Separating the Schrödinger Equation for Hydrogen
A qualitative account of separation in the spin-free, nonrelativistic Coulomb model of hydrogen. Spherical symmetry permits separated spatial stationary-state basis orbitals. Their quantum numbers describe energy and orbital angular momentum, while energy differences predict photon energies.
What a learner can do afterwards
- Explains why spherical symmetry permits separated spatial stationary-state basis orbitals when spin is omitted
- Identifies n, l and m and distinguishes energy, orbital angular-momentum magnitude and axis projection
- Explains how single-photon transitions connect energy gaps to spectral lines and how experimental agreement supports the basic model
1 · Read
Use the nonrelativistic Coulomb model of isolated hydrogen, with spin and external fields omitted. Its potential depends only on radius. Spherical symmetry permits a basis of spatial stationary states with radial and angular factors. A basis is a set of states used to build other states by superposition. Arbitrary superpositions need not be separable.
A rectangular box permits spatial stationary-state basis functions X(x)Y(y)Z(z), with one factor per coordinate. Hydrogen instead uses R(r)Y(angles): R depends on radius and Y on angular coordinates. Both problems let you practise choosing product factors, solving their equations and applying boundary conditions. Their energy spectra are different.
For a chosen axis, a separated spatial bound state has labels n, l and m. The angular equation fixes l and m. Here l determines orbital angular-momentum magnitude, while m determines its projection on that axis, not a definite classical vector tilt. The radial equation also contains l; its bound-state conditions determine n and energy. All three labels specify a basis orbital when spin is omitted.
For bound states, n = 1, 2, 3, ...; l = 0, 1, ..., n - 1; and m = -l, ..., +l in integer steps. The number n gives a characteristic radial scale, but the radial distribution also depends on l.
In this spin-free, nonrelativistic Coulomb model, hydrogen's bound-state energy depends on n alone: E_n ≈ -(13.6 eV)/n². Here eV means electronvolt. Zero is the ionization threshold, a free electron at rest infinitely far from the proton. The radial equation contains l, but its allowed energies have no l dependence in this model.
In a single-photon transition, the photon's energy equals the positive difference between the two level energies. A downward transition emits a photon; absorption supplies the gap for an upward transition. For emission, subtract the final energy from the higher initial energy. A spectral line corresponds to the transition; it is not itself an energy.
For a downward transition from E4 to E2, the rounded model energies are -0.85 eV and -3.40 eV. Subtract final from initial: E_photon = -0.85 - (-3.40) = 2.55 eV. The leftward hop on the energy scale shows the energy lost by the atom.
To test the model, compare predicted photon energies with measured spectral lines, allowing for measurement uncertainty. Agreement supports the predicted energy differences; it does not prove every part of the theory. The basic model predicts the main hydrogen lines closely. Finer splittings require corrections involving spin and relativity.
With spin omitted, separated basis orbitals use n, l and m for energy, angular-momentum magnitude and axis projection. Photon energies equal transition gaps; spectral agreement supports the model.
The lesson, questions, worked solutions and flashcards were written for this dataset.
The three readings were written for this dataset. "Quantum numbers" is a Khan Academy video.
More to read
Hydrogen's symmetry and quantum numbers
Consider isolated hydrogen in the nonrelativistic Coulomb model, with spin and external fields omitted. Its potential depends only on distance from the proton. Spherical coordinates permit a basis of spatial stationary states formed from radial and angular factors. Other states can be superpositions of these basis orbitals and need not factor themselves.
Boundary conditions select n, l and m. The angular equations determine l and m. The radial equation contains l, and its bound-state conditions determine n and the allowed energy. In this model, energy depends on n alone. Orbital angular-momentum magnitude depends on l, while its projection on a chosen axis depends on m. That projection is not a definite classical vector tilt. The three labels specify a separated spatial basis orbital; they do not include spin.
Practising separation with a rectangular box
For a particle confined to a rectangular box, spatial stationary-state basis functions can be written as X(x)Y(y)Z(z). Each factor depends on one Cartesian coordinate. Substitution gives separate equations, and the boundary conditions restrict their solutions.
Hydrogen's central Coulomb potential instead permits a basis with a radial factor and an angular factor in spherical coordinates. This provides a useful comparison of separation methods: choose product factors, solve the factor equations and impose boundary conditions. The box and hydrogen have different potentials, boundaries and spectra. In either system, a general superposition need not retain a single product form. For hydrogen here, the model omits spin.
Testing hydrogen energies with light
Transitions reveal atomic energy levels through light. When an electron moves downward, the emitted photon's energy equals the difference between the initial and final levels. Absorption supplies the energy for an upward transition. Each spectral line therefore corresponds to a particular energy difference.
Compare the predicted photon energies with measured hydrogen lines to test the calculation. The separated equations predict the line colours closely. Small additional splittings require effects involving spin and relativity beyond the basic model.
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Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.