--- title: "Separating the Schrödinger Equation for Hydrogen" description: "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 des" canonical: https://lightmysky.com/learn/science/separating-the-schrodinger-equation-for-hydrogen-mt_Yk2qHB0_v2 source: https://lightmysky.com/learn/science/separating-the-schrodinger-equation-for-hydrogen-mt_Yk2qHB0_v2.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # 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. Subject: Science · Area: Quantum & Modern Physics · Ages 20 to 22 Page: https://lightmysky.com/learn/science/separating-the-schrodinger-equation-for-hydrogen-mt_Yk2qHB0_v2 ## Ready when they can - 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 ## Lesson: Hydrogen separation: spatial states and quantum numbers 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. **Example.** 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. **Recap.** 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.* ## Practice 8 questions on this page, each with its working shown. ## Needs first - [Operators, Eigenvalues and Measurement](https://lightmysky.com/learn/science/operators-eigenvalues-and-measurement-mt_E9C_9DlwD2) - [Energy Levels and Line Spectra](https://lightmysky.com/learn/science/energy-levels-and-line-spectra-mt_RlMCSDxE5q) ## Optional connections - [Triple Integrals and Coordinates for Solids](https://lightmysky.com/learn/mathematics/triple-integrals-and-coordinates-for-solids-mt_RFeK8LD_jT) ## Opens up - [Orbital Angular Momentum and the Quantum Numbers](https://lightmysky.com/learn/science/orbital-angular-momentum-and-the-quantum-numbers-mt_OXlbat0vat) - [Stimulated Emission and How a Laser Works](https://lightmysky.com/learn/science/stimulated-emission-and-how-a-laser-works-mt_xgjD0mzEAV)