Hydrogen Atom

The hydrogen atom is the canonical exactly solvable three-dimensional bound-state problem. In the nonrelativistic approximation, the electron and proton interact through an attractive Coulomb potential. After separating the center-of-mass motion, the relative coordinate obeys a one-body central-potential problem.

Hydrogen is important because it is both concrete and structurally rich: it introduces principal, orbital, and magnetic quantum numbers; it connects spherical harmonics to radial special functions; and it exposes degeneracies that later corrections split.

Scope of the Model

This page treats the spinless nonrelativistic Coulomb Hamiltonian for a point electron and point proton, with reduced mass. It does not include fine structure, hyperfine structure, the Lamb shift, finite nuclear size, external fields, radiative transitions, or many-electron effects.

Those corrections are physically essential for precision spectroscopy, but they should not be folded into the basic Coulomb solution. The purpose here is to solve the standard Schrödinger eigenvalue problem cleanly.

Two-Body Reduction

Let the electron and proton masses be $m_e$ and $m_p$. The reduced mass is

$$\mu = \frac{m_em_p}{m_e+m_p}.$$

After separating the free center-of-mass motion, the relative-coordinate Hamiltonian is

$$\hat H = -\frac{\hbar^2}{2\mu}\nabla^2 - \frac{e^2}{4\pi\epsilon_0r}.$$

The stationary Schrödinger equation is

$$\hat H\psi(\mathbf r)=E\psi(\mathbf r).$$

It is useful to introduce the reduced-mass Bohr radius

$$a_0 = \frac{4\pi\epsilon_0\hbar^2}{\mu e^2}.$$

If one uses the electron mass instead of $\mu$, this is the infinite-proton-mass approximation. The numerical difference is small for hydrogen but conceptually important.

Separation and Quantum Numbers

Because the potential depends only on $r$, write

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

The angular quantum numbers satisfy

$$\ell=0,1,\ldots,n-1, \qquad m=-\ell,-\ell+1,\ldots,\ell.$$

The principal quantum number is

$$n=1,2,3,\ldots.$$

The angular meaning of the $n,\ell,m$ labels, parity, and shell multiplets is developed in Hydrogen Atom Angular Structure.

Within the spinless bound-state sector, the commuting observables $H$, $L^2$, and $L_z$ distinguish the states by $n$, $\ell$, and $m$. Their role as a sector-dependent complete set is explained in Complete Sets of Commuting Observables.

The radial equation is the Coulomb specialization of the general radial equation:

$$-\frac{\hbar^2}{2\mu}\frac{d^2u}{dr^2} + \left[ -\frac{e^2}{4\pi\epsilon_0r} + \frac{\hbar^2\ell(\ell+1)}{2\mu r^2} \right]u =Eu, \qquad u=rR.$$

Normalizable bound-state solutions exist only for the allowed combinations of $n$ and $\ell$ above. The angular part is supplied by spherical harmonics, and the radial polynomial part is supplied by Laguerre polynomials.

Energy Spectrum

The bound-state energies are

$$E_n = -\frac{\mu e^4}{2(4\pi\epsilon_0)^2\hbar^2} \frac{1}{n^2} = -\frac{\hbar^2}{2\mu a_0^2} \frac{1}{n^2}.$$

For ordinary hydrogen this is approximately

$$E_n\approx-\frac{13.6\,\mathrm{eV}}{n^2},$$

with the reduced-mass correction included in high-precision values.

The energy depends only on $n$, not on $\ell$ or $m$, in the ideal nonrelativistic Coulomb problem. The $m$ degeneracy follows from rotational symmetry for any central potential. The additional $\ell$ degeneracy is special to the Coulomb potential and is explained in Degeneracy of the Hydrogen Atom.

For a fixed $n$, the number of spatial bound states is

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

If spin is included without spin-dependent interactions, this doubles to $2n^2$. Fine structure, hyperfine structure, Lamb-shift physics, and external fields split this ideal degeneracy.

Radial Wavefunctions

The detailed reference for the functions in this section is Radial Wavefunctions.

With

$$\rho=\frac{2r}{na_0},$$

one common normalized convention is

$$R_{n\ell}(r) = \sqrt{ \left( \frac{2}{na_0} \right)^3 \frac{(n-\ell-1)!}{2n[(n+\ell)!]} } e^{-\rho/2} \rho^\ell L_{n-\ell-1}^{(2\ell+1)}(\rho).$$

The full wavefunction is $R_{n\ell}Y_\ell^m$. The normalization convention is

$$\int_0^\infty \lvert R_{n\ell}(r)\rvert^2r^2\,dr =1, \qquad \int_{S^2} \lvert Y_\ell^m\rvert^2\,d\Omega =1.$$

Equivalently, if $u_{n\ell}=rR_{n\ell}$, then

$$\int_0^\infty \lvert u_{n\ell}(r)\rvert^2\,dr =1.$$

The ground state has $n=1$, $\ell=0$, and $m=0$. Its radial and full wavefunctions are

$$R_{10}(r) = 2a_0^{-3/2}e^{-r/a_0}, \qquad \psi_{100}(r,\theta,\phi) = \frac{1}{\sqrt{\pi a_0^3}}e^{-r/a_0}.$$

The wavefunction is largest at the origin, but the radial probability density includes the volume factor $r^2$:

$$P_{10}(r) = \lvert R_{10}(r)\rvert^2r^2 = \frac{4r^2}{a_0^3}e^{-2r/a_0}.$$

This radial probability density is maximal at $r=a_0$. This is one precise sense in which the Bohr radius sets the size of the ground-state atom.

