University Quantum Mechanics I · topic UQMI-11.03
Operators as matrices: A_mn = <m|A|n> and Hermiticity
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.
Operators as matrices: Aₘₙ = ⟨m|Â|n⟩ and Hermiticity: predict a finite change
In an orthonormal basis an operator becomes the table Aₘₙ = ⟨m|Â|n⟩, whose column n lists the components of Â|n⟩; it can represent an observable only if the table equals its own conjugate transpose, Aₘₙ = Aₙₘ*. Use the supplied topic model Y_out = 1+sqrt(d*d+v*v), where Y_out is LARGER EIGENVALUE a₀ + |a⃗| (meV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x. d is Half-splitting a₃ = (A₁₁ − A₂₂)/2, measured in meV. Keep v (Coupling size |A₁₂|) = 2 meV; beta (Coupling phase β = arg A₁₂) = 60 ° fixed. For this exercise use only -2.3625 meV ≤ d ≤ -0.1575 meV 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 d = -1.8774 meV and -1.5687 meV. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δd across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
LARGER EIGENVALUE a₀ + |a⃗| against d
Predictions from the supplied Operators as matrices: Aₘₙ = ⟨m|Â|n⟩ and Hermiticity 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): LARGER EIGENVALUE a₀ + |a⃗| / meV
Horizontal axis (x): d: Half-splitting a₃ = (A₁₁ − A₂₂)/2 / meV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| d: Half-splitting a₃ = (A₁₁ − A₂₂)/2 / meV | LARGER EIGENVALUE a₀ + |a⃗| / meV |
|---|---|
| -2.3625 | 4.09538466914857 |
| -1.8112499999999998 | 3.6982636199044747 |
| -1.2599999999999998 | 3.363810483097154 |
| -0.7087499999999998 | 3.1218686487386536 |
| -0.15749999999999975 | 3.0061919773541117 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of d separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At d = -1.8774 meV, substitute into Y_out = 1+sqrt(d*d+v*v): Y₀ = 3.74311 meV.
- At d = -1.5687 meV, the same fixed-parameter model gives Y₁ = 3.54181 meV.
- Signed change: ΔY_out = Y₁ − Y₀ = -0.201292 meV; the input change is Δd = 0.3087 meV.
- Average sensitivity is ΔY_out/Δd = -0.652063 meV per meV. 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 = 3.74311 meV, Y₁ = 3.54181 meV; ΔY_out = -0.201292 meV; average sensitivity = -0.652063 meV per meV.
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.
Operators as matrices: Aₘₙ = ⟨m|Â|n⟩ and Hermiticity: infer an input with uncertainty
In an orthonormal basis an operator becomes the table Aₘₙ = ⟨m|Â|n⟩, whose column n lists the components of Â|n⟩; it can represent an observable only if the table equals its own conjugate transpose, Aₘₙ = Aₙₘ*. Use the supplied topic model Y_out = 1+sqrt(d*d+v*v), where Y_out is LARGER EIGENVALUE a₀ + |a⃗| (meV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x. d is Half-splitting a₃ = (A₁₁ − A₂₂)/2, measured in meV. Keep v (Coupling size |A₁₂|) = 2 meV; beta (Coupling phase β = arg A₁₂) = 60 ° fixed. For this exercise use only -2.3625 meV ≤ d ≤ -0.1575 meV 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 = 3.2439261 meV, with a bounded reading uncertainty ±0.03566 meV. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown d by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0002205 meV, 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.
LARGER EIGENVALUE a₀ + |a⃗| against d
Predictions from the supplied Operators as matrices: Aₘₙ = ⟨m|Â|n⟩ and Hermiticity 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): LARGER EIGENVALUE a₀ + |a⃗| / meV
Horizontal axis (x): d: Half-splitting a₃ = (A₁₁ − A₂₂)/2 / meV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| d: Half-splitting a₃ = (A₁₁ − A₂₂)/2 / meV | LARGER EIGENVALUE a₀ + |a⃗| / meV |
|---|---|
| -2.3625 | 4.09538466914857 |
| -1.8112499999999998 | 3.6982636199044747 |
| -1.2599999999999998 | 3.363810483097154 |
| -0.7087499999999998 | 3.1218686487386536 |
| -0.15749999999999975 | 3.0061919773541117 |
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(d) = 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(-2.3625) = 4.09538 meV and Y_out(-0.1575) = 3.00619 meV. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives d_line = -2.3625 + (3.24393 − 4.09538)(-0.1575 − -2.3625)/(3.00619 − 4.09538) = -0.638777 meV. 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 d = -1.01745 meV; substituting back gives Y_out = 3.2439261 meV.
- The interpolation error in the input is |d_line − d| = 0.378673 meV. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 3.2082661 meV and 3.2795861 meV. Sort the roots even if the response decreases: -1.09385 meV ≤ d ≤ -0.936183 meV.
- The lower and upper input deviations are 0.0764023 meV and 0.0812666 meV. 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 d ≈ -1.01745 meV; compatible interval [-1.09385, -0.936183] meV. Endpoint interpolation gives -0.638777 meV.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.