Hydrogen atom bound states

Intuition [Beginner]

A hydrogen atom is a proton and an electron held together by electric attraction. The proton is heavy and sits nearly still. The electron moves around it. In classical physics the electron would spiral inward, radiating away its energy, and the atom would collapse in a fraction of a second. Atoms do not collapse. Quantum mechanics explains why.

The electron is not a tiny ball orbiting the nucleus like a planet around the sun. It is described by a wave function $\psi$, a cloud that gives the probability of finding the electron at each point in space. For bound states — the states where the electron stays near the proton — only certain wave shapes satisfy the equations. Each allowed shape corresponds to a specific energy, and those energies form a discrete set.

The lowest allowed energy is $E_1=-13.6$ eV. The minus sign means the electron is bound: you would need to add 13.6 eV to rip it away (this is the ionisation energy). The next energies are $E_2=-3.4$ eV, $E_3=-1.51$ eV, and so on, following the pattern $E_n=-13.6/n^2$ eV where $n=1,2,3,\dots$ is called the principal quantum number. The energies squeeze closer together as $n$ increases, approaching zero from below.

When the electron jumps from a higher level to a lower one, it emits a photon whose energy equals the difference. The jump from $n=2$ to $n=1$ produces the Lyman-alpha line at wavelength 121.6 nm, in the ultraviolet. Jumps to $n=2$ from higher levels produce the Balmer series, visible as distinct coloured lines in a spectroscope — red, blue-green, violet. These spectral lines were measured decades before quantum mechanics explained them.

The ground state ($n=1$) is not a point at the nucleus. The electron cloud is spread out, and the most probable distance from the nucleus is the Bohr radius $a_0=0.529$ Angstroms, about half a ten-billionth of a metre. The electron is not at $r=0$. It is most likely to be found near $r=a_0$, forming a diffuse spherical cloud around the proton.

The formula $E_n=-13.6/n^2$ is one of the most tested results in all of physics. It was first derived by Bohr in 1913 using a semi-classical argument (quantised circular orbits), then derived correctly by Schrödinger in 1926 from the wave equation, and again by Pauli in 1926 using an algebraic method that revealed a hidden symmetry. All three approaches produce the same energies. Only the Schrödinger and Pauli methods produce the correct wave functions and degeneracy structure.

Visual [Beginner]

The energy ladder on the left shows the $1/n^2$ spacing: the rungs are far apart at the bottom and crowd together at the top. The Lyman series (transitions down to $n=1$) spans large energy gaps and produces ultraviolet photons. The Balmer series (transitions down to $n=2$) spans smaller gaps and produces visible light — the four lines at 656 nm (red), 486 nm (blue-green), 434 nm (violet), and 410 nm (violet). The Paschen series (transitions down to $n=3$) produces infrared photons.

The inset on the right shows the radial probability density for the ground state. The horizontal axis is distance from the nucleus $r$. The curve rises from zero at the nucleus, peaks at $r=a_0$, and falls off smoothly at large $r$. The peak is the most probable distance for finding the electron — not the nucleus, not infinity, but one specific radius.

Worked example [Beginner]

First three energy levels. Using $E_n=-13.6/n^2$ eV:

$$E_1=-13.6\text{ eV},E_2=-13.6/4=-3.40\text{ eV},E_3=-13.6/9\approx -1.51\text{ eV}.$$

The spacing from $n=1$ to $n=2$ is $10.2$ eV; from $n=2$ to $n=3$ it is $1.89$ eV. The levels squeeze together.

Lyman-alpha wavelength. The electron drops from $n=2$ to $n=1$, emitting a photon with energy $\Delta E=E_2-E_1=-3.40-(-13.6)=10.2$ eV. Convert to joules: $10.2\times 1.602\times 10^{-19}=1.63\times 10^{-18}$ J. The wavelength is

$$\lambda =\frac{hc}{\Delta E}=\frac{6.626\times 10^{-34}\times 2.998\times 10^8}{1.63\times 10^{-18}}\approx 1.22\times 10^{-7}\text{ m}=121.6\text{ nm}.$$

This is ultraviolet, the strongest line in the hydrogen spectrum.

Ground-state most probable radius. The radial probability density for the ground state peaks at $r=a_0$ where $a_0={\hslash }^2/(m_e{e}^2)=0.529\times 10^{-10}$ m $=0.529$ Angstroms. The electron is most likely to be found about half an Angstrom from the nucleus — not at the nucleus, not far away.

Check your understanding [Beginner]

Formal definition [Intermediate+]

The hydrogen atom consists of an electron (mass $m_e$, charge $-e$) bound by the Coulomb attraction to a proton (mass $m_p$, charge $+e$). The reduced mass $\mu =m_e{m}_p/(m_e+m_p)\approx m_e$ to one part in 2000, so we approximate $\mu \approx m_e$ throughout and return to the reduced-mass correction in the Master tier.

The Coulomb Hamiltonian in the centre-of-mass frame is

$$\hat{H}=-\frac{{\hslash }^2}{2\mu }{\nabla }^2-\frac{e^2}{4\pi {ϵ}_0}\frac{1}{r},$$

where $r=|\mathbf{r}|$ is the electron-proton distance and ${\nabla }^2$ is the three-dimensional Laplacian. The first term is kinetic energy; the second is the Coulomb potential $V(r)=-e^2/(4\pi {ϵ}_{0}r)$.

In spherical coordinates $(r,\theta ,\phi )$ the Laplacian separates into radial and angular parts. The Hilbert space is $\mathcal{H}=L^2({\mathbb{R}}^3)$, and the Hamiltonian commutes with the angular-momentum operators ${\hat{L}}^2$ and ${\hat{L}}_z$. A simultaneous eigenstate of $\hat{H}$, ${\hat{L}}^2$, and ${\hat{L}}_z$ has the factored form

$${\psi }_{n\ell m}(r,\theta ,\phi )=R_{n\ell }(r)Y_{\ell }^m(\theta ,\phi ),$$

where $Y_{\ell }^m$ are the spherical harmonics 12.05.01 pending and $R_{n\ell }(r)$ is the radial wave function.

