The Hydrogen Atom

The hydrogen atom is the only atom whose bound-state problem admits a closed solution, and every feature of atomic structure is calibrated against it. A single electron of charge $-e$ moves in the Coulomb field of a proton of charge $+e$. The potential is central, so the machinery of the radial equation applies directly: the angular part is a spherical harmonic, and the whole problem reduces to a one-dimensional radial equation on the half-line. What is special is what the $1/r$ potential does to the spectrum: the energies depend only on a single principal quantum number, and the resulting degeneracy exceeds what rotational invariance alone requires.

The Coulomb radial equation

The electron-proton interaction is $V(r)=-\frac{e^2}{4\pi {ϵ}_0}\frac{1}{r}$ in SI units. Because the proton is roughly $1836$ times heavier than the electron, both orbit their common center of mass and the correct one-body mass is the reduced mass $\mu =m_e{m}_p/(m_e+m_p)\approx 0.99946m_e$; the approximation $\mu \approx m_e$ is used throughout and corrected where a part-per-thousand number matters. With $u=rR$ the radial equation is

Bound states have $E<0$. Two abbreviations make the structure visible. Define the positive decay constant $\kappa$ and the dimensionless radius $\rho$,

so that $\kappa$ carries the (negative) energy and $\rho$ is measured in units of $1/\kappa$. Dividing the equation by $E=-{\hslash }^2{\kappa }^2/2\mu$ and collecting constants,

The single dimensionless combination ${\rho }_0$ measures the strength of the Coulomb attraction relative to the energy. Quantization will come from requiring that the solution stay normalizable, which restricts ${\rho }_0$ to a discrete set.

Asymptotic behavior and the series solution

The solution is peeled apart by extracting its behavior at the two ends of the half-line, exactly as for the harmonic oscillator.

• As $\rho \to \infty$ the bracket approaches $1$, so $u^{\prime \prime }\approx u$ and the normalizable solution is $u\sim e^{-\rho }$.
• As $\rho \to 0$ the centrifugal term dominates, giving $u^{\prime \prime }\approx \ell (\ell +1){\rho }^{-2}u$ and the regular solution $u\sim {\rho }^{\ell +1}$, the origin behavior established for every central potential.

Factor out both limits and let a power series carry the rest:

Substituting into the radial equation and cancelling the common factor ${\rho }^{\ell }{e}^{-\rho }$ produces a second-order equation for $v$,

Matching powers of $\rho$ term by term gives a two-term recursion for the coefficients,

Quantization of the energy

Truncation requires the numerator of the recursion to vanish at some finite $j=j_{\max }$, killing $c_{j_{\max }+1}$ and all higher terms:

The integer combination on the left is the principal quantum number,

Since $j_{\max }\ge 0$, the angular momentum is bounded by $\ell \le n-1$: a level with principal number $n$ admits $\ell =0,1,\dots ,n-1$. The truncation condition ${\rho }_0=2n$ feeds straight back into the definition of ${\rho }_0$ and $\kappa$. Solving $\frac{\mu e^2}{2\pi {ϵ}_0{\hslash }^2\kappa }=2n$ for $\kappa$ gives $\kappa =\frac{1}{n{a}_0}$ with the natural length scale

the Bohr radius. Because $E=-{\hslash }^2{\kappa }^2/2\mu$, the energy is

The constant $E_1$ is the Rydberg energy, $\text{Ry}=13.6057\text{eV}$; it can be written $E_1=\frac{1}{2}{\alpha }^2\mu c^2$ with $\alpha \approx 1/137$ the fine-structure constant, exposing hydrogen binding as an order-${\alpha }^2$ correction to the electron rest energy. The spectral lines follow from energy differences: a transition from $n_i$ to $n_f$ emits or absorbs a photon of energy

the Rydberg formula, with $n_f=1$ the Lyman series (ultraviolet), $n_f=2$ the Balmer series (visible), $n_f=3$ the Paschen series (infrared).

The bound-state wavefunctions

The truncated series $v$ is, up to normalization, an associated Laguerre polynomial. With the standard definitions

the radial factor is $v(\rho )\propto L_{n-\ell -1}^{2\ell +1}(2\rho )$, a polynomial of degree $n-\ell -1$. Restoring $u={\rho }^{\ell +1}{e}^{-\rho }v$ and $R=u/r$, and normalizing so that ${\int }_0^{\infty }|R_{n\ell }{|}^2{r}^2\mathbf{d}r=1$, gives

The full stationary state is the product with the angular harmonic,

labelled by the three quantum numbers $(n,\ell ,m)$: principal, azimuthal, and magnetic. The lowest few radial functions, with $a\equiv a_0$:

The ground state ${\psi }_{100}=(\pi a^3{)}^{-1/2}{e}^{-r/a}$ is nodeless, spherically symmetric, and peaks at the origin. Each increase of $n$ at fixed $\ell$ adds one radial node; the polynomial degree $n-\ell -1$ equals the number of interior nodes.

Radial probability and characteristic radii

The probability of finding the electron in a shell of thickness $\mathbf{d}r$ at radius $r$ is the radial distribution

the extra $r^2$ from the volume element competing against the exponential decay to place the density at a finite radius. For the ground state, $P_{10}(r)=\frac{4}{a^3}{r}^2{e}^{-2r/a}$; setting $P_{10}^{\prime }(r)=0$ gives the most probable radius $r=a_0$, reproducing the Bohr radius as the peak of the quantum-mechanical distribution rather than a sharp orbit. The mean radius is larger,

growing as $n^2$: highly excited states are physically large, the wave-mechanical content of the classical scaling of orbit size with energy.

The n-squared degeneracy

For each principal number $n$, the allowed values are $\ell =0,1,\dots ,n-1$, and for each $\ell$ there are $2\ell +1$ magnetic sublevels. Summing,

The $n^2$ states with the same $n$ all share the single energy $E_n$. Part of this is expected: the $(2\ell +1)$-fold $m$-degeneracy holds for any central potential, being a direct consequence of rotational invariance — nothing in $H$ singles out a direction, so states differing only in the orientation of $\vec{L}$ must be degenerate. The remainder is not expected. That levels of different $\ell$ (say $2s$ and $2p$) coincide is a property of the Coulomb $1/r$ potential specifically; a spherical box or a screened potential splits them.

Orbital shapes

The probability density $|{\psi }_{n\ell m}{|}^2$ has an angular shape from $|Y_{\ell }^m{|}^2$ and a radial envelope from $|R_{n\ell }{|}^2$. The $s$ states ($\ell =0$) are spherically symmetric; the $p$ states ($\ell =1$) have a two-lobe angular profile; the $d$ states ($\ell =2$) have four lobes or the pinched $d_{z^2}$ shape. Multiplying by the radial nodes of $R_{n\ell }$ produces the shells-within-lobes structure familiar from chemistry.

The size of the correction terms

The solution here treats the electron as a spinless particle in a static Coulomb field, and it reproduces the gross spectrum to within a part in $10^4$. The neglected effects each carry a definite scale.

The reduced-mass correction is included exactly by using $\mu$ in $a_0$ and $E_n$. The others enter as perturbations on the degenerate $n^2$ multiplets and break the accidental degeneracy: the fine structure splits levels of different $j$, and external fields produce the Zeeman and Stark effects. The unperturbed spectrum derived here is the platform every one of those corrections is measured against.

The exactness of the Coulomb solution and its oversized degeneracy point to a structure beyond rotational symmetry. The next lesson identifies the conserved Runge–Lenz vector whose algebra with $\vec{L}$ closes into $\mathrm{SO}(4)$ and forces exactly the $n^2$ degeneracy found here.¹²³