University Physics IV · unit UPIV-05
Foundations of Quantum Mechanics
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.
Probability amplitudes and superposition: infer an input with uncertainty
A probability amplitude is a complex number attached to one way an outcome can happen; amplitudes for alternatives that stay indistinguishable are added first, and the probability is the squared modulus of the sum. Use the supplied topic model Y_out = sqrt(max(1-(2*amp*ov/(1+amp*amp))^2,0)), where Y_out is DISTINGUISHABILITY D (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x; max(a,b) selects the larger value. ov is Marker overlap |⟨d₁|d₂⟩|, dimensionless. Keep phi (Relative phase φ) = 0 °; amp (Amplitude ratio |ψ₂|/|ψ₁|) = 1 fixed. For this exercise use only 0.10625 ≤ ov ≤ 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 = 0.8098587, with a bounded reading uncertainty ±0.027672. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown ov 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.
DISTINGUISHABILITY D against ov
Predictions from the supplied Probability amplitudes and superposition 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): DISTINGUISHABILITY D
Horizontal axis (x): ov: Marker overlap |⟨d₁|d₂⟩|
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| ov: Marker overlap |⟨d₁|d₂⟩| | DISTINGUISHABILITY D |
|---|---|
| 0.10625 | 0.9943394478245344 |
| 0.303125 | 0.9529508037537929 |
| 0.5 | 0.8660254037844386 |
| 0.6968749999999999 | 0.717192606190973 |
| 0.8937499999999999 | 0.44856542164995294 |
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(ov) = 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) = 0.994339 and Y_out(0.89375) = 0.448565. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives ov_line = 0.10625 + (0.809859 − 0.994339)(0.89375 − 0.10625)/(0.448565 − 0.994339) = 0.372438. 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 ov = 0.586625; substituting back gives Y_out = 0.8098587.
- The interpolation error in the input is |ov_line − ov| = 0.214187. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 0.7821867 and 0.8375307. Sort the roots even if the response decreases: 0.54639 ≤ ov ≤ 0.623044.
- The lower and upper input deviations are 0.0402347 and 0.0364191. 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 ov ≈ 0.586625; compatible interval [0.54639, 0.623044]. Endpoint interpolation gives 0.372438.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
The wave function as the state: infer an input with uncertainty
The wave function is a complex, square-integrable function of position and time that holds everything predictable about one spinless non-relativistic particle — with its overall phase holding nothing at all. Use the supplied topic model Y_out = 0.66490*exp(-d*d/2.88)*(1+sind(2*p)*cosd(t))/max(1+sind(2*p)*cosd(t)*exp(-d*d/2.88),0.05), where Y_out is |ψ(0)|² SUPERPOSITION (nm⁻¹). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; exp(x) means eˣ; max(a,b) selects the larger value; sind/cosd/tand use degrees. p is Mixing angle φ, measured in °. Keep t (Relative phase θ) = 0 °; d (Packet separation d) = 2 nm fixed. For this exercise use only 47.3625 ° ≤ p ≤ 80.4375 ° 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 = 0.24052035 nm⁻¹, with a bounded reading uncertainty ±0.0036925 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 p by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0033075 °, 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.
|ψ(0)|² SUPERPOSITION against p
Predictions from the supplied The wave function as the state 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): |ψ(0)|² SUPERPOSITION / nm⁻¹
Horizontal axis (x): p: Mixing angle φ / °
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| p: Mixing angle φ / ° | |ψ(0)|² SUPERPOSITION / nm⁻¹ |
|---|---|
| 47.3625 | 0.265137245125507 |
| 55.631249999999994 | 0.2599061470850576 |
| 63.9 | 0.24794552940328637 |
| 72.16875 | 0.22914242220302958 |
| 80.4375 | 0.20348936776290164 |
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(p) = 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(47.3625) = 0.265137 nm⁻¹ and Y_out(80.4375) = 0.203489 nm⁻¹. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives p_line = 47.3625 + (0.24052 − 0.265137)(80.4375 − 47.3625)/(0.203489 − 0.265137) = 60.5698 °. 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 p = 67.5383 °; substituting back gives Y_out = 0.24052035 nm⁻¹.
- The interpolation error in the input is |p_line − p| = 6.96842 °. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 0.23682785 nm⁻¹ and 0.24421285 nm⁻¹. Sort the roots even if the response decreases: 65.8102 ° ≤ p ≤ 69.1405 °.