The time-independent Schrödinger equation $\hat{H}\psi =E\psi$ reduces, after substituting $\psi =R(r)Y_{\ell }^m(\theta ,\phi )$ and cancelling the angular part, to the radial equation

$$-\frac{{\hslash }^2}{2\mu }\frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{dR}{dr}\right)+\left[\frac{{\hslash }^2\ell (\ell +1)}{2\mu r^2}-\frac{e^2}{4\pi {ϵ}_{0}r}\right]R=ER.$$

The substitution $u(r)=rR(r)$ simplifies this to a one-dimensional form:

$$-\frac{{\hslash }^2}{2\mu }\frac{d^{2}u}{d{r}^2}+\left[\frac{{\hslash }^2\ell (\ell +1)}{2\mu r^2}-\frac{e^2}{4\pi {ϵ}_{0}r}\right]u=Eu,u(0)=0.$$

The term in square brackets is the effective potential: the centrifugal barrier ${\hslash }^2\ell (\ell +1)/(2\mu r^2)$ plus the Coulomb well $-e^2/(4\pi {ϵ}_{0}r)$. For $\ell =0$ the centrifugal barrier vanishes and the potential is purely attractive.

Introduce the Bohr radius $a_0=4\pi {ϵ}_0{\hslash }^2/(\mu e^2)\approx 0.529$ Angstroms and the Rydberg energy $E_R=\mu e^4/(2(4\pi {ϵ}_0{)}^2{\hslash }^2)=13.6$ eV. Define the dimensionless variable $\rho =r/(a_{0}n)$ and the parameter $\ell$. The radial equation becomes a form of the associated Laguerre equation. Square-integrable solutions (normalisable wave functions) exist only when $n$ is a positive integer satisfying $n\ge \ell +1$. This quantisation condition produces the energy levels

$$\boxed{E_n=-\frac{E_R}{n^2}=-\frac{13.6\text{ eV}}{n^2},n=1,2,3,\dots }$$

The quantum number $\ell$ takes values $0,1,2,\dots ,n-1$, and for each $\ell$ the magnetic quantum number $m$ takes values $-\ell ,-\ell +1,\dots ,\ell$. The total number of states with the same $n$ is

$$\sum _{\ell =0}^{n-1}(2\ell +1)=n^2.$$

All $n^2$ states share the same energy. This is accidental degeneracy: the Coulomb potential has more symmetry than generic central potentials, which would give energies depending on both $n$ and $\ell$. The hidden symmetry is the $\mathrm{SO}(4)$ symmetry discovered by Pauli and Fock, developed in the Master tier.

The radial eigenfunctions are

$$R_{n\ell }(r)=\sqrt{{\left(\frac{2}{n{a}_0}\right)}^3\frac{(n-\ell -1)!}{2n[(n+\ell )!{]}^3}}{\left(\frac{2r}{n{a}_0}\right)}^{\ell }{e}^{-r/(n{a}_0)}{L}_{n-\ell -1}^{2\ell +1}\left(\frac{2r}{n{a}_0}\right),$$

where $L_p^q$ are the associated Laguerre polynomials. For the ground state ($n=1$, $\ell =0$): $R_{10}(r)=2a_0^{-3/2}{e}^{-r/a_0}$. For the first excited states ($n=2$): $R_{20}(r)=(2a_0{)}^{-3/2}(2-r/a_0)e^{-r/(2a_0)}$ and $R_{21}(r)=(2a_0{)}^{-3/2}(r/\sqrt{3}{a}_0)e^{-r/(2a_0)}$.

Counterexamples to common slips

• The Bohr model (1913) produces the correct energies $E_n=-13.6/n^2$ eV but for the wrong reasons: it assumes classical circular orbits with ad hoc angular-momentum quantisation $L=n\hslash$. The full quantum treatment shows $L=\ell \hslash$ with $\ell =0,1,\dots ,n-1$ — the ground state has $\ell =0$ (zero orbital angular momentum), contradicting Bohr's picture which requires $n=1$ to carry angular momentum.
• The principal quantum number $n$ does not measure angular momentum. It measures the total number of radial plus angular nodes in the wave function. Angular momentum is set by $\ell$.
• The degeneracy $n^2$ counts spatial states only. Including electron spin (two states per spatial orbital), the total degeneracy is $2n^2$.

Key theorem with proof [Intermediate+]

Theorem (Bound-state spectrum of the hydrogen atom). The Coulomb Hamiltonian $\hat{H}=-\frac{{\hslash }^2}{2\mu }{\nabla }^2-\frac{e^2}{4\pi {ϵ}_{0}r}$ on $L^2({\mathbb{R}}^3)$ has discrete spectrum $E_n=-E_R/n^2$ for $n=1,2,3,\dots$ with degeneracy $n^2$. The corresponding eigenfunctions ${\psi }_{n\ell m}(r,\theta ,\phi )=R_{n\ell }(r)Y_{\ell }^m(\theta ,\phi )$ form a complete orthonormal basis for the bound-state subspace of $L^2({\mathbb{R}}^3)$. The continuous spectrum occupies $[0,\infty )$.

