Separate relative motion from the fixed-nucleus spectral approximation
A one-electron atom is a two-body problem before the heavy nucleus is approximated as fixed. With electron mass m and nuclear mass M, transform to center-of-mass and relative coordinates. The kinetic energy separates into center motion plus relative kinetic energy with reduced mass mu=mM/(m+M). The Coulomb potential depends only on the separation, so the relative-coordinate spectrum uses mu rather than m.
Learning outcome: Derive the center/relative kinetic separation, calculate two finite-mass line positions and freeze the predicted difference. Add a common wavelength offset to both indications and compare the behavior of difference and ratio. Retain one unresolved multiplicative calibration alternative.
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Review finite sums, exponential and logarithmic functions, radial integration measures, first-order coupled equations and mass-energy units. Keep state counts, occupations, calibration records and unused tests distinct.
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Let M/m=2000 and n=3. The reduced-mass ratio is 2000/2001; the wavelength is lambdaInfinity×1.0005. At doubled mass it is lambdaInfinity×1.00025, so heavy-minus-light shift is −lambdaInfinity/4000. This is a small differential correction, not a factor-two wavelength change.
Open the experimentPredict
What happens to the wavelength excess above the infinite-mass value when nuclear mass doubles?
Experiment
Derive the center/relative kinetic separation, calculate two finite-mass line positions and freeze the predicted difference. Add a common wavelength offset to both indications and compare the behavior of difference and ratio. Retain one unresolved multiplicative calibration alternative.
Separate a mass-dependent line from a wavelength offset
Worked teaching model · synthetic values · Nonrelativistic one-electron Z=1 Coulomb model with supplied infinite-mass binding coefficient 13.6057 eV and hc=1239.841984 eV nm. Nuclear/electron relative coordinate uses reduced mass. Fine structure, recoil beyond this separation, nuclear size and actual isotope identification are omitted.
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Reduced mass changes a relative-coordinate spectrum
Graph axes rescale independently; read units and values, not screen angles. 1: same transition with finite mass (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Reduced mass changes a relative-coordinate spectrum
same transition with finite mass
| nuclear to electron mass ratio | transition wavelength (nm) |
|---|---|
| 1000 | 656.7681 |
| 1116.667 | 656.6995 |
| 1233.333 | 656.6439 |
| 1350 | 656.598 |
| 1466.667 | 656.5593 |
| 1583.333 | 656.5263 |
| 1700 | 656.4979 |
| 1816.667 | 656.4731 |
| 1933.333 | 656.4513 |
| 2050 | 656.432 |
| 2166.667 | 656.4148 |
| 2283.333 | 656.3993 |
| 2400 | 656.3853 |
| 2516.667 | 656.3727 |
| 2633.333 | 656.3611 |
| 2750 | 656.3505 |
| 2866.667 | 656.3408 |
| 2983.333 | 656.3319 |
| 3100 | 656.3236 |
| 3216.667 | 656.3159 |
| 3333.333 | 656.3088 |
| 3450 | 656.3021 |
| 3566.667 | 656.2959 |
| 3683.333 | 656.2901 |
| 3800 | 656.2846 |
| 3916.667 | 656.2795 |
| 4033.333 | 656.2746 |
| 4150 | 656.27 |
| 4266.667 | 656.2657 |
| 4383.333 | 656.2616 |
| 4500 | 656.2577 |
| 4616.667 | 656.2541 |
| 4733.333 | 656.2506 |
| 4850 | 656.2472 |
| 4966.667 | 656.244 |
| 5083.333 | 656.241 |
| 5200 | 656.2381 |
| 5316.667 | 656.2354 |
| 5433.333 | 656.2327 |
| 5550 | 656.2302 |
| 5666.667 | 656.2277 |
| 5783.333 | 656.2254 |
| 5900 | 656.2232 |
| 6016.667 | 656.221 |
| 6133.333 | 656.2189 |
| 6250 | 656.2169 |
| 6366.667 | 656.215 |
| 6483.333 | 656.2131 |
| 6600 | 656.2114 |
| 6716.667 | 656.2096 |
| 6833.333 | 656.208 |
| 6950 | 656.2063 |
| 7066.667 | 656.2048 |
| 7183.333 | 656.2033 |
| 7300 | 656.2018 |
| 7416.667 | 656.2004 |
| 7533.333 | 656.199 |
| 7650 | 656.1977 |
| 7766.667 | 656.1964 |
| 7883.333 | 656.1952 |
| 8000 | 656.194 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
| Quantity | Value | Unit |
|---|---|---|
| Reduced mass divided by electron mass | 0.9995002 | ratio |
| Transition photon energy | 1.888736 | eV |
| Transition wavelength | 656.44 | nm |
| Doubled-nuclear-mass wavelength | 656.276 | nm |
| Heavy minus light wavelength shift | -0.164028 | nm |
| Infinite-nuclear-mass wavelength | 656.1119 | nm |
| Relative wavelength excess over infinite mass | 0.0005 | ratio |
Finite nuclear motion changes the spectral scale; a common additive wavelength offset cancels from a measured difference but not a ratio.
Separate center-of-mass and relative kinetic energies: reduced mass mu=mM/(m+M), so the Coulomb binding coefficient is multiplied by mu/m. For transition n to 2, photon energy is 13.6057(mu/m)(1/4−1/n²). Wavelength equals the infinite-mass value times(1+m/M). Doubling M halves that excess; it does not halve the total wavelength. Multi-electron screening and fine-structure shifts are not described by this model.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
Finite nuclear motion changes the spectral scale; a common additive wavelength offset cancels from a measured difference but not a ratio.
Separate center-of-mass and relative kinetic energies: reduced mass mu=mM/(m+M), so the Coulomb binding coefficient is multiplied by mu/m. For transition n to 2, photon energy is 13.6057(mu/m)(1/4−1/n²). Wavelength equals the infinite-mass value times(1+m/M). Doubling M halves that excess; it does not halve the total wavelength. Multi-electron screening and fine-structure shifts are not described by this model.
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