University Quantum Field Theory I · topic UQFTI-05.06
The non-relativistic limit: Quantum Mechanics II's Born amplitude and the Yukawa potential
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 non-relativistic limit and the Yukawa potential: predict a finite change
The non-relativistic limit of an exchange amplitude names the potential whose first Born amplitude equals ℳ/(8π√s) at low speed; for scalar exchange between particles of masses M and M′ that potential has Ṽ(q) = −ℳ/(4MM′), a Yukawa potential. Use the supplied topic model Y_out = 1+(1+4*(p/m)^2)*ln(1+4*(p/m)^2)/(4*(p/m)^2*(1+2*(p/m)^2)), where Y_out is σ(ψψ) ÷ σ(ψψ′) (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; ln(x) means the natural logarithm. m is Exchanged φ mass m, measured in MeV. Keep p (CM momentum p) = 100 MeV fixed. For this exercise use only 97.8125 MeV ≤ m ≤ 452.187 MeV 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 m = 175.775 MeV and 225.387 MeV. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δm across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
σ(ψψ) ÷ σ(ψψ′) against m
Predictions from the supplied The non-relativistic limit and the Yukawa potential 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): σ(ψψ) ÷ σ(ψψ′)
Horizontal axis (x): m: Exchanged φ mass m / MeV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| m: Exchanged φ mass m / MeV | σ(ψψ) ÷ σ(ψψ′) |
|---|---|
| 97.8125 | 1.6595890605912653 |
| 186.40625 | 1.9085084922672746 |
| 275 | 1.9705773877854629 |
| 363.59374999999994 | 1.9884483113436007 |
| 452.18749999999994 | 1.994699342283551 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of m separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At m = 175.775 MeV, substitute into Y_out = 1+(1+4*(p/m)^2)*ln(1+4*(p/m)^2)/(4*(p/m)^2*(1+2*(p/m)^2)): Y₀ = 1.89365.
- At m = 225.387 MeV, the same fixed-parameter model gives Y₁ = 1.94592.
- Signed change: ΔY_out = Y₁ − Y₀ = 0.0522711; the input change is Δm = 49.6125 MeV.
- Average sensitivity is ΔY_out/Δm = 0.00105359 1 per MeV. 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 = 1.89365, Y₁ = 1.94592; ΔY_out = 0.0522711; average sensitivity = 0.00105359 1 per MeV.
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 non-relativistic limit and the Yukawa potential: infer an input with uncertainty
The non-relativistic limit of an exchange amplitude names the potential whose first Born amplitude equals ℳ/(8π√s) at low speed; for scalar exchange between particles of masses M and M′ that potential has Ṽ(q) = −ℳ/(4MM′), a Yukawa potential. Use the supplied topic model Y_out = 1+(1+4*(p/m)^2)*ln(1+4*(p/m)^2)/(4*(p/m)^2*(1+2*(p/m)^2)), where Y_out is σ(ψψ) ÷ σ(ψψ′) (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; ln(x) means the natural logarithm. m is Exchanged φ mass m, measured in MeV. Keep p (CM momentum p) = 100 MeV fixed. For this exercise use only 97.8125 MeV ≤ m ≤ 452.187 MeV 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 = 1.9809274, with a bounded reading uncertainty ±0.0020658. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown m by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0354375 MeV, 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.
σ(ψψ) ÷ σ(ψψ′) against m
Predictions from the supplied The non-relativistic limit and the Yukawa potential 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): σ(ψψ) ÷ σ(ψψ′)
Horizontal axis (x): m: Exchanged φ mass m / MeV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| m: Exchanged φ mass m / MeV | σ(ψψ) ÷ σ(ψψ′) |
|---|---|
| 97.8125 | 1.6595890605912653 |
| 186.40625 | 1.9085084922672746 |
| 275 | 1.9705773877854629 |
| 363.59374999999994 | 1.9884483113436007 |
| 452.18749999999994 | 1.994699342283551 |
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(m) = 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(97.8125) = 1.65959 and Y_out(452.187) = 1.9947. The target lies between them. On the selected branch the response is strictly increasing.
- Endpoint interpolation gives m_line = 97.8125 + (1.98093 − 1.65959)(452.187 − 97.8125)/(1.9947 − 1.65959) = 437.624 MeV. 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 m = 313.981 MeV; substituting back gives Y_out = 1.9809274.
- The interpolation error in the input is |m_line − m| = 123.643 MeV. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 1.9788616 and 1.9829932. Sort the roots even if the response decreases: 304.427 MeV ≤ m ≤ 324.87 MeV.
- The lower and upper input deviations are 9.55391 MeV and 10.8893 MeV. 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 m ≈ 313.981 MeV; compatible interval [304.427, 324.87] MeV. Endpoint interpolation gives 437.624 MeV.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.