Proof. We solve the radial equation for $u(r)=rR(r)$:

$$-\frac{{\hslash }^2}{2\mu }{u}^{\prime \prime }+\left[\frac{{\hslash }^2\ell (\ell +1)}{2\mu r^2}-\frac{e^2}{4\pi {ϵ}_{0}r}\right]u=Eu.$$

For bound states, $E<0$. Introduce dimensionless variables. Define $\kappa =\sqrt{-2\mu E}/\hslash >0$ and $\rho =2\kappa r$. Also define $\gamma =\mu e^2/(4\pi {ϵ}_0{\hslash }^2\kappa )$. The equation becomes

$$\frac{d^{2}u}{d{\rho }^2}=\left[1-\frac{\gamma }{\rho }+\frac{\ell (\ell +1)}{{\rho }^2}\right]u.$$

Asymptotic analysis. As $\rho \to \infty$, the equation reduces to $u^{\prime \prime }=u$, so $u\sim e^{-\rho }$ (discarding the growing exponential). As $\rho \to 0$, the dominant term is $\ell (\ell +1)/{\rho }^2$, giving $u\sim {\rho }^{\ell +1}$ (discarding the non-normalisable solution ${\rho }^{-\ell }$).

Series solution. Write $u(\rho )={\rho }^{\ell +1}{e}^{-\rho }v(\rho )$ and substitute. The equation for $v$ is

$$\rho v^{\prime \prime }+[2(\ell +1)-2\rho ]v^{\prime }+[2\gamma -2(\ell +1)]v=0.$$

This is the associated Laguerre equation. For a normalisable solution, the series for $v$ must terminate — otherwise $v$ would grow as $e^{2\rho }$ and overwhelm the decaying exponential. Termination requires

$$2\gamma -2(\ell +1)=2(n_r),n_r=0,1,2,\dots$$

where $n_r$ is the number of radial nodes. Setting $n=n_r+\ell +1$ (the principal quantum number), the quantisation condition $\gamma =n$ gives

$$\kappa =\frac{1}{n{a}_0},E_n=-\frac{{\hslash }^2}{2\mu a_0^2{n}^2}=-\frac{E_R}{n^2}.$$

The degeneracy count: for a given $n$, the angular momentum $\ell$ ranges from $0$ to $n-1$ (from the constraint $n_r=n-\ell -1\ge 0$). For each $\ell$, $m$ ranges from $-\ell$ to $+\ell$, giving $2\ell +1$ states. Summing: ${\sum }_{\ell =0}^{n-1}(2\ell +1)=n^2$.

Completeness. The bound-state eigenfunctions are a subset of the full spectrum. The Hamiltonian is self-adjoint on its domain in $L^2({\mathbb{R}}^3)$, so by the spectral theorem the eigenfunctions of the discrete spectrum plus the generalised eigenfunctions of the continuous spectrum form a complete set. The bound-state subspace is spanned by $\{{\psi }_{n\ell m}\}$. $\square$

Corollary (Rydberg formula). The wavelength of a photon emitted in the transition $n_i\to n_f$ (with $n_i>n_f$) is

$$\frac{1}{\lambda }=R_H\left(\frac{1}{n_f^2}-\frac{1}{n_i^2}\right),$$

where $R_H=E_R/(hc)=1.097\times 10^7{\text{ m}}^{-1}$ is the Rydberg constant for hydrogen.

Bridge. The spectral decomposition $E_n=-E_R/n^2$ builds toward the perturbation-theory treatment of fine structure in 12.07.01 pending, where spin-orbit coupling, the relativistic kinetic-energy correction, and the Darwin term split each $n$-level into sub-levels labelled by total angular momentum $j$. The foundational reason for the $n^2$-fold degeneracy — the hidden $\mathrm{SO}(4)$ symmetry of the Coulomb Hamiltonian — appears again in the Master-tier analysis below, where Pauli's algebraic method derives the spectrum from representation theory without solving a differential equation. This is exactly the structural fact that underlies the linear Stark effect, the parabolic coordinate separation, and Fock's identification of the bound-state Hilbert space with representations of $\mathrm{SO}(4)$ acting on $S^3$.

Exercises [Intermediate+]

Lean formalization [Intermediate+]

The hydrogen atom is one of the most challenging targets for formal verification in quantum mechanics. The key objects to formalize are the Coulomb Hamiltonian as a self-adjoint operator on $L^2({\mathbb{R}}^3)$, its spectral decomposition into discrete and continuous parts, and the explicit eigenfunctions ${\psi }_{n\ell m}$.

Mathlib has the spherical harmonics $Y_{\ell }^m$ (via the harmonic polynomial API) and the general theory of Sturm-Liouville operators on compact intervals. It does not have: the Coulomb Hamiltonian as a named operator; the associated Laguerre equation and its polynomial solutions; the spectral theorem applied to produce the discrete spectrum $E_n=-E_R/n^2$; or the completeness of the bound-state eigenfunctions in $L^2({\mathbb{R}}^3)$. The radial ODE machinery exists in principle (second-order linear ODEs with regular singular points are formalizable), but the specific analysis of the confluent hypergeometric function and its polynomial truncation has not been carried out.

A minimal formalization would prove the eigenvalue equation $\hat{H}{\psi }_{n\ell m}=E_n{\psi }_{n\ell m}$ by direct computation for fixed $n$, verify orthonormality via integration identities, and state completeness as an axiom pending the full spectral-theory development. This unit ships with lean_status: none and awaits the Sturm-Liouville and unbounded-operator infrastructure in Mathlib.

Advanced results [Master]

The accidental SO(4) symmetry

The degeneracy of the hydrogen energy levels — all $n^2$ states with the same $n$ share the same energy regardless of $\ell$ — is not explained by rotational symmetry $\mathrm{SO}(3)$ alone. A generic central potential $V(r)$ produces energies that depend on both $n$ and $\ell$ (as seen in the three-dimensional harmonic oscillator, where $E\propto 2n_r+\ell +3/2$). The Coulomb potential has an extra conserved quantity that enlarges the symmetry group.

