University Physics IV · topic UPIV-10.05
The hydrogen spectrum, Bohr radius, and Rydberg energy
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.
Hydrogen spectrum, Bohr radius and Rydberg energy: predict a finite change
A hydrogenic atom's bound energies are E_n = −Z²E_R/n² and its sizes are near n²a₀/Z, where the Rydberg energy E_R ≈ 13.606 eV and the Bohr radius a₀ ≈ 0.0529 nm are built entirely from ħ, the reduced mass, the electron charge, and ε₀. Use the supplied topic model Y_out = 1239.84/max(13.6057*(m/(m+1))*(1/(nf^2)-1/((nf+dn)^2)),0.01), where Y_out is PHOTON λ (nm). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; max(a,b) selects the larger value. m is Partner mass M, measured in mₑ. Keep nf (Lower level n_f) = 2; dn (Jump Δn = n_i − n_f) = 1 fixed. For this exercise use only 195.969 mₑ ≤ m ≤ 1641.03 mₑ 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 m = 513.882 mₑ and 716.191 mₑ. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δm across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
PHOTON λ against m
Predictions from the supplied Hydrogen spectrum, Bohr radius and Rydberg energy 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): PHOTON λ / nm
Horizontal axis (x): m: Partner mass M / mₑ
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| m: Partner mass M / mₑ | PHOTON λ / nm |
|---|---|
| 195.96875 | 659.4589330704722 |
| 557.234375 | 657.2883362278814 |
| 918.4999999999999 | 656.8252233808519 |
| 1279.7656249999998 | 656.6235752096339 |
| 1641.0312499999998 | 656.510710908443 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of m separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At m = 513.882 mₑ, substitute into Y_out = 1239.84/max(13.6057*(m/(m+1))*(1/(nf^2)-1/((nf+dn)^2)),0.01): Y₀ = 657.388 nm.
- At m = 716.191 mₑ, the same fixed-parameter model gives Y₁ = 657.027 nm.
- Signed change: ΔY_out = Y₁ − Y₀ = -0.360661 nm; the input change is Δm = 202.309 mₑ.
- Average sensitivity is ΔY_out/Δm = -0.00178273 nm per mₑ. 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 = 657.388 nm, Y₁ = 657.027 nm; ΔY_out = -0.360661 nm; average sensitivity = -0.00178273 nm per mₑ.
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.
Hydrogen spectrum, Bohr radius and Rydberg energy: infer an input with uncertainty
A hydrogenic atom's bound energies are E_n = −Z²E_R/n² and its sizes are near n²a₀/Z, where the Rydberg energy E_R ≈ 13.606 eV and the Bohr radius a₀ ≈ 0.0529 nm are built entirely from ħ, the reduced mass, the electron charge, and ε₀. Use the supplied topic model Y_out = 1239.84/max(13.6057*(m/(m+1))*(1/(nf^2)-1/((nf+dn)^2)),0.01), where Y_out is PHOTON λ (nm). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; max(a,b) selects the larger value. m is Partner mass M, measured in mₑ. Keep nf (Lower level n_f) = 2; dn (Jump Δn = n_i − n_f) = 1 fixed. For this exercise use only 195.969 mₑ ≤ m ≤ 1641.03 mₑ 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 = 656.71984 nm, with a bounded reading uncertainty ±0.031369 nm. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown m by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.144506 mₑ, 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.
PHOTON λ against m
Predictions from the supplied Hydrogen spectrum, Bohr radius and Rydberg energy 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): PHOTON λ / nm
Horizontal axis (x): m: Partner mass M / mₑ
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| m: Partner mass M / mₑ | PHOTON λ / nm |
|---|---|
| 195.96875 | 659.4589330704722 |
| 557.234375 | 657.2883362278814 |
| 918.4999999999999 | 656.8252233808519 |
| 1279.7656249999998 | 656.6235752096339 |
| 1641.0312499999998 | 656.510710908443 |
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(m) = 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(195.969) = 659.459 nm and Y_out(1641.03) = 656.511 nm. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives m_line = 195.969 + (656.72 − 659.459)(1641.03 − 195.969)/(656.511 − 659.459) = 1538.53 mₑ. 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 m = 1077.45 mₑ; substituting back gives Y_out = 656.71984 nm.
- The interpolation error in the input is |m_line − m| = 461.073 mₑ. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 656.68847 nm and 656.75121 nm. Sort the roots even if the response decreases: 1024.67 mₑ ≤ m ≤ 1135.97 mₑ.
- The lower and upper input deviations are 52.7845 mₑ and 58.5181 mₑ. 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 m ≈ 1077.45 mₑ; compatible interval [1024.67, 1135.97] mₑ. Endpoint interpolation gives 1538.53 mₑ.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.