University Physics V · unit UPV-01
Mathematical Foundations of Quantum Physics
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.
Extra challenging questions
A selection from this pathway. Open a topic below for its complete practice and challenge set.
Complex amplitudes and the phase that survives: infer an input with uncertainty
A complex amplitude z = r e^(iφ) carries a modulus, which fixes the probability through |z|² = z*z = r², and a phase, which is measurable only as a difference — when two amplitudes for the same outcome are added before the modulus is squared. Use the supplied topic model Y_out = 2*r2/max(1 + r2*r2, 0.01), where Y_out is VISIBILITY (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; max(a,b) selects the larger value. r2 is Modulus r₂ of z₂, dimensionless. Keep d (Relative phase δ) = 60 °; a (Global phase α) = 0 ° fixed. For this exercise use only 0.10625 ≤ r2 ≤ 0.89375 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 = 0.87287016, with a bounded reading uncertainty ±0.018128. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown r2 by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00007875, 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.
VISIBILITY against r2
Predictions from the supplied Complex amplitudes and the phase that survives 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): VISIBILITY
Horizontal axis (x): r2: Modulus r₂ of z₂
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| r2: Modulus r₂ of z₂ | VISIBILITY |
|---|---|
| 0.10625 | 0.21012785352852564 |
| 0.303125 | 0.5552325841390228 |
| 0.5 | 0.8 |
| 0.6968749999999999 | 0.93815117433231 |
| 0.8937499999999999 | 0.9937240765271774 |
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(r2) = 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(0.10625) = 0.210128 and Y_out(0.89375) = 0.993724. The target lies between them. On the selected branch the response is strictly increasing.
- Endpoint interpolation gives r2_line = 0.10625 + (0.87287 − 0.210128)(0.89375 − 0.10625)/(0.993724 − 0.210128) = 0.772294. 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 r2 = 0.586625; substituting back gives Y_out = 0.87287016.
- The interpolation error in the input is |r2_line − r2| = 0.185669. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 0.85474216 and 0.89099816. Sort the roots even if the response decreases: 0.562681 ≤ r2 ≤ 0.612788.
- The lower and upper input deviations are 0.023944 and 0.0261631. 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 r2 ≈ 0.586625; compatible interval [0.562681, 0.612788]. Endpoint interpolation gives 0.772294.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
Probability distributions, moments and expectation values: infer an input with uncertainty
A probability distribution assigns each outcome its long-run share over identical trials — dimensionless Pₙ for discrete outcomes, a density ρ(x) with the inverse unit of x for continuous ones — and moments ⟨xᵏ⟩ = ∫xᵏρ dx compress that weight into location, spread and shape. Use the supplied topic model Y_out = ((1-w)*exp(-2*(s*w/sig)^2)+w*exp(-2*(s*(1-w)/sig)^2))/(sig*sqrt(2*pi)), where Y_out is DENSITY AT MEAN (nm⁻¹). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x; exp(x) means eˣ. s is Hump position ±s, measured in nm. Keep w (Right-hump weight w) = 0.5; sig (Hump width σₕ) = 0.35 nm fixed. For this exercise use only 0.2125 nm ≤ s ≤ 1.7875 nm 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 = 0.0041379811 nm⁻¹, with a bounded reading uncertainty ±0.00062033 nm⁻¹. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown s by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0001575 nm, 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.
DENSITY AT MEAN against s
Predictions from the supplied Probability distributions, moments and expectation values 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): DENSITY AT MEAN / nm⁻¹
Horizontal axis (x): s: Hump position ±s / nm
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| s: Hump position ±s / nm | DENSITY AT MEAN / nm⁻¹ |
|---|---|
| 0.2125 | 0.9479745404095744 |
| 0.60625 | 0.2542910383900601 |
| 1 | 0.019240284214943552 |
| 1.3937499999999998 | 0.0004106179753637183 |
| 1.7874999999999999 | 0.0000024717837787474024 |
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(s) = 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(0.2125) = 0.947975 nm⁻¹ and Y_out(1.7875) = 0.00000247178 nm⁻¹. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives s_line = 0.2125 + (0.00413798 − 0.947975)(1.7875 − 0.2125)/(0.00000247178 − 0.947975) = 1.78063 nm. 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 s = 1.17325 nm; substituting back gives Y_out = 0.0041379811 nm⁻¹.
- The interpolation error in the input is |s_line − s| = 0.607379 nm. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 0.0035176511 nm⁻¹ and 0.0047583111 nm⁻¹. Sort the roots even if the response decreases: 1.15857 nm ≤ s ≤ 1.19009 nm.