The classical Kepler problem has the Laplace-Runge-Lenz vector

$$\mathbf{A}=\mathbf{p}\times \mathbf{L}-mk\hat{\mathbf{r}},$$

which points along the major axis of the orbit and is conserved for the $1/r$ potential and no other central potential. Its quantum-mechanical counterpart, discovered by Pauli in 1926, is

$$\hat{\mathbf{A}}=\frac{1}{2\mu }(\hat{\mathbf{p}}\times \hat{\mathbf{L}}-\hat{\mathbf{L}}\times \hat{\mathbf{p}})-\frac{e^2}{4\pi {ϵ}_0}\frac{\hat{\mathbf{r}}}{r}.$$

The symmetrised cross product is needed because $\hat{\mathbf{p}}$ and $\hat{\mathbf{L}}$ do not commute. The quantum Runge-Lenz vector commutes with the Hamiltonian: $[\hat{\mathbf{A}},\hat{H}]=0$.

The six operators $\hat{\mathbf{L}}$ and $\hat{\mathbf{A}}$ generate a Lie algebra. On the subspace of energy $E_n<0$, rescaling $\hat{\mathbf{K}}=\hat{\mathbf{A}}/\sqrt{-2\mu E_n}$ gives commutation relations

$$[{\hat{L}}_i,{\hat{L}}_j]=i\hslash {ϵ}_{ijk}{\hat{L}}_k,[{\hat{L}}_i,{\hat{K}}_j]=i\hslash {ϵ}_{ijk}{\hat{K}}_k,[{\hat{K}}_i,{\hat{K}}_j]=i\hslash {ϵ}_{ijk}{\hat{L}}_k.$$

Defining ${\hat{\mathbf{J}}}^{(\pm )}=(\hat{\mathbf{L}}\pm \hat{\mathbf{K}})/2$, the algebra splits into two independent $\mathfrak{su}(2)$ factors: $[{\hat{J}}_i^{(\pm )},{\hat{J}}_j^{(\pm )}]=i\hslash {ϵ}_{ijk}{\hat{J}}_k^{(\pm )}$ and $[{\hat{J}}_i^{(+)},{\hat{J}}_j^{(-)}]=0$. This is the Lie algebra of $\mathrm{SO}(4)\cong \mathrm{SU}(2)\times \mathrm{SU}(2)$.

Representations of $\mathrm{SO}(4)$ are labelled by two half-integers $(j_{+},j_{-})$. The Casimir ${\hat{\mathbf{L}}}^2+{\hat{\mathbf{K}}}^2=2({\hat{\mathbf{J}}}^{(+)2}+{\hat{\mathbf{J}}}^{(-)2})$ is related to the Hamiltonian by $\hat{H}=-\mu e^4/[2{\hslash }^2(2j_{+}+2j_{-}+1{)}^2(4\pi {ϵ}_0{)}^2]$. Setting $n=2j_{+}+2j_{-}+1$ and requiring the representation to be physical ($j_{+}=j_{-}=(n-1)/2$ for the maximal degeneracy) gives $E_n=-E_R/n^2$ with degeneracy $(2j_{+}+1)(2j_{-}+1)=n^2$.

This is Pauli's algebraic solution of the hydrogen atom, completed in 1926 before Schrödinger's wave-equation solution was published. It derives the energy levels and degeneracy entirely from symmetry, without solving any differential equation.

The representation-theoretic content can be stated precisely. For each energy $E_n<0$, the bound-state eigenspace carries an irreducible representation of $\mathrm{SO}(4)$. The two $\mathfrak{su}(2)$ factors are generated by ${\hat{\mathbf{J}}}^{(\pm )}=(\hat{\mathbf{L}}\pm \hat{\mathbf{K}})/2$, and the irreducible representation is the tensor product $D_{j_{+}}\otimes D_{j_{-}}$ where $j_{+}=j_{-}=(n-1)/2$. The dimension $(2j_{+}+1)(2j_{-}+1)=n^2$ is the degeneracy. The angular momentum $\hat{\mathbf{L}}={\hat{\mathbf{J}}}^{(+)}+{\hat{\mathbf{J}}}^{(-)}$ decomposes this representation into a direct sum $D_0\oplus D_1\oplus \cdots \oplus D_{n-1}$ of representations of $\mathrm{SO}(3)$, recovering the multiplet structure $\ell =0,1,\dots ,n-1$. This Clebsch-Gordan decomposition of the $\mathrm{SO}(4)$ representation into $\mathrm{SO}(3)$ representations is the algebraic origin of the $\ell$-multiplets within each energy shell.

The commutation relations $[{\hat{K}}_i,{\hat{K}}_j]=i\hslash {ϵ}_{ijk}{\hat{L}}_k$ have a direct physical meaning: the three components of the Runge-Lenz vector do not commute among themselves, but their commutators reproduce the angular momentum algebra. This is the hallmark of a dynamical symmetry — a symmetry that mixes coordinates and momenta, unlike the geometric rotation symmetry $\mathrm{SO}(3)$ which acts on coordinates alone. The classical LRL vector is conserved for the $1/r$ potential and no other central potential, and this uniqueness carries over to the quantum theory. The dynamical symmetry is destroyed by any perturbation that deviates from $1/r$, which is why the degeneracy is lifted in multi-electron atoms where electron-electron repulsion modifies the central potential.

Hydrogen in momentum space: Fock's SO(4) symmetry

Fock (1935) gave an even more geometric account. In momentum space, the Coulomb Schrodinger equation for bound states with energy $E_n=-p_0^2/(2\mu )$ (where $p_0=\hslash /(n{a}_0)$) can be transformed via stereographic projection from ${\mathbb{R}}^3$ (momentum space) onto the three-sphere $S^3\subset {\mathbb{R}}^4$ of radius $p_0$. The eigenvalue equation becomes the Laplace-Beltrami equation on $S^3$, whose eigenfunctions are the four-dimensional spherical harmonics with total angular momentum $n-1$. The degeneracy $n^2$ is the dimension of the $(n-1)$-th representation of $\mathrm{SO}(4)$ acting on $S^3$.

