University Physics V · topic UPV-17.06
Baryons, colour, and total exchange antisymmetry
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.
Baryons, colour and exchange antisymmetry: predict a finite change
Total exchange antisymmetry is the demand that a baryon's whole wavefunction — space, spin, flavour and colour together — change sign when any two of its three quarks are swapped, a sign supplied entirely by the colour factor. Use the supplied topic model Y_out = 104.252*(8/max(mu,1)+1/max(md,1)), where Y_out is PREDICTED μ_p (μ_N). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; max(a,b) selects the larger value. mu is Constituent mass m_u, measured in MeV. Keep md (Constituent mass m_d) = 335 MeV fixed. For this exercise use only 277 MeV ≤ mu ≤ 403 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 mu = 304.72 MeV and 322.36 MeV. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δmu across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
PREDICTED μ_p against mu
Predictions from the supplied Baryons, colour and exchange antisymmetry 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): PREDICTED μ_p / μ_N
Horizontal axis (x): mu: Constituent mass m_u / MeV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| mu: Constituent mass m_u / MeV | PREDICTED μ_p / μ_N |
|---|---|
| 277 | 3.3220880866425997 |
| 308.5 | 3.0146554294975685 |
| 340 | 2.7641882352941174 |
| 371.5 | 2.5561959623149395 |
| 403 | 2.380718610421836 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of mu separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At mu = 304.72 MeV, substitute into Y_out = 104.252*(8/max(mu,1)+1/max(md,1)): Y₀ = 3.04819 μ_N.
- At mu = 322.36 MeV, the same fixed-parameter model gives Y₁ = 2.89842 μ_N.
- Signed change: ΔY_out = Y₁ − Y₀ = -0.149772 μ_N; the input change is Δmu = 17.64 MeV.
- Average sensitivity is ΔY_out/Δmu = -0.00849048 μ_N per MeV. 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 = 3.04819 μ_N, Y₁ = 2.89842 μ_N; ΔY_out = -0.149772 μ_N; average sensitivity = -0.00849048 μ_N 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.
Baryons, colour and exchange antisymmetry: infer an input with uncertainty
Total exchange antisymmetry is the demand that a baryon's whole wavefunction — space, spin, flavour and colour together — change sign when any two of its three quarks are swapped, a sign supplied entirely by the colour factor. Use the supplied topic model Y_out = 104.252*(8/max(mu,1)+1/max(md,1)), where Y_out is PREDICTED μ_p (μ_N). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; max(a,b) selects the larger value. mu is Constituent mass m_u, measured in MeV. Keep md (Constituent mass m_d) = 335 MeV fixed. For this exercise use only 277 MeV ≤ mu ≤ 403 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 = 2.6681095 μ_N, with a bounded reading uncertainty ±0.043109 μ_N. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown mu by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0126 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.
PREDICTED μ_p against mu
Predictions from the supplied Baryons, colour and exchange antisymmetry 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): PREDICTED μ_p / μ_N
Horizontal axis (x): mu: Constituent mass m_u / MeV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| mu: Constituent mass m_u / MeV | PREDICTED μ_p / μ_N |
|---|---|
| 277 | 3.3220880866425997 |
| 308.5 | 3.0146554294975685 |
| 340 | 2.7641882352941174 |
| 371.5 | 2.5561959623149395 |
| 403 | 2.380718610421836 |
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(mu) = 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(277) = 3.32209 μ_N and Y_out(403) = 2.38072 μ_N. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives mu_line = 277 + (2.66811 − 3.32209)(403 − 277)/(2.38072 − 3.32209) = 364.533 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 mu = 353.86 MeV; substituting back gives Y_out = 2.6681095 μ_N.
- The interpolation error in the input is |mu_line − mu| = 10.6734 MeV. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 2.6250005 μ_N and 2.7112185 μ_N. Sort the roots even if the response decreases: 347.504 MeV ≤ mu ≤ 360.453 MeV.
- The lower and upper input deviations are 6.35601 MeV and 6.59285 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 mu ≈ 353.86 MeV; compatible interval [347.504, 360.453] MeV. Endpoint interpolation gives 364.533 MeV.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.