The Hydrogen Atom Schrödinger Equation

The reduced-mass, spherical-coordinate hydrogen Hamiltonian is not a historical curiosity; it is the computational and conceptual seed of essentially all of atomic and molecular electronic structure theory. Every basis set used in quantum chemistry software — Gaussian-type orbitals, Slater-type orbitals, numerical grids in density-functional codes — is built to mimic the exact hydrogenic radial and angular functions that emerge once this equation is separated. Muonic hydrogen spectroscopy, a direct descendant of the reduced-mass correction in Example 2, has been used to measure the proton's charge radius to a precision that created the "proton radius puzzle" debated actively in precision atomic physics. Positronium spectroscopy — governed by exactly this Hamiltonian with $\mu = m_e/2$ — tests bound-state quantum electrodynamics and probes possible new short-range interactions. Rydberg atom physics, used in quantum sensing and in circuit-QED platforms for quantum computing, relies on the same central-potential Schrödinger equation extended to high principal quantum number, where the electron sits so far from the ionic core that hydrogenic formulas apply almost unmodified even to multi-electron atoms. In astrophysics, the reduced-mass-corrected Rydberg constant sets the precise rest-frame wavelengths of the Lyman and Balmer series used to measure redshifts of distant galaxies and to detect deuterium abundances left over from Big Bang nucleosynthesis. Every one of these applications begins from the same starting point developed here: strip away the translational motion of the whole atom, keep the internal relative motion with its reduced mass, and set that one-body Schrödinger equation up in the spherical coordinates matched to the Coulomb potential's symmetry.

Ask the tutor: How does separating this equation in spherical coordinates actually produce the radial and angular quantum numbers n, ℓ, and mℓ?

Picture a proton and an electron connected by an invisible Coulomb spring — not a mechanical spring, but a $1/r$ attraction that never vanishes and never saturates. Classically, both particles orbit their common center of mass. Quantum mechanically, both particles are represented by a joint wavefunction $\Psi(\mathbf r_e, \mathbf r_N)$ depending on six coordinates: three for the electron's position and three for the nucleus's position. The full non-relativistic Hamiltonian is

The two kinetic-energy terms act independently on the electron and nuclear coordinates, but the potential energy couples them together through the separation vector $\mathbf r = \mathbf r_e - \mathbf r_N$. This coupling is exactly what makes the six-dimensional problem look intractable — until a change of variables is introduced. The trick, familiar from classical two-body mechanics (planets orbiting a star, or a diatomic vibrating and rotating), is to trade $(\mathbf r_e, \mathbf r_N)$ for the center-of-mass coordinate $\mathbf R$ and the relative coordinate $\mathbf r$. Because the potential depends only on $\mathbf r$, this substitution decouples the Hamiltonian exactly, with no approximation, into a free-particle piece describing the atom's total translation through space, and a one-body piece describing internal relative motion of an electron "around" the nucleus, carrying an effective mass — the reduced mass $\mu$ — instead of the bare electron mass. Since chemists care about internal structure (orbital shapes, energy levels, spectra) rather than the atom's flight through the lab, the translational piece is dropped and all attention focuses on the internal Hamiltonian,

This is now a genuinely one-body problem: a single fictitious particle of mass $\mu$, position $\mathbf r$ relative to a fixed origin, moving in the central potential $V(r) = -e^2/4\pi\varepsilon_0 r$. Central potentials are special because they depend only on the magnitude $r = |\mathbf r|$, never on direction. This spherical symmetry is why spherical polar coordinates $(r,\theta,\phi)$ — not Cartesian $(x,y,z)$ — are the natural coordinate system: in spherical coordinates the potential energy operator is simply a function of the single coordinate $r$, while the entire angular dependence is isolated inside the kinetic-energy operator's Laplacian. The diagram below shows the two-body-to-one-body reduction schematically, followed by the coordinate system in which the resulting one-body problem is solved.

The physical content of this picture is that the Coulomb potential's spherical symmetry guarantees a conserved angular momentum. Because $\hat H_{\text{rel}}$ commutes with both $\hat L^2$ and $\hat L_z$, every eigenstate of $\hat H_{\text{rel}}$ can be chosen simultaneously to be an eigenstate of $\hat L^2$ and $\hat L_z$. This is the deep reason the eventual solutions factor into a radial function times a spherical harmonic, and it is why the quantum numbers $\ell$ and $m_\ell$ appear naturally, well before the radial equation is even written down.