Fock's construction makes the $\mathrm{SO}(4)$ symmetry visible: the bound-state Hilbert space at each energy is a representation space of $\mathrm{SO}(4)$, and the stereographic map is the canonical identification between the Coulomb problem and the free particle on $S^3$.

The stereographic projection is explicit. For a bound state with energy $E_n=-{\hslash }^2/(2\mu a_0^2{n}^2)$, define $p_0=\hslash /(n{a}_0)$ and map each momentum $\mathbf{p}\in {\mathbb{R}}^3$ to a point $\mathbf{u}\in S^3\subset {\mathbb{R}}^4$ via

$$\mathbf{u}=\frac{1}{p_0^2+p^2}\left(2p_0\mathbf{p},p_0^2-p^2\right).$$

Under this map, the Coulomb Schrödinger equation in momentum space transforms into ${\Delta }_{S^3}\Phi =-n^2\Phi$, where ${\Delta }_{S^3}$ is the Laplace-Beltrami operator on $S^3$. The eigenfunctions of ${\Delta }_{S^3}$ are the four-dimensional spherical harmonics ${\mathcal{Y}}_{n-1}(\mathbf{u})$, labelled by an integer $n-1$ (the total angular momentum quantum number on $S^3$) and additional magnetic quantum numbers. The degeneracy $n^2$ is the dimension of the $(n-1)$-th representation of $\mathrm{SO}(4)$ on these harmonics. The inverse stereographic projection sends ${\mathcal{Y}}_{n-1}$ back to the momentum-space hydrogen wave functions, and a Fourier transform recovers the position-space functions $R_{n\ell }(r)Y_{\ell }^m(\theta ,\phi )$.

The Stark effect: hydrogen in an external electric field

When a uniform electric field $\mathcal{E}$ is applied along the $z$-axis, the Hamiltonian gains the perturbation ${\hat{H}}^{\prime }=e\mathcal{E}\hat{z}$. Because the hydrogen Hamiltonian has $\mathrm{SO}(4)$ symmetry (not merely $\mathrm{SO}(3)$), the degenerate subspace at each $n$ must be diagonalised carefully. The perturbation $\hat{z}$ connects states within the same $n$-multiplet, producing first-order energy shifts proportional to $\mathcal{E}$ — the linear Stark effect.

For $n=2$ (four-fold degenerate: $2s$, $2p_{-1}$, $2p_0$, $2p_1$), the only nonzero matrix elements of $\hat{z}$ are $⟨2s|\hat{z}|2p_0⟩=\pm 3a_0$. The perturbation splits the four-fold level into three: $E=E_2$ (doubly degenerate, $m=\pm 1$) and $E=E_2\pm 3e{a}_0\mathcal{E}$ (non-degenerate). This linear splitting in $\mathcal{E}$ is unique to hydrogen; all other atoms exhibit only a quadratic (second-order) Stark effect because they lack the accidental degeneracy. For $n=3$ (nine-fold degenerate), the linear Stark effect produces five sub-levels, with splittings proportional to $\mathcal{E}$ that depend on the parabolic quantum numbers $n_1$, $n_2$. At large field strengths the Stark effect destabilises the atom: the potential $-e^2/(4\pi {ϵ}_{0}r)+e\mathcal{E}z$ is unbounded below along the field direction, and there are no longer true bound states — only metastable resonances with finite lifetimes. This field ionisation sets a practical upper limit on the electric fields that can be applied to hydrogen before the atom is torn apart.

The Stark effect is developed fully in perturbation theory (unit 12.07.01).

Parabolic coordinates and the Runge-Lenz vector as a separation constant

The hydrogen Schrödinger equation also separates in parabolic coordinates $\xi =r+z$, $\eta =r-z$, $\phi =\arctan (y/x)$. The surfaces of constant $\xi$ and $\eta$ are confocal paraboloids of revolution about the $z$-axis, opening in opposite directions. This coordinate system makes the Runge-Lenz component $A_z$ a separation constant. The parabolic quantum numbers $n_1$, $n_2$, $m$ satisfy $n=n_1+n_2+|m|+1$, and the energy depends only on $n$ (not on $n_1$ and $n_2$ separately), reflecting the accidental degeneracy.

Parabolic coordinates are the natural basis for the Stark effect (the perturbation $\hat{z}=(\xi -\eta )/2$ is diagonal in parabolic coordinates) and for Coulomb scattering states (the continuum analogue of the bound-state problem).

Selection rules and electric dipole transitions

The transition rate for spontaneous emission from state ${\psi }_{n^{\prime }{\ell }^{\prime }{m}^{\prime }}$ to ${\psi }_{n\ell m}$ is governed by the matrix element of the electric dipole operator $\hat{\mathbf{d}}=-e\hat{\mathbf{r}}$. Fermi's golden rule gives

$${\Gamma }_{n^{\prime }{\ell }^{\prime }{m}^{\prime }\to n\ell m}=\frac{4\alpha {\omega }^3}{3c^2}|⟨n\ell m|\hat{\mathbf{r}}|n^{\prime }{\ell }^{\prime }{m}^{\prime }⟩{|}^2,$$

where $\alpha \approx 1/137$ is the fine-structure constant and $\omega =(E_{n^{\prime }}-E_n)/\hslash$. The matrix element vanishes unless the selection rules $\Delta \ell =\ell -{\ell }^{\prime }=\pm 1$ and $\Delta m=m-m^{\prime }=0,\pm 1$ are satisfied. The parity argument is decisive: the dipole operator $\hat{\mathbf{r}}$ is odd under spatial inversion, so it connects states of opposite parity. Since $Y_{\ell }^m$ has parity $(-1{)}^{\ell }$, the initial and final states must have different parity, forcing $\Delta \ell$ to be odd. Combined with the angular momentum selection rule from the Wigner-Eckart theorem (the dipole operator is a rank-1 spherical tensor), only $\Delta \ell =\pm 1$ survives.

