University Quantum Mechanics II · topic UQMII-15.07
Resonances and the Ramsauer-Townsend effect
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.
Resonances and the Ramsauer-Townsend effect: predict a finite change
A scattering resonance is an energy E_r where one phase shift δₗ climbs steeply through π/2, so that partial wave reaches its unitarity limit and the projectile is held for about ħ/Γ; the Ramsauer-Townsend effect is δ₀ passing through π, which removes the s wave. Use the supplied topic model Y_out = 0.6582/w, where Y_out is LIFETIME τ = ħ/Γ (fs). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. w is Width Γ, measured in eV. Keep b (Background phase δ_bg) = 0 ° fixed. For this exercise use only 0.07825 eV ≤ w ≤ 0.36175 eV 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 w = 0.14062 eV and 0.18031 eV. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δw across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
LIFETIME τ = ħ/Γ against w
Predictions from the supplied Resonances and the Ramsauer-Townsend effect 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): LIFETIME τ = ħ/Γ / fs
Horizontal axis (x): w: Width Γ / eV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| w: Width Γ / eV | LIFETIME τ = ħ/Γ / fs |
|---|---|
| 0.07825000000000001 | 8.411501597444088 |
| 0.149125 | 4.413746856663872 |
| 0.22 | 2.9918181818181817 |
| 0.290875 | 2.262827675118178 |
| 0.36175 | 1.8194885970974428 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of w separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At w = 0.14062 eV, substitute into Y_out = 0.6582/w: Y₀ = 4.6807 fs.
- At w = 0.18031 eV, the same fixed-parameter model gives Y₁ = 3.65038 fs.
- Signed change: ΔY_out = Y₁ − Y₀ = -1.03032 fs; the input change is Δw = 0.03969 eV.
- Average sensitivity is ΔY_out/Δw = -25.9592 fs per eV. 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 = 4.6807 fs, Y₁ = 3.65038 fs; ΔY_out = -1.03032 fs; average sensitivity = -25.9592 fs per eV.
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.
Resonances and the Ramsauer-Townsend effect: infer an input with uncertainty
A scattering resonance is an energy E_r where one phase shift δₗ climbs steeply through π/2, so that partial wave reaches its unitarity limit and the projectile is held for about ħ/Γ; the Ramsauer-Townsend effect is δ₀ passing through π, which removes the s wave. Use the supplied topic model Y_out = 0.6582/w, where Y_out is LIFETIME τ = ħ/Γ (fs). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. w is Width Γ, measured in eV. Keep b (Background phase δ_bg) = 0 ° fixed. For this exercise use only 0.07825 eV ≤ w ≤ 0.36175 eV 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 = 2.6203794 fs, with a bounded reading uncertainty ±0.12013 fs. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown w by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00002835 eV, 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.
LIFETIME τ = ħ/Γ against w
Predictions from the supplied Resonances and the Ramsauer-Townsend effect 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): LIFETIME τ = ħ/Γ / fs
Horizontal axis (x): w: Width Γ / eV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| w: Width Γ / eV | LIFETIME τ = ħ/Γ / fs |
|---|---|
| 0.07825000000000001 | 8.411501597444088 |
| 0.149125 | 4.413746856663872 |
| 0.22 | 2.9918181818181817 |
| 0.290875 | 2.262827675118178 |
| 0.36175 | 1.8194885970974428 |
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(w) = 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(0.07825) = 8.4115 fs and Y_out(0.36175) = 1.81949 fs. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives w_line = 0.07825 + (2.62038 − 8.4115)(0.36175 − 0.07825)/(1.81949 − 8.4115) = 0.327306 eV. 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 w = 0.251185 eV; substituting back gives Y_out = 2.6203794 fs.
- The interpolation error in the input is |w_line − w| = 0.0761214 eV. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 2.5002494 fs and 2.7405094 fs. Sort the roots even if the response decreases: 0.240174 eV ≤ w ≤ 0.263254 eV.
- The lower and upper input deviations are 0.0110107 eV and 0.0120687 eV. 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 w ≈ 0.251185 eV; compatible interval [0.240174, 0.263254] eV. Endpoint interpolation gives 0.327306 eV.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.