University Physics IV · topic UPIV-05.08
Commutators and the general uncertainty relation
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.
Commutators and the uncertainty relation: predict a finite change
The commutator [A, B] = AB − BA records how much the order of two operations matters, and Robertson's theorem says that in any state the product of the two standard deviations is at least half the modulus of that commutator's expectation value. Use the supplied topic model Y_out = 98.66/max(s,0.1), where Y_out is MOMENTUM SPREAD σ_p (eV/c). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; max(a,b) selects the larger value. s is Initial width σ_0, measured in nm. Keep r (Elapsed time t / τ) = 0.8 τ fixed. For this exercise use only 0.659375 nm ≤ s ≤ 1.84062 nm 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 s = 0.91925 nm and 1.08462 nm. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δs across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
MOMENTUM SPREAD σ_p against s
Predictions from the supplied Commutators and the uncertainty relation 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): MOMENTUM SPREAD σ_p / eV/c
Horizontal axis (x): s: Initial width σ_0 / nm
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| s: Initial width σ_0 / nm | MOMENTUM SPREAD σ_p / eV/c |
|---|---|
| 0.659375 | 149.62654028436017 |
| 0.9546875 | 103.34271685761047 |
| 1.25 | 78.928 |
| 1.5453125 | 63.844691607684524 |
| 1.840625 | 53.60135823429542 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of s separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At s = 0.91925 nm, substitute into Y_out = 98.66/max(s,0.1): Y₀ = 107.327 eV/c.
- At s = 1.08462 nm, the same fixed-parameter model gives Y₁ = 90.9623 eV/c.
- Signed change: ΔY_out = Y₁ − Y₀ = -16.3643 eV/c; the input change is Δs = 0.165375 nm.
- Average sensitivity is ΔY_out/Δs = -98.9527 eV/c per nm. 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 = 107.327 eV/c, Y₁ = 90.9623 eV/c; ΔY_out = -16.3643 eV/c; average sensitivity = -98.9527 eV/c per nm.
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.
Commutators and the uncertainty relation: infer an input with uncertainty
The commutator [A, B] = AB − BA records how much the order of two operations matters, and Robertson's theorem says that in any state the product of the two standard deviations is at least half the modulus of that commutator's expectation value. Use the supplied topic model Y_out = 98.66/max(s,0.1), where Y_out is MOMENTUM SPREAD σ_p (eV/c). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; max(a,b) selects the larger value. s is Initial width σ_0, measured in nm. Keep r (Elapsed time t / τ) = 0.8 τ fixed. For this exercise use only 0.659375 nm ≤ s ≤ 1.84062 nm 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 = 71.495992 eV/c, with a bounded reading uncertainty ±2.6842 eV/c. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown s by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000118125 nm, 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.
MOMENTUM SPREAD σ_p against s
Predictions from the supplied Commutators and the uncertainty relation 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): MOMENTUM SPREAD σ_p / eV/c
Horizontal axis (x): s: Initial width σ_0 / nm
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| s: Initial width σ_0 / nm | MOMENTUM SPREAD σ_p / eV/c |
|---|---|
| 0.659375 | 149.62654028436017 |
| 0.9546875 | 103.34271685761047 |
| 1.25 | 78.928 |
| 1.5453125 | 63.844691607684524 |
| 1.840625 | 53.60135823429542 |
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(s) = 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(0.659375) = 149.627 eV/c and Y_out(1.84062) = 53.6014 eV/c. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives s_line = 0.659375 + (71.496 − 149.627)(1.84062 − 0.659375)/(53.6014 − 149.627) = 1.62049 nm. 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 s = 1.37994 nm; substituting back gives Y_out = 71.495992 eV/c.
- The interpolation error in the input is |s_line − s| = 0.240557 nm. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 68.811792 eV/c and 74.180192 eV/c. Sort the roots even if the response decreases: 1.33 nm ≤ s ≤ 1.43377 nm.
- The lower and upper input deviations are 0.0499328 nm and 0.0538284 nm. 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 s ≈ 1.37994 nm; compatible interval [1.33, 1.43377] nm. Endpoint interpolation gives 1.62049 nm.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.