For the $2p\to 1s$ transition (the fastest decay in hydrogen), the rate is $\Gamma \approx 6.27\times 10^8{\text{s}}^{-1}$, corresponding to a lifetime of about 1.6 ns. The $2s\to 1s$ transition violates $\Delta \ell =\pm 1$ (both states have $\ell =0$) and is forbidden to single-photon electric dipole decay. The $2s$ level instead decays via a two-photon process with a lifetime of about 0.12 s — eight orders of magnitude slower. This metastability is the basis of hydrogen masers and is relevant to the 21-cm hyperfine line in astrophysics, where the analogous forbidden transition between the two hyperfine sub-levels of the ground state has a lifetime of $\sim 10^7$ years.

The Coulomb scattering continuum

The hydrogen Hamiltonian also has a continuous spectrum $E>0$ describing scattering states: an electron with positive energy approaches the proton from infinity, is deflected by the Coulomb field, and escapes to infinity. The radial equation for $E>0$ has solutions in terms of confluent hypergeometric functions rather than associated Laguerre polynomials. The scattering wave functions are

$${\psi }_{\mathbf{k}}^{(+)}(\mathbf{r})=e^{i\mathbf{k}\cdot \mathbf{r}}{}_1{F}_1\left(-i\eta ,1,i(kr-\mathbf{k}\cdot \mathbf{r})\right),$$

where $\mathbf{k}$ is the asymptotic momentum, $E={\hslash }^2{k}^2/(2\mu )$, and $\eta =1/(k{a}_0)$ is the Sommerfeld parameter. These Coulomb wave functions reduce to plane waves $e^{i\mathbf{k}\cdot \mathbf{r}}$ when the Coulomb potential is screened or in the limit $\eta \to 0$, but the long-range nature of the $1/r$ potential modifies the asymptotic behaviour: the phase shifts acquire a logarithmic term $\eta \ln (kr)$ absent in short-range scattering.

The differential cross section for Coulomb scattering in the centre-of-mass frame is the Rutherford formula

$$\frac{d\sigma }{d\Omega }={\left(\frac{e^2}{8\pi {ϵ}_{0}E}\right)}^2\frac{1}{{\sin }^4(\theta /2)},$$

which Rutherford derived classically in 1911 and which quantum mechanics reproduces exactly in the Born approximation — a coincidence unique to the $1/r$ potential. The continuum wave functions are orthogonal to the bound-state wave functions and together they form a complete set in $L^2({\mathbb{R}}^3)$, realising the spectral decomposition of the Coulomb Hamiltonian into discrete and continuous parts.

Reduced mass and the Rydberg constant

Throughout this unit we used $\mu \approx m_e$. The exact reduced mass $\mu =m_e{m}_p/(m_e+m_p)$ differs from $m_e$ by about one part in 2000, producing a Rydberg constant $R_H=R_{\infty }\cdot \mu /m_e$ where $R_{\infty }=1.097373\times 10^7$ m${}^{-1}$ is the Rydberg constant for infinite nuclear mass. The difference between $R_H$ and $R_{\infty }$ is measurable and was historically important in confirming the existence of isotopes (deuterium has $\mu$ closer to $m_e$ because the deuteron is twice the proton mass, shifting its spectral lines by about 0.03 nm relative to hydrogen — the Pickering series).

Synthesis. The hydrogen atom is the foundational example of a quantum system whose full symmetry group exceeds the manifest geometric symmetry. The central insight is the $\mathrm{SO}(4)$ accidental degeneracy: the Coulomb Hamiltonian commutes not only with $\hat{\mathbf{L}}$ (rotations) but with the Runge-Lenz vector $\hat{\mathbf{A}}$ (dynamical symmetry), and the combined algebra identifies the bound-state eigenspaces with irreducible representations of $\mathrm{SO}(4)$. Putting these together with Fock's stereographic construction — which identifies the momentum-space hydrogen problem with the free particle on $S^3$ — the degeneracy $n^2$ becomes the dimension of a representation of $\mathrm{SO}(4)$, and the selection rules for electric dipole transitions become a consequence of the tensor-product decomposition of these representations under the $\mathrm{SO}(3)$ subgroup.

This is exactly the pattern that generalises to higher-dimensional Coulomb problems, to the relativistic Dirac equation (where the symmetry enlarges further for the Kepler-Dirac problem), and to the dynamical symmetries of superintegrable systems in mathematical physics. The bridge is between the purely algebraic derivation of the hydrogen spectrum and the analytic solution of the Schrödinger equation: both reach the same destination, but the algebraic route reveals the hidden symmetry that makes the result possible.

Full proof set [Master]

Proposition (SO(4) commutation relations). The rescaled Runge-Lenz vector $\hat{\mathbf{K}}=\hat{\mathbf{A}}/\sqrt{-2\mu E_n}$ satisfies $[{\hat{K}}_i,{\hat{K}}_j]=i\hslash {ϵ}_{ijk}{\hat{L}}_k$, $[{\hat{L}}_i,{\hat{K}}_j]=i\hslash {ϵ}_{ijk}{\hat{K}}_k$, and $[{\hat{K}}_i,\hat{H}]=0$ on the subspace of energy $E_n$.

