University Quantum Mechanics II · topic UQMII-03.05
Stern-Gerlach as a dynamical process
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.
Stern-Gerlach as a dynamical process: predict a finite change
In the dynamical model a Stern-Gerlach magnet is the Hamiltonian −γ(B₀ + αz)S_z acting for a time T; it kicks the spin-up and spin-down parts of one packet by opposite momenta ±γαħT/2, and so ties spin to momentum rather than measuring it. Use the supplied topic model Y_out = exp(-k^2/2), where Y_out is PACKET OVERLAP (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; exp(x) means eˣ. k is Kick Δp in units of σ_p, measured in σ_p. Keep th (Spin tilt θ from +z) = 60 ° fixed. For this exercise use only 0.425 σ_p ≤ k ≤ 3.575 σ_p 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 k = 1.118 σ_p and 1.559 σ_p. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δk across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
PACKET OVERLAP against k
Predictions from the supplied Stern-Gerlach as a dynamical process 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): PACKET OVERLAP
Horizontal axis (x): k: Kick Δp in units of σ_p / σ_p
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| k: Kick Δp in units of σ_p / σ_p | PACKET OVERLAP |
|---|---|
| 0.425 | 0.913645626396729 |
| 1.2125 | 0.4794679990742081 |
| 2 | 0.1353352832366127 |
| 2.7874999999999996 | 0.020546223525013892 |
| 3.5749999999999997 | 0.0016777318282638824 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of k separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At k = 1.118 σ_p, substitute into Y_out = exp(-k^2/2): Y₀ = 0.535282.
- At k = 1.559 σ_p, the same fixed-parameter model gives Y₁ = 0.296639.
- Signed change: ΔY_out = Y₁ − Y₀ = -0.238643; the input change is Δk = 0.441 σ_p.
- Average sensitivity is ΔY_out/Δk = -0.541141 1 per σ_p. 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 = 0.535282, Y₁ = 0.296639; ΔY_out = -0.238643; average sensitivity = -0.541141 1 per σ_p.
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.
Stern-Gerlach as a dynamical process: infer an input with uncertainty
In the dynamical model a Stern-Gerlach magnet is the Hamiltonian −γ(B₀ + αz)S_z acting for a time T; it kicks the spin-up and spin-down parts of one packet by opposite momenta ±γαħT/2, and so ties spin to momentum rather than measuring it. Use the supplied topic model Y_out = exp(-k^2/2), where Y_out is PACKET OVERLAP (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; exp(x) means eˣ. k is Kick Δp in units of σ_p, measured in σ_p. Keep th (Spin tilt θ from +z) = 60 ° fixed. For this exercise use only 0.425 σ_p ≤ k ≤ 3.575 σ_p 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 = 0.063734381, with a bounded reading uncertainty ±0.0093085. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown k by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000315 σ_p, 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.
PACKET OVERLAP against k
Predictions from the supplied Stern-Gerlach as a dynamical process 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): PACKET OVERLAP
Horizontal axis (x): k: Kick Δp in units of σ_p / σ_p
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| k: Kick Δp in units of σ_p / σ_p | PACKET OVERLAP |
|---|---|
| 0.425 | 0.913645626396729 |
| 1.2125 | 0.4794679990742081 |
| 2 | 0.1353352832366127 |
| 2.7874999999999996 | 0.020546223525013892 |
| 3.5749999999999997 | 0.0016777318282638824 |
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(k) = 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(0.425) = 0.913646 and Y_out(3.575) = 0.00167773. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives k_line = 0.425 + (0.0637344 − 0.913646)(3.575 − 0.425)/(0.00167773 − 0.913646) = 3.36065 σ_p. 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 k = 2.3465 σ_p; substituting back gives Y_out = 0.063734381.
- The interpolation error in the input is |k_line − k| = 1.01415 σ_p. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 0.054425881 and 0.073042881. Sort the roots even if the response decreases: 2.28767 σ_p ≤ k ≤ 2.41285 σ_p.
- The lower and upper input deviations are 0.0588337 σ_p and 0.0663471 σ_p. 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 k ≈ 2.3465 σ_p; compatible interval [2.28767, 2.41285] σ_p. Endpoint interpolation gives 3.36065 σ_p.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.