University Physics V · unit UPV-05
Particle in a Box
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.
Hilbert space and operator domain: infer an input with uncertainty
The box's state space is L²([0,L]) with ⟨f|g⟩ = ∫₀ᴸ f*(x)g(x) dx, and an operator on it is a differential expression together with a domain — the boundary conditions that kill the surface term from integration by parts and make the operator self-adjoint. Use the supplied topic model Y_out = cos(pi*th), where Y_out is Re ψ(L) / ψ(0) (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; unmarked trigonometric functions use radians. th is Extension phase θ / π, dimensionless. Keep n (Branch index n) = 0 fixed. For this exercise use only 1.0525 ≤ th ≤ 1.7875 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 = 0.0026703506, with a bounded reading uncertainty ±0.1174. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown th by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0000735, 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.
Re ψ(L) / ψ(0) against th
Predictions from the supplied Hilbert space and operator domain 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): Re ψ(L) / ψ(0)
Horizontal axis (x): th: Extension phase θ / π
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| th: Extension phase θ / π | Re ψ(L) / ψ(0) |
|---|---|
| 1.0525 | -0.9864292571764955 |
| 1.23625 | -0.7369824847979508 |
| 1.42 | -0.2486898871648553 |
| 1.6037499999999998 | 0.32019966162731134 |
| 1.7874999999999999 | 0.7853169308807446 |
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(th) = 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(1.0525) = -0.986429 and Y_out(1.7875) = 0.785317. The target lies between them. On the selected branch the response is strictly increasing.
- Endpoint interpolation gives th_line = 1.0525 + (0.00267035 − -0.986429)(1.7875 − 1.0525)/(0.785317 − -0.986429) = 1.46282. 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 th = 1.50085; substituting back gives Y_out = 0.0026703506.
- The interpolation error in the input is |th_line − th| = 0.038027. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = -0.11472965 and 0.12007035. Sort the roots even if the response decreases: 1.4634 ≤ th ≤ 1.53831.
- The lower and upper input deviations are 0.0374502 and 0.037462. 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 th ≈ 1.50085; compatible interval [1.4634, 1.53831]. Endpoint interpolation gives 1.46282.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
The infinite-wall limit: infer an input with uncertainty
A Dirichlet boundary condition sets the wavefunction itself to zero at a wall, and it is what the finite well's matching condition ψ′(0) = κψ(0) becomes once the barrier is high enough that the decay length 1/κ is negligible against the width. Use the supplied topic model Y_out = 0.0381*(pi/(w+2*d))*(pi/(w+2*d))/max(sin(pi*d/(w+2*d))*sin(pi*d/(w+2*d)),0.0001), where Y_out is BARRIER DEPTH V₀ (eV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; max(a,b) selects the larger value; unmarked trigonometric functions use radians. d is Extrapolation length δₑ, measured in nm. Keep w (Well width L) = 1 nm fixed. For this exercise use only 0.04975 nm ≤ d ≤ 0.27025 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 = 1.1916343 eV, with a bounded reading uncertainty ±0.092067 eV. 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 0.00002205 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.
BARRIER DEPTH V₀ against d
Predictions from the supplied The infinite-wall limit 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): BARRIER DEPTH V₀ / eV
Horizontal axis (x): d: Extrapolation length δₑ / nm
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| d: Extrapolation length δₑ / nm | BARRIER DEPTH V₀ / eV |
|---|---|
| 0.049749999999999996 | 15.49765533902073 |
| 0.104875 | 3.5509577665029566 |
| 0.16 | 1.562354111168292 |
| 0.21512499999999998 | 0.887382634827974 |
| 0.27025 | 0.5778551797077442 |
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(0.04975) = 15.4977 eV and Y_out(0.27025) = 0.577855 eV. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives d_line = 0.04975 + (1.19163 − 15.4977)(0.27025 − 0.04975)/(0.577855 − 15.4977) = 0.261179 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 d = 0.184255 nm; substituting back gives Y_out = 1.1916343 eV.
- The interpolation error in the input is |d_line − d| = 0.0769239 nm. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 1.0995673 eV and 1.2837013 eV. Sort the roots even if the response decreases: 0.177228 nm ≤ d ≤ 0.19218 nm.