Proof. The unscaled Runge-Lenz vector is

$${\hat{A}}_i=\frac{1}{2\mu }(\hat{\mathbf{p}}\times \hat{\mathbf{L}}-\hat{\mathbf{L}}\times \hat{\mathbf{p}}{)}_i-\frac{e^2}{4\pi {ϵ}_0}\frac{{\hat{r}}_i}{r}.$$

The commutator $[{\hat{A}}_i,\hat{H}]$ vanishes on the energy shell $E_n$ by direct calculation: each term in $[{\hat{A}}_i,\hat{H}]$ is proportional to $(\hat{H}-E_n)$, so projecting onto the eigenspace of $E_n$ gives zero. The rescaling ${\hat{K}}_i={\hat{A}}_i/\sqrt{-2\mu E_n}$ with $E_n<0$ ensures the algebra closes on the eigenspace. Computing $[{\hat{K}}_i,{\hat{K}}_j]$ requires the canonical commutation relations $[{\hat{r}}_i,{\hat{p}}_j]=i\hslash {\delta }_{ij}$, the angular momentum algebra $[{\hat{L}}_i,{\hat{L}}_j]=i\hslash {ϵ}_{ijk}{\hat{L}}_k$, and the identities for operator cross products. The mixed commutator $[{\hat{L}}_i,{\hat{K}}_j]=i\hslash {ϵ}_{ijk}{\hat{K}}_k$ follows because $\hat{\mathbf{K}}$ transforms as a vector under rotations (it is built from $\hat{\mathbf{p}}$, $\hat{\mathbf{L}}$, and $\hat{\mathbf{r}}$, all of which are vectors). The $\hat{K}$-$\hat{K}$ commutator requires projection onto the energy shell: the unprojected commutator $[{\hat{A}}_i,{\hat{A}}_j]$ contains terms proportional to $\hat{H}$, which become $E_n$ on the eigenspace, yielding the rescaled result $i\hslash {ϵ}_{ijk}{\hat{L}}_k$. $\square$

Proposition (Dipole selection rule). The matrix element $⟨n\ell m|{\hat{r}}_q|n^{\prime }{\ell }^{\prime }{m}^{\prime }⟩$, where ${\hat{r}}_q$ ($q=0,\pm 1$) are the spherical components of the position operator, vanishes unless $\Delta \ell =\pm 1$ and $\Delta m=q$.

Proof. The spherical components ${\hat{r}}_q$ are rank-1 spherical tensors: under rotation by $R\in \mathrm{SO}(3)$, they transform as $D_{q{q}^{\prime }}^{(1)}(R)$. By the Wigner-Eckart theorem,

$$⟨n\ell m|{\hat{r}}_q|n^{\prime }{\ell }^{\prime }{m}^{\prime }⟩=⟨\ell ||\hat{r}||{\ell }^{\prime }⟩\cdot C_{{\ell }^{\prime }{m}^{\prime }1q}^{\ell m},$$

where $C_{{\ell }^{\prime }{m}^{\prime }1q}^{\ell m}$ is a Clebsch-Gordan coefficient. The coefficient is nonzero only if $\ell$ appears in the Clebsch-Gordan decomposition ${\ell }^{\prime }\otimes 1=({\ell }^{\prime }-1)\oplus {\ell }^{\prime }\oplus ({\ell }^{\prime }+1)$, giving $\ell ={\ell }^{\prime }\pm 1$, and if $m=m^{\prime }+q$. The parity constraint excludes $\ell ={\ell }^{\prime }$: ${\hat{r}}_q$ is odd under inversion while both bra and ket would carry the same parity $(-1{)}^{\ell }=(-1{)}^{{\ell }^{\prime }}$, so the matrix element would require an odd integrand to produce an even (nonzero) result. Since $(-1{)}^{\ell }\cdot (-1)\cdot (-1{)}^{{\ell }^{\prime }}=(-1{)}^{\ell +{\ell }^{\prime }+1}=1$ only when $\ell +{\ell }^{\prime }$ is odd, the case $\ell ={\ell }^{\prime }$ is forbidden. $\square$

Connections [Master]

The hydrogen atom is the bridge between quantum mechanics and atomic physics, chemistry, and spectroscopy. Every multi-electron atom is treated as a perturbation of hydrogen: the Hartree-Fock method and density functional theory both start from hydrogenic orbitals as a basis. The periodic table is built from hydrogenic quantum numbers $n$, $\ell$, $m$, and spin, with the Aufbau principle filling these orbitals according to the Pauli exclusion principle.

In spectroscopy (14.12.01), the Rydberg formula $1/\lambda =R_H(1/n_f^2-1/n_i^2)$ is the starting point for every atomic spectroscopy experiment. The Lyman series ($n_f=1$) dominates the interstellar medium and is the principal absorption feature in quasar spectra. The Balmer series ($n_f=2$) gave Balmer his empirical formula in 1885, the first quantitative regularity in atomic spectra, decades before its quantum-mechanical explanation.

The connection to perturbation theory (12.07.01) is the hydrogen atom's most important technical legacy. Fine structure (spin-orbit coupling, relativistic kinetic energy correction, Darwin term) splits each $n$-level into sub-levels labelled by total angular momentum $j$. The Lamb shift (quantum electrodynamic correction) further splits the $2s_{1/2}$ and $2p_{1/2}$ levels, which are degenerate even after fine structure. Hyperfine structure (interaction with the nuclear magnetic moment) splits the ground state into two sub-levels separated by $5.9\times 10^{-6}$ eV, corresponding to the 21-cm radio line used in radio astronomy.

The algebraic method — solving a quantum system by identifying the symmetry group and its representations rather than by solving differential equations — is the foundation of angular-momentum theory, the Wigner-Eckart theorem, and the classification of elementary particles by group representations. The hydrogen atom is the first and richest example of this method.

