University Quantum Mechanics II · topic UQMII-03.07
Gauge transformations and the phase of the wavefunction
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.
Gauge transformations and the phase of the wavefunction: predict a finite change
A gauge transformation replaces A by A + ∇Λ and φ by φ − ∂Λ/∂t for a smooth, single-valued Λ(r, t), and multiplies ψ by e^(iqΛ/ħ); the new potentials and wavefunction describe exactly the same physics as the old. Use the supplied topic model Y_out = k0+a*kap*exp(-1.125*kap^2), where Y_out is CANONICAL ⟨p⟩/ħ (nm⁻¹). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; exp(x) means eˣ. kap is Gauge wavenumber κ, measured in nm⁻¹. Keep a (Gauge phase amplitude qΛ₀/ħ) = 1 rad; k0 (Mechanical wavenumber k₀) = 2 nm⁻¹ fixed. For this exercise use only 0.69875 nm⁻¹ ≤ kap ≤ 1.84062 nm⁻¹ 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 kap = 0.949963 nm⁻¹ and 1.10982 nm⁻¹. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δkap across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
CANONICAL ⟨p⟩/ħ against kap
Predictions from the supplied Gauge transformations and the phase of the wavefunction 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): CANONICAL ⟨p⟩/ħ / nm⁻¹
Horizontal axis (x): kap: Gauge wavenumber κ / nm⁻¹
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| kap: Gauge wavenumber κ / nm⁻¹ | CANONICAL ⟨p⟩/ħ / nm⁻¹ |
|---|---|
| 0.69875 | 2.4034328345207454 |
| 0.9842187499999999 | 2.330985912127612 |
| 1.2696874999999999 | 2.207038847568097 |
| 1.55515625 | 2.102361070531059 |
| 1.840625 | 2.0407099497179675 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of kap separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At kap = 0.949963 nm⁻¹, substitute into Y_out = k0+a*kap*exp(-1.125*kap^2): Y₀ = 2.34419 nm⁻¹.
- At kap = 1.10982 nm⁻¹, the same fixed-parameter model gives Y₁ = 2.27763 nm⁻¹.
- Signed change: ΔY_out = Y₁ − Y₀ = -0.0665607 nm⁻¹; the input change is Δkap = 0.159862 nm⁻¹.
- Average sensitivity is ΔY_out/Δkap = -0.416362 nm⁻¹ per nm⁻¹. 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 = 2.34419 nm⁻¹, Y₁ = 2.27763 nm⁻¹; ΔY_out = -0.0665607 nm⁻¹; average sensitivity = -0.416362 nm⁻¹ per nm⁻¹.
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.
Gauge transformations and the phase of the wavefunction: infer an input with uncertainty
A gauge transformation replaces A by A + ∇Λ and φ by φ − ∂Λ/∂t for a smooth, single-valued Λ(r, t), and multiplies ψ by e^(iqΛ/ħ); the new potentials and wavefunction describe exactly the same physics as the old. Use the supplied topic model Y_out = k0+a*kap*exp(-1.125*kap^2), where Y_out is CANONICAL ⟨p⟩/ħ (nm⁻¹). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; exp(x) means eˣ. kap is Gauge wavenumber κ, measured in nm⁻¹. Keep a (Gauge phase amplitude qΛ₀/ħ) = 1 rad; k0 (Mechanical wavenumber k₀) = 2 nm⁻¹ fixed. For this exercise use only 0.69875 nm⁻¹ ≤ kap ≤ 1.84062 nm⁻¹ 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 = 2.1561255 nm⁻¹, with a bounded reading uncertainty ±0.017312 nm⁻¹. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown kap by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000114188 nm⁻¹, 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.
CANONICAL ⟨p⟩/ħ against kap
Predictions from the supplied Gauge transformations and the phase of the wavefunction 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): CANONICAL ⟨p⟩/ħ / nm⁻¹
Horizontal axis (x): kap: Gauge wavenumber κ / nm⁻¹
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| kap: Gauge wavenumber κ / nm⁻¹ | CANONICAL ⟨p⟩/ħ / nm⁻¹ |
|---|---|
| 0.69875 | 2.4034328345207454 |
| 0.9842187499999999 | 2.330985912127612 |
| 1.2696874999999999 | 2.207038847568097 |
| 1.55515625 | 2.102361070531059 |
| 1.840625 | 2.0407099497179675 |
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(kap) = 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(0.69875) = 2.40343 nm⁻¹ and Y_out(1.84062) = 2.04071 nm⁻¹. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives kap_line = 0.69875 + (2.15613 − 2.40343)(1.84062 − 0.69875)/(2.04071 − 2.40343) = 1.47729 nm⁻¹. 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 kap = 1.39529 nm⁻¹; substituting back gives Y_out = 2.1561255 nm⁻¹.
- The interpolation error in the input is |kap_line − kap| = 0.0819957 nm⁻¹. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 2.1388135 nm⁻¹ and 2.1734375 nm⁻¹. Sort the roots even if the response decreases: 1.35075 nm⁻¹ ≤ kap ≤ 1.44254 nm⁻¹.
- The lower and upper input deviations are 0.0445408 nm⁻¹ and 0.0472434 nm⁻¹. 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 kap ≈ 1.39529 nm⁻¹; compatible interval [1.35075, 1.44254] nm⁻¹. Endpoint interpolation gives 1.47729 nm⁻¹.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.