University Physics V · topic UPV-06.07
WKB, the Gamow exponent, and the turning-point connection
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.
WKB and the Gamow exponent: predict a finite change
The WKB approximation solves the one-dimensional Schrödinger equation by putting the unknown in the exponent, ψ ≈ p(x)^(−1/2) exp(±(i/ħ)∫p dx), and is valid wherever the local de Broglie wavelength changes by little over its own length. Use the supplied topic model Y_out = w*sqrt(max(1-en/v0,0)), where Y_out is TURNING POINT x_t (nm). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x; max(a,b) selects the larger value. en is Electron energy E, measured in eV. Keep w (Barrier half-width w) = 0.35 nm; v0 (Barrier height V₀) = 5 eV fixed. For this exercise use only 1.08438 eV ≤ en ≤ 4.9825 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 en = 1.94196 eV and 2.4877 eV. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δen across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
TURNING POINT x_t against en
Predictions from the supplied WKB and the Gamow exponent 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): TURNING POINT x_t / nm
Horizontal axis (x): en: Electron energy E / eV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| en: Electron energy E / eV | TURNING POINT x_t / nm |
|---|---|
| 1.084375 | 0.30973022535748745 |
| 2.05890625 | 0.26843397116423245 |
| 3.0334375 | 0.21950121013333845 |
| 4.00796875 | 0.15589985768114095 |
| 4.9825 | 0.020706279240848828 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of en separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At en = 1.94196 eV, substitute into Y_out = w*sqrt(max(1-en/v0,0)): Y₀ = 0.273719 nm.
- At en = 2.4877 eV, the same fixed-parameter model gives Y₁ = 0.248095 nm.
- Signed change: ΔY_out = Y₁ − Y₀ = -0.0256232 nm; the input change is Δen = 0.545738 eV.
- Average sensitivity is ΔY_out/Δen = -0.0469516 nm 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 = 0.273719 nm, Y₁ = 0.248095 nm; ΔY_out = -0.0256232 nm; average sensitivity = -0.0469516 nm 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.
WKB and the Gamow exponent: infer an input with uncertainty
The WKB approximation solves the one-dimensional Schrödinger equation by putting the unknown in the exponent, ψ ≈ p(x)^(−1/2) exp(±(i/ħ)∫p dx), and is valid wherever the local de Broglie wavelength changes by little over its own length. Use the supplied topic model Y_out = w*sqrt(max(1-en/v0,0)), where Y_out is TURNING POINT x_t (nm). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x; max(a,b) selects the larger value. en is Electron energy E, measured in eV. Keep w (Barrier half-width w) = 0.35 nm; v0 (Barrier height V₀) = 5 eV fixed. For this exercise use only 1.08438 eV ≤ en ≤ 4.9825 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 = 0.19410135 nm, with a bounded reading uncertainty ±0.017344 nm. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown en by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000389812 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.
TURNING POINT x_t against en
Predictions from the supplied WKB and the Gamow exponent 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): TURNING POINT x_t / nm
Horizontal axis (x): en: Electron energy E / eV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| en: Electron energy E / eV | TURNING POINT x_t / nm |
|---|---|
| 1.084375 | 0.30973022535748745 |
| 2.05890625 | 0.26843397116423245 |
| 3.0334375 | 0.21950121013333845 |
| 4.00796875 | 0.15589985768114095 |
| 4.9825 | 0.020706279240848828 |
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(en) = 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(1.08438) = 0.30973 nm and Y_out(4.9825) = 0.0207063 nm. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives en_line = 1.08438 + (0.194101 − 0.30973)(4.9825 − 1.08438)/(0.0207063 − 0.30973) = 2.64389 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 en = 3.46223 eV; substituting back gives Y_out = 0.19410135 nm.
- The interpolation error in the input is |en_line − en| = 0.818346 eV. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 0.17675735 nm and 0.21144535 nm. Sort the roots even if the response decreases: 3.17514 eV ≤ en ≤ 3.72477 eV.
- The lower and upper input deviations are 0.287094 eV and 0.262538 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 en ≈ 3.46223 eV; compatible interval [3.17514, 3.72477] eV. Endpoint interpolation gives 2.64389 eV.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.