University Quantum Mechanics I · topic UQMI-01.05
de Broglie matter waves and electron diffraction
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.
de Broglie matter waves and electron diffraction: predict a finite change
A particle of momentum p behaves, in any interference or diffraction experiment, as a wave of wavelength λ = h/p, so a periodic structure diffracts it through the same angles as it would diffract light of that wavelength. Use the supplied topic model Y_out = sqrt(kv)*135*tan(2*asin(19.39/(d*sqrt(kv)))), where Y_out is R₁√V, CONSTANT IF λ ∝ 1/√V (mm √kV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x; unmarked trigonometric functions use radians. kv is Accelerating voltage V, measured in kV. Keep d (Plane spacing d (inner ring)) = 213 pm fixed. For this exercise use only 1.97812 kV ≤ kv ≤ 5.52187 kV 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 kv = 2.75775 kV and 3.25387 kV. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δkv across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
R₁√V, CONSTANT IF λ ∝ 1/√V against kv
Predictions from the supplied de Broglie matter waves and electron diffraction 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): R₁√V, CONSTANT IF λ ∝ 1/√V / mm √kV
Horizontal axis (x): kv: Accelerating voltage V / kV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| kv: Accelerating voltage V / kV | R₁√V, CONSTANT IF λ ∝ 1/√V / mm √kV |
|---|---|
| 1.978125 | 24.73457662895243 |
| 2.8640624999999997 | 24.68614434943705 |
| 3.75 | 24.660693717927042 |
| 4.6359375 | 24.645003881906636 |
| 5.521875 | 24.63436317368194 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of kv separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At kv = 2.75775 kV, substitute into Y_out = sqrt(kv)*135*tan(2*asin(19.39/(d*sqrt(kv)))): Y₀ = 24.6903 mm √kV.
- At kv = 3.25387 kV, the same fixed-parameter model gives Y₁ = 24.6732 mm √kV.
- Signed change: ΔY_out = Y₁ − Y₀ = -0.0170734 mm √kV; the input change is Δkv = 0.496125 kV.
- Average sensitivity is ΔY_out/Δkv = -0.0344135 mm √kV per kV. 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 = 24.6903 mm √kV, Y₁ = 24.6732 mm √kV; ΔY_out = -0.0170734 mm √kV; average sensitivity = -0.0344135 mm √kV per kV.
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.
de Broglie matter waves and electron diffraction: infer an input with uncertainty
A particle of momentum p behaves, in any interference or diffraction experiment, as a wave of wavelength λ = h/p, so a periodic structure diffracts it through the same angles as it would diffract light of that wavelength. Use the supplied topic model Y_out = sqrt(kv)*135*tan(2*asin(19.39/(d*sqrt(kv)))), where Y_out is R₁√V, CONSTANT IF λ ∝ 1/√V (mm √kV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x; unmarked trigonometric functions use radians. kv is Accelerating voltage V, measured in kV. Keep d (Plane spacing d (inner ring)) = 213 pm fixed. For this exercise use only 1.97812 kV ≤ kv ≤ 5.52187 kV 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 = 24.65296 mm √kV, with a bounded reading uncertainty ±0.0027895 mm √kV. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown kv by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000354375 kV, 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.
R₁√V, CONSTANT IF λ ∝ 1/√V against kv
Predictions from the supplied de Broglie matter waves and electron diffraction 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): R₁√V, CONSTANT IF λ ∝ 1/√V / mm √kV
Horizontal axis (x): kv: Accelerating voltage V / kV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| kv: Accelerating voltage V / kV | R₁√V, CONSTANT IF λ ∝ 1/√V / mm √kV |
|---|---|
| 1.978125 | 24.73457662895243 |
| 2.8640624999999997 | 24.68614434943705 |
| 3.75 | 24.660693717927042 |
| 4.6359375 | 24.645003881906636 |
| 5.521875 | 24.63436317368194 |
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(kv) = 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(1.97812) = 24.7346 mm √kV and Y_out(5.52187) = 24.6344 mm √kV. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives kv_line = 1.97812 + (24.653 − 24.7346)(5.52187 − 1.97812)/(24.6344 − 24.7346) = 4.86425 kV. 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 kv = 4.13979 kV; substituting back gives Y_out = 24.65296 mm √kV.
- The interpolation error in the input is |kv_line − kv| = 0.72446 kV. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 24.650171 mm √kV and 24.655749 mm √kV. Sort the roots even if the response decreases: 3.99016 kV ≤ kv ≤ 4.30114 kV.
- The lower and upper input deviations are 0.149638 kV and 0.161348 kV. 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 kv ≈ 4.13979 kV; compatible interval [3.99016, 4.30114] kV. Endpoint interpolation gives 4.86425 kV.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.