Hydrogenic energies and degeneracy

For an electron of mass $m_e$ and nucleus of mass $M$ and charge $+Ze$, separate the centre-of-mass motion. The internal relative-coordinate Hamiltonian contains reduced mass $\mu=m_eM/(m_e+M)$ and potential $-Ze^2/(4\pi\epsilon_0r)$. Its bound energies are measured below the zero-energy ionization threshold at infinite separation. A negative eigenvalue gives binding; the emitted photon in a downward transition carries a positive difference between two negative level energies.

The characteristic Coulomb length is $a_\mu=4\pi\epsilon_0\hbar^2/(\mu e^2)$. Increasing nuclear charge contracts hydrogenic orbitals roughly by a factor Z, while increasing reduced mass contracts them further. The binding scale is proportional to $\mu Z^2$. Replacing reduced mass by electron mass is usually a useful heavy-nucleus approximation, but it is not exact and fails badly for equal-mass constituents. The energy and length conventions must be changed together.

For a given principal quantum number n, allowed orbital angular momenta are $\ell=0,1,\ldots,n-1$. Each has $m=-\ell,\ldots,\ell$ and therefore $2\ell+1$ magnetic substates. Adding these multiplicities yields $n^2$ spatial states with the same Coulomb energy. These are distinct orthogonal eigenvectors, not $n^2$ different energy values. Each fixed-ℓ radial function is shared by its m states because the central radial equation contains no m.

Rotational symmetry explains equality between m states at fixed angular momentum. Equality between different ℓ values in this ideal Coulomb spectrum reflects an additional dynamical symmetry, often expressed through the Runge–Lenz operator. A generic central potential preserves m degeneracy but makes energy depend on radial and angular labels separately. Consequently, the degeneracy formula for hydrogen must not be transplanted to an arbitrary screened atomic potential, finite spherical well, or many-electron atom.

A transition energy follows by subtracting final from initial internal energy. Its photon wavelength is $hc/\Delta E$ when recoil is negligible at the precision of the estimate. The existence of two levels alone does not guarantee a strong radiative line: transition amplitudes and selection rules determine whether a particular coupling connects the states. An energy-level calculation provides possible energy differences, not their intensities or decay lifetimes.

Real spectra resolve corrections to the ideal model. Electron spin and relativistic dynamics produce fine structure, radiative effects can distinguish otherwise degenerate states, and external electric or magnetic fields remove further equalities. Finite nuclear mass already changes the gross binding scale through reduced mass. A useful statement therefore names both the Hamiltonian that has the degeneracy and the resolution at which omitted effects matter, instead of asserting that every state with the same principal number is exactly equal in an actual atom.