University Physics V · unit UPV-13
Atomic Spectroscopy
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.
Emission, absorption and term analysis: infer an input with uncertainty
Term analysis is the least-squares inversion that turns a list of measured line wavenumbers into a set of term values T_n = −E_n/hc, each line modelled as a difference σ = T_l − T_u of two eigenvalues of one Hermitian atomic Hamiltonian. Use the supplied topic model Y_out = t2+g, where Y_out is σ 1→3 = σ 1→2 + σ 2→3 (10³ cm⁻¹). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. g is Gap from level 2 up to level 3, measured in 10³ cm⁻¹. Keep t2 (Term of level 2 above ground) = 15.4 10³ cm⁻¹; c (Zero offset c added to every level) = 0 10³ cm⁻¹ fixed. For this exercise use only 6.16875 10³ cm⁻¹ ≤ g ≤ 14.8312 10³ cm⁻¹ 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 = 26.852875 10³ cm⁻¹, with a bounded reading uncertainty ±0.50676 10³ cm⁻¹. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown g by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00086625 10³ cm⁻¹, 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.
σ 1→3 = σ 1→2 + σ 2→3 against g
Predictions from the supplied Emission, absorption and term analysis 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): σ 1→3 = σ 1→2 + σ 2→3 / 10³ cm⁻¹
Horizontal axis (x): g: Gap from level 2 up to level 3 / 10³ cm⁻¹
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| g: Gap from level 2 up to level 3 / 10³ cm⁻¹ | σ 1→3 = σ 1→2 + σ 2→3 / 10³ cm⁻¹ |
|---|---|
| 6.16875 | 21.56875 |
| 8.334375 | 23.734375 |
| 10.5 | 25.9 |
| 12.665624999999999 | 28.065624999999997 |
| 14.831249999999997 | 30.231249999999996 |
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(g) = 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(6.16875) = 21.5688 10³ cm⁻¹ and Y_out(14.8312) = 30.2312 10³ cm⁻¹. The target lies between them. On the selected branch the response is strictly increasing.
- Endpoint interpolation gives g_line = 6.16875 + (26.8529 − 21.5688)(14.8312 − 6.16875)/(30.2312 − 21.5688) = 11.4529 10³ cm⁻¹. 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 g = 11.4529 10³ cm⁻¹; substituting back gives Y_out = 26.852875 10³ cm⁻¹.
- The interpolation error in the input is |g_line − g| = 0 10³ cm⁻¹. This is within the requested numerical tolerance.
- Invert both limits Y_out = 26.346115 10³ cm⁻¹ and 27.359635 10³ cm⁻¹. Sort the roots even if the response decreases: 10.9461 10³ cm⁻¹ ≤ g ≤ 11.9596 10³ cm⁻¹.
- The lower and upper input deviations are 0.50676 10³ cm⁻¹ and 0.50676 10³ cm⁻¹. They agree on this branch; this does not establish linearity for changes in other inputs or outside the interval. This interval includes reading uncertainty only, not model error or uncertainty in fixed inputs.
Answer g ≈ 11.4529 10³ cm⁻¹; compatible interval [10.9461, 11.9596] 10³ cm⁻¹. Endpoint interpolation gives 11.4529 10³ cm⁻¹.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
Atom–field coupling and the dipole matrix element: infer an input with uncertainty
The electric-dipole matrix element ⟨f|d⋅ε|i⟩ is the single complex number, with units of charge times length, through which the interaction −d⋅E lets light polarised along ε carry an atom from |i⟩ to |f⟩. Use the supplied topic model Y_out = sin(8.04*d*f*t/2)^2, where Y_out is EXACT P AT δ = 0 (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; unmarked trigonometric functions use radians. f is Field amplitude E₀, measured in 10⁵ V m⁻¹. Keep d (Matrix element ⟨f|d⋅ε|i⟩) = 0.75 e a₀; t (Drive duration t) = 0.55 ns fixed. For this exercise use only 0.46375 10⁵ V m⁻¹ ≤ f ≤ 0.93625 10⁵ V m⁻¹ 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 = 0.89874659, with a bounded reading uncertainty ±0.015138. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown f by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00004725 10⁵ V m⁻¹, 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.
EXACT P AT δ = 0 against f
Predictions from the supplied Atom–field coupling and the dipole matrix element 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): EXACT P AT δ = 0
Horizontal axis (x): f: Field amplitude E₀ / 10⁵ V m⁻¹
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| f: Field amplitude E₀ / 10⁵ V m⁻¹ | EXACT P AT δ = 0 |
|---|---|
| 0.46375 | 0.4836182063658465 |
| 0.5818749999999999 | 0.6756653870589105 |
| 0.7 | 0.8410950103363651 |
| 0.818125 | 0.954840484711103 |
| 0.9362499999999999 | 0.9996666180623222 |
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(f) = 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(0.46375) = 0.483618 and Y_out(0.93625) = 0.999667. The target lies between them. On the selected branch the response is strictly increasing.
- Endpoint interpolation gives f_line = 0.46375 + (0.898747 − 0.483618)(0.93625 − 0.46375)/(0.999667 − 0.483618) = 0.843846 10⁵ V m⁻¹. 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 f = 0.751975 10⁵ V m⁻¹; substituting back gives Y_out = 0.89874659.
