University Physics IV · topic UPIV-04.08
Fourier reciprocity and the 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.
Fourier reciprocity and uncertainty: predict a finite change
Fourier reciprocity is the rule that a function and its transform cannot both be narrow: the product of their root-mean-square widths, Δx Δk, can never fall below one half. Use the supplied topic model Y_out = 9.53*(1+r*r)/max(s*s,0.01), where Y_out is ⟨T⟩ (electron) (meV). 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 Position width Δx, measured in nm. Keep r (Phase curvature (chirp) r) = 0 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.
⟨T⟩ (electron) against s
Predictions from the supplied Fourier reciprocity and uncertainty 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): ⟨T⟩ (electron) / meV
Horizontal axis (x): s: Position width Δx / nm
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| s: Position width Δx / nm | ⟨T⟩ (electron) / meV |
|---|---|
| 0.659375 | 21.91936389568967 |
| 0.9546875 | 10.45611685386035 |
| 1.25 | 6.0992 |
| 1.5453125 | 3.9908027738899374 |
| 1.840625 | 2.812951651816984 |
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 = 9.53*(1+r*r)/max(s*s,0.01): Y₀ = 11.2778 meV.
- At s = 1.08462 nm, the same fixed-parameter model gives Y₁ = 8.10091 meV.
- Signed change: ΔY_out = Y₁ − Y₀ = -3.17692 meV; the input change is Δs = 0.165375 nm.
- Average sensitivity is ΔY_out/Δs = -19.2104 meV 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 = 11.2778 meV, Y₁ = 8.10091 meV; ΔY_out = -3.17692 meV; average sensitivity = -19.2104 meV 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.
Fourier reciprocity and uncertainty: infer an input with uncertainty
Fourier reciprocity is the rule that a function and its transform cannot both be narrow: the product of their root-mean-square widths, Δx Δk, can never fall below one half. Use the supplied topic model Y_out = 9.53*(1+r*r)/max(s*s,0.01), where Y_out is ⟨T⟩ (electron) (meV). 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 Position width Δx, measured in nm. Keep r (Phase curvature (chirp) r) = 0 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 = 5.0046541 meV, with a bounded reading uncertainty ±0.32876 meV. 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.
⟨T⟩ (electron) against s
Predictions from the supplied Fourier reciprocity and uncertainty 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): ⟨T⟩ (electron) / meV
Horizontal axis (x): s: Position width Δx / nm
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| s: Position width Δx / nm | ⟨T⟩ (electron) / meV |
|---|---|
| 0.659375 | 21.91936389568967 |
| 0.9546875 | 10.45611685386035 |
| 1.25 | 6.0992 |
| 1.5453125 | 3.9908027738899374 |
| 1.840625 | 2.812951651816984 |
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) = 21.9194 meV and Y_out(1.84062) = 2.81295 meV. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives s_line = 0.659375 + (5.00465 − 21.9194)(1.84062 − 0.659375)/(2.81295 − 21.9194) = 1.70512 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 = 5.0046541 meV.
- The interpolation error in the input is |s_line − s| = 0.325186 nm. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 4.6758941 meV and 5.3334141 meV. Sort the roots even if the response decreases: 1.33673 nm ≤ s ≤ 1.42762 nm.
- The lower and upper input deviations are 0.0432072 nm and 0.0476874 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 [1.33673, 1.42762] nm. Endpoint interpolation gives 1.70512 nm.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.