University Advanced Quantum Mechanics · topic UAQM-02.03
Polarisation by scattering, and analysing power equal to it by time reversal
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.
Polarisation by scattering and the analysing power: predict a finite change
Polarisation by scattering is the net spin alignment along the normal n = k × k′/|k × k′| that an unpolarised beam acquires in one scattering; the analysing power is the left-right asymmetry per unit of beam polarisation, and time reversal makes the two equal. Use the supplied topic model Y_out = (1 + 2*sind(d0)*(sind(dp)*sind(d0-dp) - sind(dm)*sind(d0-dm))/max(sind(d0)^2 + sind(dp)^2 + sind(dm)^2 - 2*sind(dp)*sind(dm)*cosd(dp-dm), 0.0001))/2, where Y_out is FRACTION WITH SPIN ALONG n (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; max(a,b) selects the larger value; sind/cosd/tand use degrees. dm is p₁/₂ phase shift δ₁/₂, measured in °. Keep d0 (s-wave phase shift δ₀) = -30 °; dp (p₃/₂ phase shift δ₃/₂) = 80 ° fixed. For this exercise use only -17.25 ° ≤ dm ≤ 70.95 ° 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 = 2.154 ° and 14.502 °. 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.
FRACTION WITH SPIN ALONG n against dm
Predictions from the supplied Polarisation by scattering and the analysing power 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): FRACTION WITH SPIN ALONG n
Horizontal axis (x): dm: p₁/₂ phase shift δ₁/₂ / °
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| dm: p₁/₂ phase shift δ₁/₂ / ° | FRACTION WITH SPIN ALONG n |
|---|---|
| -17.25 | 0.9014599966235702 |
| 4.799999999999997 | 0.8703998210611346 |
| 26.849999999999994 | 0.807340519879384 |
| 48.89999999999999 | 0.6799031987558322 |
| 70.94999999999999 | 0.4952538179848161 |
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 = 2.154 °, substitute into Y_out = (1 + 2*sind(d0)*(sind(dp)*sind(d0-dp) - sind(dm)*sind(d0-dm))/max(sind(d0)^2 + sind(dp)^2 + sind(dm)^2 - 2*sind(dp)*sind(dm)*cosd(dp-dm), 0.0001))/2: Y₀ = 0.875481.
- At dm = 14.502 °, the same fixed-parameter model gives Y₁ = 0.847816.
- Signed change: ΔY_out = Y₁ − Y₀ = -0.0276649; the input change is Δdm = 12.348 °.
- Average sensitivity is ΔY_out/Δdm = -0.00224043 1 per °. 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 = 0.875481, Y₁ = 0.847816; ΔY_out = -0.0276649; average sensitivity = -0.00224043 1 per °.
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.
Polarisation by scattering and the analysing power: infer an input with uncertainty
Polarisation by scattering is the net spin alignment along the normal n = k × k′/|k × k′| that an unpolarised beam acquires in one scattering; the analysing power is the left-right asymmetry per unit of beam polarisation, and time reversal makes the two equal. Use the supplied topic model Y_out = (1 + 2*sind(d0)*(sind(dp)*sind(d0-dp) - sind(dm)*sind(d0-dm))/max(sind(d0)^2 + sind(dp)^2 + sind(dm)^2 - 2*sind(dp)*sind(dm)*cosd(dp-dm), 0.0001))/2, where Y_out is FRACTION WITH SPIN ALONG n (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; max(a,b) selects the larger value; sind/cosd/tand use degrees. dm is p₁/₂ phase shift δ₁/₂, measured in °. Keep d0 (s-wave phase shift δ₀) = -30 °; dp (p₃/₂ phase shift δ₃/₂) = 80 ° fixed. For this exercise use only -17.25 ° ≤ dm ≤ 70.95 ° 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 = 0.76215986, with a bounded reading uncertainty ±0.020895. 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 0.00882 °, 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.
FRACTION WITH SPIN ALONG n against dm
Predictions from the supplied Polarisation by scattering and the analysing power 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): FRACTION WITH SPIN ALONG n
Horizontal axis (x): dm: p₁/₂ phase shift δ₁/₂ / °
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| dm: p₁/₂ phase shift δ₁/₂ / ° | FRACTION WITH SPIN ALONG n |
|---|---|
| -17.25 | 0.9014599966235702 |
| 4.799999999999997 | 0.8703998210611346 |
| 26.849999999999994 | 0.807340519879384 |
| 48.89999999999999 | 0.6799031987558322 |
| 70.94999999999999 | 0.4952538179848161 |
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(-17.25) = 0.90146 and Y_out(70.95) = 0.495254. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives dm_line = -17.25 + (0.76216 − 0.90146)(70.95 − -17.25)/(0.495254 − 0.90146) = 12.9964 °. 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 = 36.552 °; substituting back gives Y_out = 0.76215986.
- The interpolation error in the input is |dm_line − dm| = 23.5556 °. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 0.74126486 and 0.78305486. Sort the roots even if the response decreases: 32.4435 ° ≤ dm ≤ 40.172 °.
- The lower and upper input deviations are 4.10847 ° and 3.61996 °. 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 ≈ 36.552 °; compatible interval [32.4435, 40.172] °. Endpoint interpolation gives 12.9964 °.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.