- The interpolation error in the input is |f_line − f| = 0.0918714 10⁵ V m⁻¹. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 0.88360859 and 0.91388459. Sort the roots even if the response decreases: 0.73731 10⁵ V m⁻¹ ≤ f ≤ 0.767652 10⁵ V m⁻¹.
- The lower and upper input deviations are 0.0146654 10⁵ V m⁻¹ and 0.0156765 10⁵ V m⁻¹. 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 f ≈ 0.751975 10⁵ V m⁻¹; compatible interval [0.73731, 0.767652] 10⁵ V m⁻¹. Endpoint interpolation gives 0.843846 10⁵ V m⁻¹.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
Einstein coefficients and oscillator strengths: infer an input with uncertainty
The Einstein coefficients are the three rate constants of a transition — A₂₁ for spontaneous emission, B₁₂ and B₂₁ for absorption and stimulated emission — and detailed balance against the Planck spectrum leaves only one of them free. Use the supplied topic model Y_out = 100*(1-exp(0-2.4933*f*n/max(w,0.05))), where Y_out is CENTRE ABSORPTION (%). 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. n is Column density N, measured in ×10¹⁵ m⁻². Keep f (Oscillator strength f) = 0.64; w (Doppler FWHM) = 1.4 GHz fixed. For this exercise use only 1.15188 ×10¹⁵ m⁻² ≤ n ≤ 8.94812 ×10¹⁵ m⁻² 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 = 99.880956 %, with a bounded reading uncertainty ±0.017298 %. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown n by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000779625 ×10¹⁵ m⁻², 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.
CENTRE ABSORPTION against n
Predictions from the supplied Einstein coefficients and oscillator strengths 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): CENTRE ABSORPTION / %
Horizontal axis (x): n: Column density N / ×10¹⁵ m⁻²
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| n: Column density N / ×10¹⁵ m⁻² | CENTRE ABSORPTION / % |
|---|---|
| 1.1518750000000002 | 73.09614343310173 |
| 3.1009374999999997 | 97.08246423720476 |
| 5.05 | 99.68361357762878 |
| 6.999062499999999 | 99.96569009726038 |
| 8.948125 | 99.99627933013946 |
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(n) = 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(1.15188) = 73.0961 % and Y_out(8.94812) = 99.9963 %. The target lies between them. On the selected branch the response is strictly increasing.
- Endpoint interpolation gives n_line = 1.15188 + (99.881 − 73.0961)(8.94812 − 1.15188)/(99.9963 − 73.0961) = 8.9147 ×10¹⁵ m⁻². 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 n = 5.90759 ×10¹⁵ m⁻²; substituting back gives Y_out = 99.880956 %.
- The interpolation error in the input is |n_line − n| = 3.00712 ×10¹⁵ m⁻². This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 99.863658 % and 99.898254 %. Sort the roots even if the response decreases: 5.78855 ×10¹⁵ m⁻² ≤ n ≤ 6.04534 ×10¹⁵ m⁻².
- The lower and upper input deviations are 0.119033 ×10¹⁵ m⁻² and 0.137756 ×10¹⁵ m⁻². 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 n ≈ 5.90759 ×10¹⁵ m⁻²; compatible interval [5.78855, 6.04534] ×10¹⁵ m⁻². Endpoint interpolation gives 8.9147 ×10¹⁵ m⁻².
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
Selection rules: parity and Wigner-Eckart: infer an input with uncertainty
A selection rule is the statement that a transition's matrix element vanishes identically by symmetry — parity under inversion, the Wigner-Eckart 3-j factor under rotation — so the line is absent whatever the radial wavefunctions happen to be. Use the supplied topic model Y_out = max(0,1-u^2)*2*clamp(m,-j,j)*clamp(m,-j,j)/max(2*j*(j+1),0.01)+max(0,1-(u-1)^2)*2*(j-clamp(m,-j,j)+1)*(j+clamp(m,-j,j)+1)/((2*j+1)*(2*j+2)), where Y_out is π BRANCH Δm = 0 (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; clamp(x,a,b) confines x to [a,b]. j is Initial J, dimensionless. Keep u (ΔJ = J′ − J) = 1; m (Initial m_J) = 0 fixed. For this exercise use only 0 ≤ j ≤ 4, 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 = 0.6, with a bounded reading uncertainty ±0.035556. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. Determine every allowed value of j 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.
π BRANCH Δm = 0 against j
Predictions from the supplied Selection rules: parity and Wigner-Eckart 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): π BRANCH Δm = 0
Horizontal axis (x): j: Initial J
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| j: Initial J | π BRANCH Δm = 0 |
|---|---|
| 0 | 1 |
| 1 | 0.6666666666666666 |
| 2 | 0.6 |
| 3 | 0.5714285714285714 |
| 4 | 0.5555555555555556 |
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(0) = 1 and Y_out(4) = 0.555556. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives j_line = 0 + (0.6 − 1)(4 − 0)/(0.555556 − 1) = 3.6. 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 j = 2; only the allowed settings 2, 3 lie inside the stated reading bound.
- The interpolation error in the input is |j_line − j| = 1.6.
- The compatible allowed values span 2 to 3. 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 j: 2, 3. Endpoint interpolation predicts 3.6; check admissibility against the discrete settings.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.