University Physics V · topic UPV-04.09
Matrix Hamiltonians and numerical eigenvalues
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.
Matrix Hamiltonians and numerical eigenvalues: predict a finite change
A matrix Hamiltonian is the finite Hermitian matrix you get by sampling H on a grid or projecting it onto N basis states, real symmetric when V and the basis are real, so that H|ψ⟩ = E|ψ⟩ becomes an ordinary matrix eigenvalue problem a computer solves to rounding precision. Use the supplied topic model Y_out = 2*(gp+1)^2*sin(n*pi/(2*gp+2))^2, where Y_out is GRID E_n (ħ²/mL²). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; unmarked trigonometric functions use radians. gp is Interior grid points N, dimensionless. Keep n (Level index n) = 10 fixed. For this exercise use only 12 ≤ gp ≤ 60, at the specified test settings separated by 1. 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 gp = 23 and 24. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δgp across this interval. State whether the response increases or decreases; explain why a finite difference between these settings does not require interpolation between them.
GRID E_n against gp
Predictions from the supplied Matrix Hamiltonians and numerical eigenvalues 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): GRID E_n / ħ²/mL²
Horizontal axis (x): gp: Interior grid points N
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| gp: Interior grid points N | GRID E_n / ħ²/mL² |
|---|---|
| 12 | 295.4983164409161 |
| 24 | 431.86437851565785 |
| 36 | 464.53630367897847 |
| 48 | 476.805893895036 |
| 60 | 482.6686181267924 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of gp separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At gp = 23, substitute into Y_out = 2*(gp+1)^2*sin(n*pi/(2*gp+2))^2: Y₀ = 426.92 ħ²/mL².
- At gp = 24, the same fixed-parameter model gives Y₁ = 431.864 ħ²/mL².
- Signed change: ΔY_out = Y₁ − Y₀ = 4.94415 ħ²/mL²; the input change is Δgp = 1.
- Average sensitivity is ΔY_out/Δgp = 4.94415 ħ²/mL² per unit input. Its positive sign means that this response increases over the specified interval.
- This finite difference compares two specified test settings without interpolating between them. A point derivative is not required. Any quantum numbers or particle counts must also obey their physical admissibility conditions.
Answer Y₀ = 426.92 ħ²/mL², Y₁ = 431.864 ħ²/mL²; ΔY_out = 4.94415 ħ²/mL²; average sensitivity = 4.94415 ħ²/mL² per unit input.
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.
Matrix Hamiltonians and numerical eigenvalues: infer an input with uncertainty
A matrix Hamiltonian is the finite Hermitian matrix you get by sampling H on a grid or projecting it onto N basis states, real symmetric when V and the basis are real, so that H|ψ⟩ = E|ψ⟩ becomes an ordinary matrix eigenvalue problem a computer solves to rounding precision. Use the supplied topic model Y_out = 2*(gp+1)^2*sin(n*pi/(2*gp+2))^2, where Y_out is GRID E_n (ħ²/mL²). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; unmarked trigonometric functions use radians. gp is Interior grid points N, dimensionless. Keep n (Level index n) = 10 fixed. For this exercise use only 12 ≤ gp ≤ 60, at the specified test settings separated by 1. 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 = 470.8965 ħ²/mL², with a bounded reading uncertainty ±14.974 ħ²/mL². Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. Determine every allowed value of gp consistent with that bound. Compare with a straight-line interpolation between the two endpoint predictions, and explain why a fractional setting may be inadmissible. Do not assume that the supplied calibration points prove monotonicity outside this branch.
GRID E_n against gp
Predictions from the supplied Matrix Hamiltonians and numerical eigenvalues 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): GRID E_n / ħ²/mL²
Horizontal axis (x): gp: Interior grid points N
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| gp: Interior grid points N | GRID E_n / ħ²/mL² |
|---|---|
| 12 | 295.4983164409161 |
| 24 | 431.86437851565785 |
| 36 | 464.53630367897847 |
| 48 | 476.805893895036 |
| 60 | 482.6686181267924 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Enumerate the permitted settings in the stated range, evaluate Y_out at each, and retain every setting satisfying |Y_out − Y_measured| ≤ u. The plotted predictions are supplied calibration evidence.
Reveal the worked solution
- Endpoint predictions are Y_out(12) = 295.498 ħ²/mL² and Y_out(60) = 482.669 ħ²/mL². The target lies between them. On the selected branch the response is strictly increasing.
- Endpoint interpolation gives gp_line = 12 + (470.897 − 295.498)(60 − 12)/(482.669 − 295.498) = 56.981. This estimates the inverse of the endpoint chord, rather than assuming that chord is the physical model.
- Evaluate the model at each permitted setting. The central reading corresponds to gp = 41; only the allowed settings 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60 lie inside the stated reading bound.
- The interpolation error in the input is |gp_line − gp| = 15.981.
- The compatible allowed values span 32 to 60. Values between specified test settings are outside this exercise's input domain even if an interpolation passes through them.
- The quoted output bound restricts which specified settings remain compatible. It does not change physical restrictions on quantum numbers or counts, and the chosen grid alone does not establish physical quantisation.
Answer Compatible gp: 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60. Endpoint interpolation predicts 56.981; check admissibility against the discrete settings.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.