Hydrogen separation: spatial states and quantum numbers

Science · Quantum & Modern Physics · ages 20-22

Name ______________________ Date ____________

In hydrogen's spin-free, nonrelativistic Coulomb model, which quantity depends on n alone?
- Orbital angular-momentum projection
- Energy
- The complete radial probability distribution
With spin omitted, what symmetry permits hydrogen's spatial stationary-state basis orbitals to factor into radial and angular parts?
- Spherical symmetry
- Radial shifts
- Mass exchange
In a single-photon emission transition in hydrogen, the photon's energy equals the initial atomic energy minus the final atomic energy.
Circle one: True False
Why is a rectangular box a useful simpler comparison when learning hydrogen separation?
- It practises coordinate separation
- It duplicates hydrogen's spectrum
- It imposes the same rectangular boundaries on hydrogen
For a separated spatial hydrogen basis state, l determines orbital angular-momentum magnitude. What does m determine?
- Energy level
- Radial distribution
- Axis projection
A hydrogen atom emits one photon as it drops from n = 5 to n = 3. Using the stated rounded energies, -0.54 eV initially and -1.51 eV finally, what is the photon's energy?
- 2.05 eV
- 0.97 eV
- 1.51 eV
With n > 1, spin omitted and an axis chosen, a student labels a separated spatial hydrogen basis orbital using only n. Which labels are missing?
- l and m
- No labels beyond n
- Exact position and momentum
What does agreement between predicted and measured hydrogen photon energies provide for the basic model?
- Proof that all states factor
- Proof that spin never matters
- Support for predicted energy gaps

Answer key

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Hydrogen separation: spatial states and quantum numbers W1-mt_Yk2qHB0_v2-s1

Energy · The model gives E_n ≈ -(13.6 eV)/n². The energy depends on n alone, while the radial distribution also depends on l.
Spherical symmetry · The potential depends only on radius. This spherical symmetry permits a separated basis, although arbitrary superpositions need not factor.
True · Energy conservation gives E_photon = E_initial - E_final. The spectral line corresponds to the transition.
It practises coordinate separation · The box problem gives you practice with product factors, separate equations and boundary conditions. Hydrogen uses different coordinates and has a different spectrum.
Axis projection · m determines orbital angular momentum's projection on a chosen axis. It does not specify a definite classical vector tilt.
0.97 eV · Subtract final from initial: -0.54 - (-1.51) = 0.97 eV. Adding the energy magnitudes gives 2.05 eV; using only the final magnitude gives 1.51 eV.
l and m · For n > 1, n identifies energy but does not fix l and m. Those labels specify the basis orbital's angular dependence.
Support for predicted energy gaps · Agreement supports the model's predicted energy differences. It does not prove the entire treatment or remove the need for finer corrections.