Separating the Schrödinger Equation for Hydrogen
In a spherically symmetric potential the wavefunction splits into a radial part and an angular part, and each piece brings its own quantum number. The energy levels of hydrogen come out of the radial equation alone.
What a learner can do afterwards
- Explains why spherical symmetry allows the wavefunction to be separated
- Names the three quantum numbers the separation produces and what each restricts
- Recovers the hydrogen energy levels and compares them with the measured spectrum
1 · Read
The hydrogen atom has a spherically symmetric pull: the electron feels the proton equally from every direction. That symmetry lets the three-dimensional wave split into a radial part times an angular part, turning one hard problem into manageable one-dimensional ones. You saw the rhythm on the simpler box, where the wave also splits into independent factors. Hydrogen repeats the trick in spherical coordinates: split, solve each piece, then let the boundary rules generate the quantum numbers.
The split produces three quantum numbers called n, l, and m. The number n restricts the energy, l restricts the size of the orbital angular momentum, and m restricts its tilt. The energies come out matching the Bohr formula, which is why this solution is the crown jewel of basic quantum mechanics. Energy levels alone come from the radial equation; the angular pieces add shape and orientation.
Levels turn visible when electrons jump. An electron dropping from E3 to E2 emits one photon carrying the gap, E3 minus E2, which appears as one sharp spectral line. Absorption runs the film backwards. The predicted gaps match the measured hydrogen colours exactly, so separation of variables is not just neat maths. It forecasts the light hydrogen actually glows.
Answer hydrogen questions with this chain. Spherical symmetry means separate into radial and angular parts. Count one quantum number per piece: n, l, m. Take energies from the radial part alone, and take every spectral line from a difference of two levels. A state labelled with only n is missing l and m from the angular pieces.
Spherical symmetry splits hydrogen into radial and angular parts, yielding n for energy, l for size, and m for tilt, with spectra confirming the gaps.
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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.