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δlmu across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
PHOTON MASS μ_γ against lmu
Predictions from the supplied Form factors in a Coulomb field: the s-state contact term 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): PHOTON MASS μ_γ / eV
Horizontal axis (x): lmu: Photon mass, log₁₀(μ_γ / 1 eV)
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| lmu: Photon mass, log₁₀(μ_γ / 1 eV) | PHOTON MASS μ_γ / eV |
|---|---|
| -2.25625 | 0.0055430653660835045 |
| -0.8781250000000003 | 0.13239604139104352 |
| 0.49999999999999956 | 3.162277660168376 |
| 1.8781249999999994 | 75.53095919585759 |
| 3.256249999999999 | 1804.055940091783 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of lmu separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At lmu = -1.0435, substitute into Y_out = 10^lmu: Y₀ = 0.090469 eV.
- At lmu = -0.27175, the same fixed-parameter model gives Y₁ = 0.534872 eV.
- Signed change: ΔY_out = Y₁ − Y₀ = 0.444403 eV; the input change is Δlmu = 0.77175.
- Average sensitivity is ΔY_out/Δlmu = 0.575838 eV 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 = 0.090469 eV, Y₁ = 0.534872 eV; ΔY_out = 0.444403 eV; average sensitivity = 0.575838 eV 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.
Form factors in a Coulomb field: the s-state contact term: infer an input with uncertainty
The high-energy contact potential is the δ³(r) term that the slope of F₁ adds to the Coulomb potential of a static nucleus; it shifts only states whose wavefunction is non-zero at the nucleus, which are the s states. Use the supplied topic model Y_out = 10^lmu, where Y_out is PHOTON MASS μ_γ (eV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. lmu is Photon mass, log₁₀(μ_γ / 1 eV), dimensionless. Keep z (Nuclear charge Z) = 1 fixed. For this exercise use only -2.25625 ≤ lmu ≤ 3.25625 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 = 12.775415 eV, with a bounded reading uncertainty ±1.9155 eV. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown lmu by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00055125, 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.
PHOTON MASS μ_γ against lmu
Predictions from the supplied Form factors in a Coulomb field: the s-state contact term 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): PHOTON MASS μ_γ / eV
Horizontal axis (x): lmu: Photon mass, log₁₀(μ_γ / 1 eV)
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| lmu: Photon mass, log₁₀(μ_γ / 1 eV) | PHOTON MASS μ_γ / eV |
|---|---|
| -2.25625 | 0.0055430653660835045 |
| -0.8781250000000003 | 0.13239604139104352 |
| 0.49999999999999956 | 3.162277660168376 |
| 1.8781249999999994 | 75.53095919585759 |
| 3.256249999999999 | 1804.055940091783 |
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(lmu) = 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(-2.25625) = 0.00554307 eV and Y_out(3.25625) = 1804.06 eV. The target lies between them. On the selected branch the response is strictly increasing.
- Endpoint interpolation gives lmu_line = -2.25625 + (12.7754 − 0.00554307)(3.25625 − -2.25625)/(1804.06 − 0.00554307) = -2.21723. 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 lmu = 1.10638; substituting back gives Y_out = 12.775415 eV.
- The interpolation error in the input is |lmu_line − lmu| = 3.32361. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 10.859915 eV and 14.690915 eV. Sort the roots even if the response decreases: 1.03583 ≤ lmu ≤ 1.16705.
- The lower and upper input deviations are 0.0705486 and 0.0606738. 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 lmu ≈ 1.10638; compatible interval [1.03583, 1.16705]. Endpoint interpolation gives -2.21723.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.