University Quantum Mechanics II · topic UQMII-03.06
Minimal coupling: H = (p - qA)^2/2m + q phi
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.
Minimal coupling: H = (p − qA)²/2m + qφ: predict a finite change
Minimal coupling places a charge q in electromagnetic fields by replacing the canonical momentum p = −iħ∇ with p − qA in the kinetic energy and adding the potential energy qφ, which makes (p − qA)/m, not p/m, the velocity. Use the supplied topic model Y_out = (d+sind(wt))*(lam*d-(1-lam)*sind(wt)) - (d+cosd(wt))*(lam*cosd(wt)-(1-lam)*d), where Y_out is CANONICAL L_z ABOUT O (mvR). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sind/cosd/tand use degrees. d is Orbit centre offset d, measured in R. Keep lam (Gauge parameter λ) = 0.5; wt (Cyclotron phase ω_c t) = 120 ° fixed. For this exercise use only 0.10625 R ≤ d ≤ 0.89375 R 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 d = 0.2795 R and 0.38975 R. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δd across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
CANONICAL L_z ABOUT O against d
Predictions from the supplied Minimal coupling: H = (p − qA)²/2m + qφ 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 L_z ABOUT O / mvR
Horizontal axis (x): d: Orbit centre offset d / R
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| d: Orbit centre offset d / R | CANONICAL L_z ABOUT O / mvR |
|---|---|
| 0.10625 | -0.48871093749999994 |
| 0.303125 | -0.40811523437499997 |
| 0.5 | -0.24999999999999994 |
| 0.6968749999999999 | -0.014365234375000077 |
| 0.8937499999999999 | 0.2987890625 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of d separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At d = 0.2795 R, substitute into Y_out = (d+sind(wt))*(lam*d-(1-lam)*sind(wt)) - (d+cosd(wt))*(lam*cosd(wt)-(1-lam)*d): Y₀ = -0.42188 mvR.
- At d = 0.38975 R, the same fixed-parameter model gives Y₁ = -0.348095 mvR.
- Signed change: ΔY_out = Y₁ − Y₀ = 0.0737848 mvR; the input change is Δd = 0.11025 R.
- Average sensitivity is ΔY_out/Δd = 0.66925 mvR per R. Its positive sign means that this response increases 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 = -0.42188 mvR, Y₁ = -0.348095 mvR; ΔY_out = 0.0737848 mvR; average sensitivity = 0.66925 mvR per R.
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.
Minimal coupling: H = (p − qA)²/2m + qφ: infer an input with uncertainty
Minimal coupling places a charge q in electromagnetic fields by replacing the canonical momentum p = −iħ∇ with p − qA in the kinetic energy and adding the potential energy qφ, which makes (p − qA)/m, not p/m, the velocity. Use the supplied topic model Y_out = (d+sind(wt))*(lam*d-(1-lam)*sind(wt)) - (d+cosd(wt))*(lam*cosd(wt)-(1-lam)*d), where Y_out is CANONICAL L_z ABOUT O (mvR). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sind/cosd/tand use degrees. d is Orbit centre offset d, measured in R. Keep lam (Gauge parameter λ) = 0.5; wt (Cyclotron phase ω_c t) = 120 ° fixed. For this exercise use only 0.10625 R ≤ d ≤ 0.89375 R 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 = -0.15587111 mvR, with a bounded reading uncertainty ±0.049926 mvR. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown d by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00007875 R, 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 L_z ABOUT O against d
Predictions from the supplied Minimal coupling: H = (p − qA)²/2m + qφ 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 L_z ABOUT O / mvR
Horizontal axis (x): d: Orbit centre offset d / R
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| d: Orbit centre offset d / R | CANONICAL L_z ABOUT O / mvR |
|---|---|
| 0.10625 | -0.48871093749999994 |
| 0.303125 | -0.40811523437499997 |
| 0.5 | -0.24999999999999994 |
| 0.6968749999999999 | -0.014365234375000077 |
| 0.8937499999999999 | 0.2987890625 |
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(d) = 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(0.10625) = -0.488711 mvR and Y_out(0.89375) = 0.298789 mvR. The target lies between them. On the selected branch the response is strictly increasing.
- Endpoint interpolation gives d_line = 0.10625 + (-0.155871 − -0.488711)(0.89375 − 0.10625)/(0.298789 − -0.488711) = 0.43909 R. 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 d = 0.586625 R; substituting back gives Y_out = -0.15587111 mvR.
- The interpolation error in the input is |d_line − d| = 0.147535 R. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = -0.20579711 mvR and -0.10594511 mvR. Sort the roots even if the response decreases: 0.542405 R ≤ d ≤ 0.627738 R.
- The lower and upper input deviations are 0.0442203 R and 0.0411129 R. 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 d ≈ 0.586625 R; compatible interval [0.542405, 0.627738] R. Endpoint interpolation gives 0.43909 R.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.