University Advanced Quantum Mechanics · topic UAQM-07.08
The ground state: a weak singularity and the size of the small component
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.
The ground state: weak singularity and small component: predict a finite change
The Dirac ground state of a hydrogen-like ion is a single power-times-exponential spinor, G ∝ r^γ e^(−Zr/a₀) with F = −(1 − γ)G/(Zα), in which γ = √(1 − (Zα)²) fixes the energy, the weight of the small component and how sharply the density rises at the nucleus. Use the supplied topic model Y_out = exp(-(2-2*sqrt(1-z*z/18778.87))*(ln(z*rp/26458.86)-0.92278+(2-2*sqrt(1-z*z/18778.87))*(0.19747+0.0283*(2-2*sqrt(1-z*z/18778.87))))), where Y_out is ρ_D/ρ_S AT PROBE r (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x; ln(x) means the natural logarithm; exp(x) means eˣ. rp is Probe radius r, measured in fm. Keep z (Nuclear charge Z) = 92 fixed. For this exercise use only 1.50937 fm ≤ rp ≤ 8.99062 fm 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 rp = 3.15525 fm and 4.20262 fm. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δrp across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
ρ_D/ρ_S AT PROBE r against rp
Predictions from the supplied The ground state: weak singularity and small component 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): ρ_D/ρ_S AT PROBE r
Horizontal axis (x): rp: Probe radius r / fm
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| rp: Probe radius r / fm | ρ_D/ρ_S AT PROBE r |
|---|---|
| 1.509375 | 23.078638405009528 |
| 3.3796874999999997 | 15.204192990577315 |
| 5.25 | 12.104042482565045 |
| 7.120312499999999 | 10.337466196563806 |
| 8.990625 | 9.161629868745981 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of rp separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At rp = 3.15525 fm, substitute into Y_out = exp(-(2-2*sqrt(1-z*z/18778.87))*(ln(z*rp/26458.86)-0.92278+(2-2*sqrt(1-z*z/18778.87))*(0.19747+0.0283*(2-2*sqrt(1-z*z/18778.87))))): Y₀ = 15.7548.
- At rp = 4.20262 fm, the same fixed-parameter model gives Y₁ = 13.582.
- Signed change: ΔY_out = Y₁ − Y₀ = -2.17285; the input change is Δrp = 1.04737 fm.
- Average sensitivity is ΔY_out/Δrp = -2.07457 1 per fm. 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 = 15.7548, Y₁ = 13.582; ΔY_out = -2.17285; average sensitivity = -2.07457 1 per fm.
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.
The ground state: weak singularity and small component: infer an input with uncertainty
The Dirac ground state of a hydrogen-like ion is a single power-times-exponential spinor, G ∝ r^γ e^(−Zr/a₀) with F = −(1 − γ)G/(Zα), in which γ = √(1 − (Zα)²) fixes the energy, the weight of the small component and how sharply the density rises at the nucleus. Use the supplied topic model Y_out = exp(-(2-2*sqrt(1-z*z/18778.87))*(ln(z*rp/26458.86)-0.92278+(2-2*sqrt(1-z*z/18778.87))*(0.19747+0.0283*(2-2*sqrt(1-z*z/18778.87))))), where Y_out is ρ_D/ρ_S AT PROBE r (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x; ln(x) means the natural logarithm; exp(x) means eˣ. rp is Probe radius r, measured in fm. Keep z (Nuclear charge Z) = 92 fixed. For this exercise use only 1.50937 fm ≤ rp ≤ 8.99062 fm 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 = 11.225079, with a bounded reading uncertainty ±0.30952. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown rp by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000748125 fm, 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.
ρ_D/ρ_S AT PROBE r against rp
Predictions from the supplied The ground state: weak singularity and small component 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): ρ_D/ρ_S AT PROBE r
Horizontal axis (x): rp: Probe radius r / fm
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| rp: Probe radius r / fm | ρ_D/ρ_S AT PROBE r |
|---|---|
| 1.509375 | 23.078638405009528 |
| 3.3796874999999997 | 15.204192990577315 |
| 5.25 | 12.104042482565045 |
| 7.120312499999999 | 10.337466196563806 |
| 8.990625 | 9.161629868745981 |
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(rp) = 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(1.50937) = 23.0786 and Y_out(8.99062) = 9.16163. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives rp_line = 1.50937 + (11.2251 − 23.0786)(8.99062 − 1.50937)/(9.16163 − 23.0786) = 7.88139 fm. 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 rp = 6.07294 fm; substituting back gives Y_out = 11.225079.
- The interpolation error in the input is |rp_line − rp| = 1.80846 fm. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 10.915559 and 11.534599. Sort the roots even if the response decreases: 5.76211 fm ≤ rp ≤ 6.40994 fm.
- The lower and upper input deviations are 0.310825 fm and 0.337002 fm. 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 rp ≈ 6.07294 fm; compatible interval [5.76211, 6.40994] fm. Endpoint interpolation gives 7.88139 fm.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.