- The lower and upper input deviations are 1.72801 ° and 1.60226 °. 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 p ≈ 67.5383 °; compatible interval [65.8102, 69.1405] °. Endpoint interpolation gives 60.5698 °.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
Born's rule and probability current: infer an input with uncertainty
Born's rule says the probability of finding the particle between x and x + dx at time t is |ψ(x,t)|² dx, which makes |ψ|² a probability per unit length and never, on its own, a probability. Use the supplied topic model Y_out = 1-(cosd(t)^2*(b/100-sind(3.6*b)/(2*pi))+sind(t)^2*(b/100-sind(7.2*b)/(4*pi))+2*cosd(t)*sind(t)*cosd(p)*(sind(1.8*b)/pi-sind(5.4*b)/(3*pi))), where Y_out is P RIGHT OF x_b (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sind/cosd/tand use degrees. b is Boundary x_b as a percentage of L, measured in %. Keep t (Mixing angle θ: c₁ = cos θ, c₂ = sin θ) = 45 °; p (Phase φ = (E₂ − E₁)t/ħ) = 60 ° fixed. For this exercise use only 14.5625 % ≤ b ≤ 85.4375 % 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 = 0.22386634, with a bounded reading uncertainty ±0.029709. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown b by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0070875 %, 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.
P RIGHT OF x_b against b
Predictions from the supplied Born's rule and probability current 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): P RIGHT OF x_b
Horizontal axis (x): b: Boundary x_b as a percentage of L / %
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| b: Boundary x_b as a percentage of L / % | P RIGHT OF x_b |
|---|---|
| 14.5625 | 0.9376172431600563 |
| 32.28125 | 0.5871711792955575 |
| 50 | 0.2877934092108061 |
| 67.71875 | 0.15307799413776624 |
| 85.4375 | 0.025808478757935793 |
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(b) = 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(14.5625) = 0.937617 and Y_out(85.4375) = 0.0258085. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives b_line = 14.5625 + (0.223866 − 0.937617)(85.4375 − 14.5625)/(0.0258085 − 0.937617) = 70.0424 %. 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 b = 57.7963 %; substituting back gives Y_out = 0.22386634.
- The interpolation error in the input is |b_line − b| = 12.2462 %. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 0.19415734 and 0.25357534. Sort the roots even if the response decreases: 53.8227 % ≤ b ≤ 62.0734 %.
- The lower and upper input deviations are 3.97355 % and 4.27716 %. 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 b ≈ 57.7963 %; compatible interval [53.8227, 62.0734] %. Endpoint interpolation gives 70.0424 %.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
Position and momentum representations: infer an input with uncertainty
The momentum representation of a state is the list of amplitudes Φ(p) got by projecting ψ onto the momentum eigenfunctions — the Fourier transform of ψ(x) with ħ setting the scale — whose squared modulus is the probability density per unit momentum. Use the supplied topic model Y_out = 0.5/max(s,0.01), where Y_out is σp MOMENTUM SPREAD (ħ/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. s is Packet width σ in x, measured in nm. Keep a (Centre shift a in x) = 0 nm; p0 (Boost phase p₀) = 0 ħ/nm fixed. For this exercise use only 0.5275 nm ≤ s ≤ 1.4725 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 = 0.45291906 ħ/nm, with a bounded reading uncertainty ±0.017004 ħ/nm. 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 0.0000945 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.
σp MOMENTUM SPREAD against s
Predictions from the supplied Position and momentum representations 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): σp MOMENTUM SPREAD / ħ/nm
Horizontal axis (x): s: Packet width σ in x / nm
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| s: Packet width σ in x / nm | σp MOMENTUM SPREAD / ħ/nm |
|---|---|
| 0.5275000000000001 | 0.947867298578199 |
| 0.7637500000000002 | 0.6546644844517183 |
| 1 | 0.5 |
| 1.23625 | 0.4044489383215369 |
| 1.4725000000000001 | 0.3395585738539898 |
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(0.5275) = 0.947867 ħ/nm and Y_out(1.4725) = 0.339559 ħ/nm. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives s_line = 0.5275 + (0.452919 − 0.947867)(1.4725 − 0.5275)/(0.339559 − 0.947867) = 1.2964 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 = 1.10395 nm; substituting back gives Y_out = 0.45291906 ħ/nm.
- The interpolation error in the input is |s_line − s| = 0.192446 nm. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 0.43591506 ħ/nm and 0.46992306 ħ/nm. Sort the roots even if the response decreases: 1.064 nm ≤ s ≤ 1.14701 nm.
- The lower and upper input deviations are 0.039946 nm and 0.0430624 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 ≈ 1.10395 nm; compatible interval [1.064, 1.14701] nm. Endpoint interpolation gives 1.2964 nm.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.