Orbitals and Nodes

Hydrogenic orbitals are stationary one-electron wavefunctions, not classical electron trajectories. The labels $s,p,d,\ldots$ correspond to $\ell=0,1,2,\ldots$. Complex orbitals are natural angular-momentum eigenstates; real orbitals are particular linear combinations within a degenerate $m$ subspace. The notation, real-versus-complex basis choice, nodal surfaces, and visualization cautions are developed in Atomic Orbitals.

For fixed $n$ and $\ell$, the number of radial nodes is

$$n-\ell-1.$$

Angular nodes come from the spherical harmonics. The total nodal structure is therefore a product of radial and angular information. A separate page can treat orbital visualization in more detail; this page uses orbitals only as labels for the exact eigenfunctions.

Physical Interpretation

The Coulomb potential supplies an infinite ladder of bound states accumulating at $E=0$ from below. States with larger $n$ are less tightly bound and extend to larger radii. The continuum above $E=0$ describes ionized scattering states; see Continuum States of the Coulomb Problem for the positive-energy sector.

The quantum numbers have distinct meanings:

• $n$ controls the energy and overall radial scale in the ideal Coulomb problem.
• $\ell$ controls orbital angular momentum and the centrifugal barrier.
• $m$ controls the projection of orbital angular momentum along the chosen $z$ axis.

The $z$ axis is a convention unless an external field or measurement apparatus selects a direction.

Common Mistakes

• Using $m_e$ without noticing the reduced-mass approximation.
• Treating orbitals as paths followed by an electron.
• Forgetting the $r^2$ measure when interpreting radial probabilities.
• Assuming the ideal $n^2$ degeneracy survives all physical corrections.
• Confusing the radial wavefunction $R_{n\ell}$ with the radial probability density $\lvert R_{n\ell}\rvert^2r^2$.
• Thinking that $m$ changes the radial equation in a rotationally invariant Coulomb problem.

Exercises

1. Normalize the ground-state radial wavefunction $R_{10}(r)=2a_0^{-3/2}e^{-r/a_0}$ using the radial measure.

Solution

Compute

$$\int_0^\infty \lvert R_{10}(r)\rvert^2r^2\,dr = \frac{4}{a_0^3} \int_0^\infty r^2e^{-2r/a_0}\,dr.$$

Using

$$\int_0^\infty r^2e^{-\alpha r}\,dr = \frac{2}{\alpha^3},$$

with $\alpha=2/a_0$, the integral becomes

$$\frac{4}{a_0^3} \frac{2}{(2/a_0)^3} =1.$$

2. Show that the spatial degeneracy of a fixed principal shell is $n^2$.

Solution

For fixed $n$, the allowed orbital quantum numbers are $\ell=0,1,\ldots,n-1$. For each $\ell$, there are $2\ell+1$ allowed values of $m$. Therefore

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

3. Find the most probable radius in the ground state.

Solution

The radial probability density is

$$P_{10}(r) = \frac{4r^2}{a_0^3}e^{-2r/a_0}.$$

Its derivative is proportional to

$$\frac{d}{dr} \left( r^2e^{-2r/a_0} \right) = 2re^{-2r/a_0} \left( 1-\frac{r}{a_0} \right).$$

The critical points are $r=0$ and $r=a_0$. The nonzero maximum occurs at

$$r=a_0.$$

Where This Is Used

• Radial Schrödinger Equation gives the general central-potential equation and boundary conventions.
• Coulomb Potential sets the sign, reduced-mass, Bohr-radius, and Rydberg-scale conventions used here.
• Hydrogenic Ions rescales this solution to one-electron ions such as $\mathrm{He}^+$ and $\mathrm{Li}^{2+}$.
• Radial Wavefunctions gives the normalized $R_{n\ell}$ functions, radial probabilities, nodes, and expectation values.
• Atomic Orbitals explains the $s,p,d,\ldots$ notation, real and complex orbital bases, nodal surfaces, and orbital visualization cautions.
• Continuum States of the Coulomb Problem explains how the positive-energy sector completes the Coulomb spectrum.
• Effective Radial Potential interprets the Coulomb centrifugal barrier and radial turning-point picture.
• Boundary Conditions for Radial Wavefunctions explains the regularity and normalizability conditions used by the Coulomb radial solution.
• Central Potentials explains why the Coulomb problem carries $\ell$ and $m$ labels.
• Hydrogen Atom Angular Structure summarizes the $n,\ell,m$ shell decomposition, parity, and angular multiplets.
• Angular and Radial Separation explains why the Coulomb eigenfunctions factor as $R_{n\ell}(r)Y_\ell^m(\theta,\phi)$.
• Spherical Coordinates fixes the angular convention and radial measure used in the wavefunctions.
• Degeneracy of the Hydrogen Atom separates ordinary rotational degeneracy from the special Coulomb degeneracy.
• Hydrogen Atom Model Card gives the short reference-library summary.
• Hydrogen as Atomic Prototype interprets this exact solution as atomic labels, spectral series, and a hierarchy of real-hydrogen corrections.
• Spherical Harmonics supplies the angular factors.
• Laguerre Polynomials gives the special-function family used in the radial solution.
• Units and Constants fixes common unit conventions.

References

• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.