University Quantum Mechanics I · topic UQMI-05.05
Fourier decomposition: position space and momentum space
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 decomposition: position space and momentum space: predict a finite change
The momentum-space wavefunction φ(k) is the Fourier transform of ψ(x): the amplitude with which each plane wave e^(ikx) appears in the state, so that |φ(k)|² dk is the probability that a momentum measurement returns a value between ħk and ħ(k + dk). Use the supplied topic model Y_out = 0.0381*(k0^2 + 1/a^2), where Y_out is ELECTRON ⟨T⟩ = ħ²⟨k²⟩/2m (eV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. a is Decay length a, measured in nm. Keep k0 (Phase-ramp wavenumber k₀) = 4 nm⁻¹; x0 (Centre x₀) = 1.5 nm fixed. For this exercise use only 0.329688 nm ≤ a ≤ 0.920312 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 a = 0.459625 nm and 0.542312 nm. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δa across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
ELECTRON ⟨T⟩ = ħ²⟨k²⟩/2m against a
Predictions from the supplied Fourier decomposition: position space and momentum space 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): ELECTRON ⟨T⟩ = ħ²⟨k²⟩/2m / eV
Horizontal axis (x): a: Decay length a / nm
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| a: Decay length a / nm | ELECTRON ⟨T⟩ = ħ²⟨k²⟩/2m / eV |
|---|---|
| 0.3296875 | 0.9601258192762966 |
| 0.47734375 | 0.7768100953335065 |
| 0.625 | 0.707136 |
| 0.77265625 | 0.6734193434145673 |
| 0.9203125 | 0.6545836129839357 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of a separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At a = 0.459625 nm, substitute into Y_out = 0.0381*(k0^2 + 1/a^2): Y₀ = 0.789951 eV.
- At a = 0.542312 nm, the same fixed-parameter model gives Y₁ = 0.739147 eV.
- Signed change: ΔY_out = Y₁ − Y₀ = -0.0508041 eV; the input change is Δa = 0.0826875 nm.
- Average sensitivity is ΔY_out/Δa = -0.614411 eV 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 = 0.789951 eV, Y₁ = 0.739147 eV; ΔY_out = -0.0508041 eV; average sensitivity = -0.614411 eV 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 decomposition: position space and momentum space: infer an input with uncertainty
The momentum-space wavefunction φ(k) is the Fourier transform of ψ(x): the amplitude with which each plane wave e^(ikx) appears in the state, so that |φ(k)|² dk is the probability that a momentum measurement returns a value between ħk and ħ(k + dk). Use the supplied topic model Y_out = 0.0381*(k0^2 + 1/a^2), where Y_out is ELECTRON ⟨T⟩ = ħ²⟨k²⟩/2m (eV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. a is Decay length a, measured in nm. Keep k0 (Phase-ramp wavenumber k₀) = 4 nm⁻¹; x0 (Centre x₀) = 1.5 nm fixed. For this exercise use only 0.329688 nm ≤ a ≤ 0.920312 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 = 0.68963245 eV, with a bounded reading uncertainty ±0.0052573 eV. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown a by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0000590625 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.
ELECTRON ⟨T⟩ = ħ²⟨k²⟩/2m against a
Predictions from the supplied Fourier decomposition: position space and momentum space 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): ELECTRON ⟨T⟩ = ħ²⟨k²⟩/2m / eV
Horizontal axis (x): a: Decay length a / nm
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| a: Decay length a / nm | ELECTRON ⟨T⟩ = ħ²⟨k²⟩/2m / eV |
|---|---|
| 0.3296875 | 0.9601258192762966 |
| 0.47734375 | 0.7768100953335065 |
| 0.625 | 0.707136 |
| 0.77265625 | 0.6734193434145673 |
| 0.9203125 | 0.6545836129839357 |
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(a) = 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(0.329688) = 0.960126 eV and Y_out(0.920312) = 0.654584 eV. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives a_line = 0.329688 + (0.689632 − 0.960126)(0.920312 − 0.329688)/(0.654584 − 0.960126) = 0.852562 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 a = 0.689969 nm; substituting back gives Y_out = 0.68963245 eV.
- The interpolation error in the input is |a_line − a| = 0.162593 nm. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 0.68437515 eV and 0.69488975 eV. Sort the roots even if the response decreases: 0.668366 nm ≤ a ≤ 0.713812 nm.
- The lower and upper input deviations are 0.0216032 nm and 0.0238432 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 a ≈ 0.689969 nm; compatible interval [0.668366, 0.713812] nm. Endpoint interpolation gives 0.852562 nm.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.