Hydrogen atom

The radial equation of the hydrogenic atom, with the behaviours at \(0\) and \(\infty\) factored out, reduces to Kummer's equation: the same confluent hypergeometric \(_1F_1\) as the harmonic oscillator, with spectrum \(E_n=-Z^2/2n^2\) and degeneracy \(n^2\). The page shows the radial part \(u(r)=rR(r)\); eigenvalues and eigenfunctions are computed by finite differences in the Python backend (via gw2py).

Hydrogen, harmonic oscillator and Morse share the same confluent branch \(_1F_1\): after the asymptotic factorization the equation is Kummer's. The quantization \(E_n\propto-1/n^2\) arises from the termination of the series.

Atomic units (\(\hbar=m_e=e=1\)); energies in hartree (1 Ha = 27.211 eV).

Radial equation

Separating \(\psi=R(r)Y_{\ell m}\) and setting \(u(r)=rR(r)\):

\[ -\tfrac12 u'' + \Big[-\tfrac{Z}{r} + \tfrac{\ell(\ell+1)}{2r^2}\Big]u = E\,u,\qquad u(0)=0. \]

Reduction to Kummer

Factoring the regular behaviour at \(0\) (\(u\sim r^{\ell+1}\)) and the bound one at \(\infty\) (\(u\sim e^{-Zr/n}\)), the remainder satisfies Kummer's equation; the normalizable solution is

\[ u_{n\ell}(r)\propto r^{\ell+1}e^{-Zr/n}\,{}_1F_1\!\Big(-(n-\ell-1);\,2\ell+2;\,\tfrac{2Zr}{n}\Big), \]

i.e. an associated Laguerre polynomial — the same confluent \(_1F_1\) as the oscillator.

Series termination (\(n-\ell-1\in\{0,1,2,\dots\}\)) gives the spectrum \[ E_n=-\frac{Z^2}{2n^2},\qquad n=\ell+1,\ell+2,\dots \] with degeneracy \(n^2\) (in \(\ell,m\)).

Physical reading

The number of radial nodes is \(n-\ell-1\). The accidental degeneracy in \(\ell\) reflects the dynamical symmetry \(SO(4)\) (Runge–Lenz vector), specific to the \(1/r\) potential.

State to represent

State type Level \(n_1\)

Level \(n_2\)

The index numbers the computed radial eigenstates (for fixed \(\ell\)).

Physical parameters and grid

Nuclear charge \(Z\)

Angular momentum \(\ell\)

The function shown is the radial part \(u(r)=rR(r)\).

Radial extent \(r_{\max}\) [a₀] = 60

The confluent branch \(_1F_1\)

Hydrogen, harmonic oscillator and Morse share the same branch: after asymptotic factorization the equation is Kummer's (\(0\) regular, \(\infty\) irregular). It is the confluent branch of the Natanzon class; the common «trick» is to isolate the behaviours at \(0\) and \(\infty\) and recognize a \(_1F_1\) in what remains.

Symmetry and degeneracy

The degeneracy in \(\ell\) is not accidental: the \(SO(4)\) group generated by angular momentum and the Runge–Lenz vector makes hydrogen «maximally symmetric», a trait absent in Morse or the anharmonic oscillator.

Computational evidence

The finite-difference engine reproduces \(E_n=-Z^2/2n^2\): for \(Z=1,\ell=0\) the first levels \(-0.4995,\,-0.1250,\,-0.0556\) Ha vs \(-0.5,\,-0.125,\,-0.0556\) (error \(\sim5\times10^{-4}\), typical of finite differences). Correct radial nodes \(0,1,2\); orthonormality \(\sim10^{-15}\).

References

NIST DLMF, ch. 13 (confluent hypergeometric) and 18 (Laguerre). dlmf.nist.gov/13.
H. A. Bethe, E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms, Springer, 1957.
L. D. Landau, E. M. Lifshitz, Quantum Mechanics, Pergamon, §36.
F. Cooper, A. Khare, U. Sukhatme, Phys. Rep. 251, 267 (1995). doi.
G. A. Natanzon, Theor. Math. Phys. 38, 146 (1979).

WebNIR · CNR-IFAC | demo interface — numerical work is provided by the Python backend (gw2py).

Keywords: hydrogen, hydrogenic atom, radial equation, confluent hypergeometric, 1F1, Laguerre, Kummer, SO(4), Runge-Lenz, quantum mechanics