The cross-domain chemistry-side treatment of the same atom is the quantum-chemistry unit 14.04.01 pending, which adapts the present mathematical solution to the orbital, term-symbol, periodic-table, and bonding context in which chemists actually work. The two units are complementary rather than redundant: the present unit is the physics-side reference that develops the radial functions, the spherical harmonics as eigenfunctions of $L^2$ and $L_z$, the accidental SO(4) symmetry and the Runge-Lenz vector, Fock's momentum-space construction, the Pauli algebraic solution, and the operator-theoretic spectral structure of the Coulomb Hamiltonian; the chemistry-side unit takes the resulting orbitals as input and develops the chemical phenomenology — orbital shapes ($s$ spherical, $p$ lobed, $d$ four-lobed cloverleaf and ring), valence framing, Aufbau ordering, atomic-radius and ionisation-energy trends, the molecular-orbital theory built on hydrogenic atomic orbitals, and the breakdown of single-particle treatment for multi-electron atoms where Hartree-Fock and DFT take over. The two perspectives are bridged by the explicit cross-domain sub-section of 14.04.01 pending; chemistry students who want the underlying SO(4) symmetry and momentum-space picture come back here, and physics students who want the bonding-and-periodic-table consequences go there.

Historical and philosophical context [Master]

The hydrogen spectrum was the testing ground for quantum mechanics from the beginning. Balmer's empirical formula (1885) fit the four known visible lines of hydrogen with remarkable precision: $\lambda =B{n}^2/(n^2-4)$ for $n=3,4,5,6$, where $B=364.56$ nm. Rydberg generalised this to $1/\lambda =R(1/n_f^2-1/n_i^2)$ in 1890, but neither Balmer nor Rydberg had a physical explanation.

Bohr's 1913 model was the first to derive the Rydberg formula from physical principles. His assumptions — quantised circular orbits with angular momentum $L=n\hslash$ — produced the correct energies but the wrong wave functions. The Bohr model predicts that the ground state carries angular momentum $\hslash$, while the Schrodinger solution shows $\ell =0$ for the ground state. Bohr's model also cannot explain the fine structure, the Stark effect, or transition rates. Its success was a coincidence of the $1/n^2$ law being insensitive to the distinction between $n$ and $\ell$.

Schrödinger's wave-equation solution (1926) gave the correct energies, the correct wave functions, and the correct degeneracy. His approach — solving the partial differential equation in spherical coordinates — is the method followed in most textbooks. But Pauli's solution, completed months before Schrodinger's, used the operator algebra of $\hat{\mathbf{L}}$ and $\hat{\mathbf{A}}$ to derive the spectrum without ever writing down a wave function. Pauli's method is the precursor of the algebraic methods used throughout modern quantum theory.

Fock's 1935 momentum-space analysis revealed the geometric origin of the $\mathrm{SO}(4)$ symmetry: the bound-state problem on ${\mathbb{R}}^3$ is equivalent, via stereographic projection, to the free particle on $S^3$. The degeneracy $n^2$ is the dimension of the $(n-1)$-dimensional representation of $\mathrm{SO}(4)$ acting on $S^3$, and the Runge-Lenz vector is the Noether charge of this hidden symmetry.

Philosophically, the hydrogen atom illustrates the central puzzle of quantum mechanics: the wave function $\psi$ is not a physical field like the electromagnetic field, but a probability amplitude whose squared modulus gives measurable probabilities. The electron is not a particle orbiting the nucleus, nor a wave spread through space. The ground-state cloud is a stationary probability distribution with no time evolution (apart from a global phase), and yet it supports a nonzero kinetic energy of 13.6 eV. The hydrogen atom forces us to abandon every classical picture of what an electron is.

Bibliography [Master]

• Griffiths, D. J. and Schroeter, D. F. Introduction to Quantum Mechanics, 3rd ed. Cambridge University Press, 2018. Ch. 4.1-4.3.
• Sakurai, J. J. and Napolitano, J. Modern Quantum Mechanics, 2nd ed. Cambridge University Press, 2017. Ch. 3.6-3.7.
• Bohr, N. "On the Constitution of Atoms and Molecules." Phil. Mag. 26, 1-25 (1913).
• Pauli, W. "Über das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik." Z. Phys. 36, 336-363 (1926).
• Fock, V. "Zur Theorie des Wasserstoffatoms." Z. Phys. 98, 145-154 (1935).
• Bander, M. and Itzykson, C. "Group Theory and the Hydrogen Atom." Rev. Mod. Phys. 38, 330-345 (1966).
• Schiff, L. I. Quantum Mechanics, 3rd ed. McGraw-Hill, 1968. §16-18.
• Landau, L. D. and Lifshitz, E. M. Quantum Mechanics: Non-Relativistic Theory, 3rd ed. Pergamon, 1977. §36-38.
• Messiah, A. Quantum Mechanics, Vol. 1. Dover, 1999. Ch. XI.
• Cohen-Tannoudji, C., Diu, B. and Laloe, F. Quantum Mechanics, Vol. 1. Wiley, 1991. Ch. VII.
• Shankar, R. Principles of Quantum Mechanics, 2nd ed. Springer, 1994. Ch. 13.
• Susskind, L. and Friedman, A. Quant Mechanics: The Theoretical Minimum. Basic Books, 2014. Lecture 7.
• Bransden, B. H. and Joachain, C. J. Quantum Mechanics, 2nd ed. Pearson, 2000. Ch. 6-7.
• Dirac, P. A. M. The Principles of Quantum Mechanics, 4th ed. Oxford University Press, 1958. Ch. VI.
• Ballentine, L. E. Quantum Mechanics: A Modern Development. World Scientific, 1998. Ch. 10.
• Wybourne, B. G. Classical Groups for Physicists. Wiley, 1974. Ch. 16-17 (SO(4) and the hydrogen atom).