University Physics V · topic UPV-17.07
Mesons: J^PC assignments and neutral-meson mixing
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.
Mesons: J^PC and neutral-meson mixing: predict a finite change
A meson's J^PC is the triple of eigenvalues of total angular momentum, parity and charge conjugation on a colour-singlet qq̄ state, fixed by its orbital L and spin S through P = (−1)^(L+1) and C = (−1)^(L+S), with C defined only when the state is its own antiparticle. Use the supplied topic model Y_out = 2*pi/max(dm, 0.01), where Y_out is OSCILLATION PERIOD 2π/Δm (τ_S). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; max(a,b) selects the larger value. dm is Mass splitting Δm / Γ_S, dimensionless. Keep rl (Width ratio Γ_L / Γ_S) = 0.002; t (Observation time t) = 2 τ_S fixed. For this exercise use only 0.2125 ≤ dm ≤ 1.7875 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 dm = 0.559 and 0.7795. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δdm across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
OSCILLATION PERIOD 2π/Δm against dm
Predictions from the supplied Mesons: J^PC and neutral-meson mixing 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): OSCILLATION PERIOD 2π/Δm / τ_S
Horizontal axis (x): dm: Mass splitting Δm / Γ_S
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| dm: Mass splitting Δm / Γ_S | OSCILLATION PERIOD 2π/Δm / τ_S |
|---|---|
| 0.2125 | 29.5679308573157 |
| 0.60625 | 10.36401700153334 |
| 1 | 6.283185307179586 |
| 1.3937499999999998 | 4.508115018604188 |
| 1.7874999999999999 | 3.5150687033172514 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of dm separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At dm = 0.559, substitute into Y_out = 2*pi/max(dm, 0.01): Y₀ = 11.24 τ_S.
- At dm = 0.7795, the same fixed-parameter model gives Y₁ = 8.06053 τ_S.
- Signed change: ΔY_out = Y₁ − Y₀ = -3.17951 τ_S; the input change is Δdm = 0.2205.
- Average sensitivity is ΔY_out/Δdm = -14.4196 τ_S per unit input. 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 = 11.24 τ_S, Y₁ = 8.06053 τ_S; ΔY_out = -3.17951 τ_S; average sensitivity = -14.4196 τ_S 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.
Mesons: J^PC and neutral-meson mixing: infer an input with uncertainty
A meson's J^PC is the triple of eigenvalues of total angular momentum, parity and charge conjugation on a colour-singlet qq̄ state, fixed by its orbital L and spin S through P = (−1)^(L+1) and C = (−1)^(L+S), with C defined only when the state is its own antiparticle. Use the supplied topic model Y_out = 2*pi/max(dm, 0.01), where Y_out is OSCILLATION PERIOD 2π/Δm (τ_S). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; max(a,b) selects the larger value. dm is Mass splitting Δm / Γ_S, dimensionless. Keep rl (Width ratio Γ_L / Γ_S) = 0.002; t (Observation time t) = 2 τ_S fixed. For this exercise use only 0.2125 ≤ dm ≤ 1.7875 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 = 5.3553678 τ_S, with a bounded reading uncertainty ±0.27604 τ_S. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown dm by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0001575, 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.
OSCILLATION PERIOD 2π/Δm against dm
Predictions from the supplied Mesons: J^PC and neutral-meson mixing 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): OSCILLATION PERIOD 2π/Δm / τ_S
Horizontal axis (x): dm: Mass splitting Δm / Γ_S
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| dm: Mass splitting Δm / Γ_S | OSCILLATION PERIOD 2π/Δm / τ_S |
|---|---|
| 0.2125 | 29.5679308573157 |
| 0.60625 | 10.36401700153334 |
| 1 | 6.283185307179586 |
| 1.3937499999999998 | 4.508115018604188 |
| 1.7874999999999999 | 3.5150687033172514 |
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(dm) = 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(0.2125) = 29.5679 τ_S and Y_out(1.7875) = 3.51507 τ_S. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives dm_line = 0.2125 + (5.35537 − 29.5679)(1.7875 − 0.2125)/(3.51507 − 29.5679) = 1.67625. 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 dm = 1.17325; substituting back gives Y_out = 5.3553678 τ_S.
- The interpolation error in the input is |dm_line − dm| = 0.502997. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 5.0793278 τ_S and 5.6314078 τ_S. Sort the roots even if the response decreases: 1.11574 ≤ dm ≤ 1.23701.
- The lower and upper input deviations are 0.0575103 and 0.0637612. 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 dm ≈ 1.17325; compatible interval [1.11574, 1.23701]. Endpoint interpolation gives 1.67625.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.