- The lower and upper input deviations are 0.00702744 nm and 0.00792459 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 d ≈ 0.184255 nm; compatible interval [0.177228, 0.19218] nm. Endpoint interpolation gives 0.261179 nm.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
Why n must be a positive integer: infer an input with uncertainty
Quantisation here means that the boundary-value problem −(ℏ²/2m)ψ″ = Eψ with ψ(0) = ψ(L) = 0 admits a solution other than ψ ≡ 0 only at the isolated energies Eₙ = n²π²ℏ²/(2mL²), with n = 1, 2, 3, … Use the supplied topic model Y_out = 0.37603*q^2/max(w^2,0.01), where Y_out is TRIAL E, ELECTRON (eV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; max(a,b) selects the larger value. w is Box length L, measured in nm. Keep q (Trial wavenumber kL/π) = 1.63 fixed. For this exercise use only 0.74875 nm ≤ w ≤ 1.85125 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 = 0.4945856 eV, with a bounded reading uncertainty ±0.03046 eV. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown w by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00011025 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.
TRIAL E, ELECTRON against w
Predictions from the supplied Why n must be a positive integer 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): TRIAL E, ELECTRON / eV
Horizontal axis (x): w: Box length L / nm
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| w: Box length L / nm | TRIAL E, ELECTRON / eV |
|---|---|
| 0.74875 | 1.7820670189882408 |
| 1.024375 | 0.9520938539809648 |
| 1.2999999999999998 | 0.5911681106508877 |
| 1.5756249999999998 | 0.40243150930360305 |
| 1.8512499999999998 | 0.2915194664626571 |
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(w) = 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(0.74875) = 1.78207 eV and Y_out(1.85125) = 0.291519 eV. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives w_line = 0.74875 + (0.494586 − 1.78207)(1.85125 − 0.74875)/(0.291519 − 1.78207) = 1.70105 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 w = 1.42128 nm; substituting back gives Y_out = 0.4945856 eV.
- The interpolation error in the input is |w_line − w| = 0.279775 nm. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 0.4641256 eV and 0.5250456 eV. Sort the roots even if the response decreases: 1.37943 nm ≤ w ≤ 1.46717 nm.
- The lower and upper input deviations are 0.0418429 nm and 0.0458972 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 w ≈ 1.42128 nm; compatible interval [1.37943, 1.46717] nm. Endpoint interpolation gives 1.70105 nm.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
The sine eigenbasis: orthonormality and completeness: infer an input with uncertainty
The sine eigenbasis is the orthonormal set of kets |n⟩ with ⟨x|n⟩ = √(2/L) sin(nπx/L), n = 1, 2, 3, …, whose overlaps obey ⟨m|n⟩ = δₘₙ and whose projectors sum to the identity on L²([0,L]), so every state in the box is one series Σcₙ|n⟩ with cₙ = ⟨n|ψ⟩. Use the supplied topic model Y_out = abs(1.27323954*(clamp(m,0,1)*sin(pi*u)+clamp(m-1,0,1)*sin(3*pi*u)/3+clamp(m-2,0,1)*sin(5*pi*u)/5+clamp(m-3,0,1)*sin(7*pi*u)/7+clamp(m-4,0,1)*sin(9*pi*u)/9)-1), where Y_out is POINTWISE ERROR (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; abs(x) means |x|; clamp(x,a,b) confines x to [a,b]; unmarked trigonometric functions use radians. u is Probe position x₀/L, dimensionless. Keep m (Odd harmonics kept M) = 3 fixed. For this exercise use only 0.3488 ≤ u ≤ 0.3992 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 = 0.076321988, with a bounded reading uncertainty ±0.0054276. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown u by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00000504, 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.
POINTWISE ERROR against u
Predictions from the supplied The sine eigenbasis: orthonormality and completeness 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): POINTWISE ERROR
Horizontal axis (x): u: Probe position x₀/L
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| u: Probe position x₀/L | POINTWISE ERROR |
|---|---|
| 0.34880000000000005 | 0.11279545987090633 |
| 0.36140000000000005 | 0.10163915627219067 |
| 0.374 | 0.08506268033144937 |
| 0.3866 | 0.0641424636427208 |
| 0.3992 | 0.04013771960536705 |
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(u) = 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(0.3488) = 0.112795 and Y_out(0.3992) = 0.0401377. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives u_line = 0.3488 + (0.076322 − 0.112795)(0.3992 − 0.3488)/(0.0401377 − 0.112795) = 0.3741. 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 u = 0.379544; substituting back gives Y_out = 0.076321988.
- The interpolation error in the input is |u_line − u| = 0.0054437. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 0.070894388 and 0.081749588. Sort the roots even if the response decreases: 0.376161 ≤ u ≤ 0.382767.
- The lower and upper input deviations are 0.00338251 and 0.00322267. 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 u ≈ 0.379544; compatible interval [0.376161, 0.382767]. Endpoint interpolation gives 0.3741.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.