- The lower and upper input deviations are 0.0146764 nm and 0.016837 nm. 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 s ≈ 1.17325 nm; compatible interval [1.15857, 1.19009] nm. Endpoint interpolation gives 1.78063 nm.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
Vector spaces over C and their inner products: infer an input with uncertainty
A complex inner-product space is a vector space over C carrying a map ⟨φ|ψ⟩ that is linear in its second argument, conjugate-linear in its first, conjugate-symmetric, and positive-definite — the minimal structure that gives each state a length and each pair an overlap. Use the supplied topic model Y_out = (a*a+b*b-2*a*b*cosd(delta))/2, where Y_out is HEADROOM (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sind/cosd/tand use degrees. a is Amplitude a on |0⟩, dimensionless. Keep b (Amplitude b on |1⟩) = 0.6; delta (Relative phase δ) = 60 ° fixed. For this exercise use only 0.338125 ≤ a ≤ 1.36187 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 = 0.35452766, with a bounded reading uncertainty ±0.03282. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown a by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000102375, 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.
HEADROOM against a
Predictions from the supplied Vector spaces over C and their inner products 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): HEADROOM
Horizontal axis (x): a: Amplitude a on |0⟩
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| a: Amplitude a on |0⟩ | HEADROOM |
|---|---|
| 0.338125 | 0.1357267578125 |
| 0.5940624999999999 | 0.17823637695312497 |
| 0.85 | 0.28624999999999995 |
| 1.1059375 | 0.4597676269531249 |
| 1.361875 | 0.6987892578124997 |
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(a) = 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(0.338125) = 0.135727 and Y_out(1.36187) = 0.698789. The target lies between them. On the selected branch the response is strictly increasing.
- Endpoint interpolation gives a_line = 0.338125 + (0.354528 − 0.135727)(1.36187 − 0.338125)/(0.698789 − 0.135727) = 0.735945. 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 a = 0.962612; substituting back gives Y_out = 0.35452766.
- The interpolation error in the input is |a_line − a| = 0.226668. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 0.32170766 and 0.38734766. Sort the roots even if the response decreases: 0.911077 ≤ a ≤ 1.01042.
- The lower and upper input deviations are 0.0515353 and 0.0478066. 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 a ≈ 0.962612; compatible interval [0.911077, 1.01042]. Endpoint interpolation gives 0.735945.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
Orthogonality, normalisation and basis expansion: infer an input with uncertainty
An orthonormal basis satisfies ⟨m|n⟩ = δₘₙ, so any state expands as |ψ⟩ = Σ cₙ|n⟩ with cₙ = ⟨n|ψ⟩ read off by projection, and normalising to ⟨ψ|ψ⟩ = 1 makes each |cₙ|² a measurement probability. Use the supplied topic model Y_out = cosd(phi-theta), where Y_out is NAIVE c₂ = ⟨v₂|ψ⟩ (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sind/cosd/tand use degrees. theta is Basis angle θ between v₁, v₂, measured in °. Keep phi (State angle φ) = 35 ° fixed. For this exercise use only 36.625 ° ≤ theta ≤ 82.5625 ° 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 = 0.86909053, with a bounded reading uncertainty ±0.019576. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown theta by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00459375 °, 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.
NAIVE c₂ = ⟨v₂|ψ⟩ against theta
Predictions from the supplied Orthogonality, normalisation and basis expansion 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): NAIVE c₂ = ⟨v₂|ψ⟩
Horizontal axis (x): theta: Basis angle θ between v₁, v₂ / °
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| theta: Basis angle θ between v₁, v₂ / ° | NAIVE c₂ = ⟨v₂|ψ⟩ |
|---|---|
| 36.625 | 0.9995978367717061 |
| 48.109375 | 0.9739388685076236 |
| 59.59375 | 0.9092815128276606 |
| 71.078125 | 0.8082147747378063 |
| 82.5625 | 0.6747855610145886 |
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(theta) = 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(36.625) = 0.999598 and Y_out(82.5625) = 0.674786. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives theta_line = 36.625 + (0.869091 − 0.999598)(82.5625 − 36.625)/(0.674786 − 0.999598) = 55.0824 °. 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 theta = 64.6469 °; substituting back gives Y_out = 0.86909053.
- The interpolation error in the input is |theta_line − theta| = 9.56451 °. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 0.84951453 and 0.88866653. Sort the roots even if the response decreases: 62.2938 ° ≤ theta ≤ 66.8411 °.
- The lower and upper input deviations are 2.35303 ° and 2.19422 °. 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 theta ≈ 64.6469 °; compatible interval [62.2938, 66.8411] °. Endpoint interpolation gives 55.0824 °.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.