Hydrogen Spectrum: Formula Reference

This reference collects the nonrelativistic hydrogenic energy spectrum, spectral-series formulas, and limiting checks for a one-electron point-Coulomb atom. The Hydrogen Atom page derives the spectrum and contains worked exercises. Use the model and Hamiltonian reference to check the physical approximation and the radial formula reference for normalized wavefunctions and moments.

Hydrogenic energies and Coulomb scales

For electron mass $m_e$, nuclear mass $M_N$, and nuclear charge $+Ze$, use SI Coulomb normalization and

$$\mu=\frac{m_eM_N}{m_e+M_N}, \qquad \alpha_{\rm fs}=\frac{e^2}{4\pi\epsilon_0\hbar c}.$$

The energy zero is the separated electron–nucleus threshold. With center-of-mass motion removed,

$$E_n=-\frac{\mu Z^2e^4}{2(4\pi\epsilon_0)^2\hbar^2n^2} =-\frac{\mu c^2(Z\alpha_{\rm fs})^2}{2n^2}, \qquad n=1,2,\ldots.$$

Quantity	Formula	Units
Reduced-mass Bohr scale	$a_\mu=4\pi\epsilon_0\hbar^2/(\mu e^2)=\hbar/(\mu c\alpha_{\rm fs})$	m
Charge-$Z$ length	$a_Z=a_\mu/Z$	m
Rydberg energy	$\operatorname{Ry}_\mu=\hbar^2/(2\mu a_\mu^2)=\mu c^2\alpha_{\rm fs}^2/2$	J or eV
Spectroscopic Rydberg constant	$R_\mu=\operatorname{Ry}_\mu/(hc)$	m$^{-1}$
Bound energy	$E_n=-Z^2\operatorname{Ry}_\mu/n^2$	J or eV
Ionization energy from shell $n$	$E_{\rm ion}(n)=-E_n$	J or eV

For ordinary hydrogen the leading estimate is $E_n\simeq-13.6\,\mathrm{eV}/n^2$. The infinite bound sequence accumulates at $E=0$ from below; there is no highest bound state. These are internal energies: free center-of-mass kinetic energy is added separately.

Quantum labels and degeneracy

Label or property	Allowed value or result
Principal quantum number	$n=1,2,\ldots$
Orbital quantum number	$\ell=0,1,\ldots,n-1$; $L^2=\hbar^2\ell(\ell+1)$
Magnetic quantum number	$m=-\ell,\ldots,\ell$; $L_z=\hbar m$
Interior radial node count	$n_r=n-\ell-1$
Parity	$(-1)^\ell$
Spatial shell degeneracy	$g_n=\sum_{\ell=0}^{n-1}(2\ell+1)=n^2$
Including uncoupled electron spin	$2n^2$, only while all spin-dependent interactions remain omitted

Rotations explain the $m$ degeneracy. Independence of the energy from $\ell$ is special to the Coulomb potential; see Degeneracy of the Hydrogen Atom. The ideal shell count is not an observed line multiplicity after fine, hyperfine, radiative, nuclear, or external-field effects are resolved.

Transition energies and series limits

For emission with $n_i>n_f$, neglecting photon recoil,

$$h\nu=E_{n_i}-E_{n_f} =Z^2\operatorname{Ry}_\mu\left(\frac1{n_f^2}-\frac1{n_i^2}\right),$$ $$\frac1\lambda=R_\mu Z^2\left(\frac1{n_f^2}-\frac1{n_i^2}\right).$$

Here $\lambda$ is the vacuum wavelength. At fixed $n_f$, the series limit $n_i\to\infty$ has photon energy $Z^2\operatorname{Ry}_\mu/n_f^2$.

Series	Final shell
Lyman	$n_f=1$
Balmer	$n_f=2$
Paschen	$n_f=3$
Brackett	$n_f=4$

Energy differences determine gross line positions, not selection rules or intensities. For the interpretation of measured lines and their corrections, see Hydrogen as Atomic Prototype.

Mass and charge scaling

Relative to an infinitely heavy nucleus,

$$\frac\mu{m_e}=\frac1{1+m_e/M_N}<1, \qquad E_n=\frac\mu{m_e}E_n^{(M_N\to\infty)}.$$

Finite nuclear mass reduces the binding magnitude and increases the length scale. At fixed reduced mass, energies scale as $Z^2$ and lengths as $1/Z$. The worked comparisons for $\mathrm H$, $\mathrm{He}^+$, and $\mathrm{Li}^{2+}$ are in Hydrogenic Ions. Neutral helium has two electrons and is outside this model.

Virial and momentum checks

For any stationary bound state of the point-Coulomb model,

$$\langle T\rangle=-E_n, \qquad \langle V\rangle=2E_n, \qquad \langle p^2\rangle=-2\mu E_n.$$

Thus the relative-motion speed scale and inverse-radius moment are

$$v_{\rm rms}=\frac{\sqrt{\langle p^2\rangle}}\mu=\frac{Z\alpha_{\rm fs}c}{n}, \qquad \left\langle\frac1r\right\rangle=\frac1{a_Zn^2}.$$

These quantities are independent of $\ell$ in the ideal model; other radial moments need not be. The Coulomb virial checks show the reasoning and a solved exercise.

Validity and conventions

The formulas omit relativistic kinetic, spin–orbit, Darwin, radiative, hyperfine, finite-nuclear-size, and external-field terms. They assume one electron and an exact point-Coulomb potential. Low-lying states require $Z\alpha_{\rm fs}\ll1$ for the nonrelativistic approximation; high-$Z$ ions require relativistic and nuclear-size treatments, and precision hydrogen requires additional recoil and QED corrections.

Do not mix SI, Gaussian, or natural-unit Coulomb conventions without translating $\alpha_{\rm fs}$. Distinguish the Rydberg energy $\operatorname{Ry}_\mu$ from the inverse-length constant $R_\mu$, the magnetic label $m$ from a mass, and the negative level energy from a positive ionization or emission energy.

References

• H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms, Springer, 1957.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Vol. 1, Wiley, 1977.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018, Ch. 4.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994, Ch. 13.