Full two-body Hamiltonian. The starting point, with the electron at $\mathbf r_e$ (mass $m_e$, charge $-e$) and the nucleus at $\mathbf r_N$ (mass $m_N$, charge $+Ze$ for a hydrogenic ion of nuclear charge $Z$):

where: $\nabla_e^2, \nabla_N^2$ are Laplacians in the electron and nuclear coordinates separately; $Z=1$ for ordinary hydrogen; the potential couples the two sets of coordinates through their difference only. Reading: this is exact and non-relativistic; it is the six-coordinate starting point that every hydrogenic derivation must reduce.

Center-of-mass and relative coordinates.

where: $\mathbf R$ locates the atom's center of mass; $\mathbf r$ is the internal separation vector whose magnitude is the electron–nucleus distance. Reading: this is a linear, invertible change of variables — no information is lost, and the transformation is exact.

Separated Hamiltonian.

where: $M = m_e + m_N$ is the total mass (governs free translation of the whole atom through space); $\mu = m_e m_N/(m_e+m_N)$ is the reduced mass (governs internal relative motion); $r = |\mathbf r|$. Reading: the Hamiltonian is now a sum of two independent pieces acting on different coordinates — a textbook signal that the total wavefunction factors as $\Psi(\mathbf R,\mathbf r) = \chi(\mathbf R)\,\psi(\mathbf r)$ and the total energy is a sum $E = E_{\text{trans}} + E_{\text{rel}}$.

Internal (relative-motion) time-independent Schrödinger equation, the one that defines atomic structure.

where: $\nabla^2$ is now understood as acting on the relative coordinate; $\psi$ is the internal (electronic) wavefunction depending on the electron's position relative to the nucleus; $E$ is the internal energy — the number tabulated as hydrogen's orbital energies. Reading: everything a chemist calls "the hydrogen atom's wavefunction" is $\psi(\mathbf r)$, the solution of this equation — never the full six-coordinate $\Psi$.

Laplacian in spherical polar coordinates, needed because $V$ depends only on $r$:

where: the first term is the radial kinetic energy piece; the bracketed $\theta,\phi$ terms, multiplied by $-\hbar^2/2\mu$, combine exactly into $\hat L^2/(2\mu r^2)$, the centrifugal (angular-momentum) energy. Reading: writing the Laplacian this way is what makes the eventual separation of variables into radial and angular factors possible — the angular momentum operator $\hat L^2$ is hiding inside the last two terms, ready to be pulled out.

Reduced mass, explicitly.

where: as $m_N \to \infty$ (infinitely heavy, immobile nucleus), $\mu \to m_e$ exactly. Reading: the reduced mass is always slightly less than the smaller of the two masses; for hydrogen it is 99.95% of $m_e$, a small but experimentally resolvable correction (it is exactly this correction that separates ordinary hydrogen's spectral lines from deuterium's).

Example 1 — Foundational: computing the reduced mass of ordinary hydrogen and its fractional deviation from $m_e$. Use $m_e = 9.10938\times10^{-31}\text{ kg}$ and the proton mass $m_p = 1.67262\times10^{-27}\text{ kg}$ (a genuinely asymmetric, non-round mass ratio — the proton is 1836.15 times heavier than the electron).

Numerator:

Denominator (the proton mass utterly dominates, but keep both terms):

so

The fractional deviation from the bare electron mass is

so $\mu$ is about 0.0545% smaller than $m_e$ — small, but this exact number is what shifts the Rydberg constant for hydrogen ($R_H$) below the infinite-mass Rydberg constant $R_\infty$ by precisely this fraction, a shift precise enough to distinguish hydrogen's Balmer lines from deuterium's in high-resolution spectroscopy. Limiting case check: if the nucleus were infinitely heavy, $m_N\to\infty$, then $\mu = m_e/(1+m_e/m_N) \to m_e$ exactly — recovering the familiar (approximate) textbook formulas that use $m_e$ in place of $\mu$. The finite-proton-mass calculation above shows quantitatively how good that approximation is (correct to four significant figures) and exactly how it fails at the fifth.

Example 2 — Applied: reduced mass and energy-level shift for a muonic hydrogen atom. Muonic hydrogen replaces the electron with a muon of mass $m_\mu = 206.768\,m_e$, orbiting a proton with $m_p = 1836.15\,m_e$ — again markedly asymmetric numbers, chosen so no cancellation hides an error. Compute the reduced mass in units of $m_e$:

Compare this to the electronic case ($\mu_e = 0.99946\,m_e$ from Example 1, expressed in units of $m_e$). Since the hydrogenic ground-state energy scales linearly with the reduced mass,

the muonic hydrogen ground-state energy is

roughly 186 times deeper than ordinary hydrogen's $-13.6$ eV ground state. Physically, the muon is far heavier than the electron, so it "carries" much more of the reduced mass and orbits the proton in a correspondingly smaller, more tightly bound orbit — the Bohr radius scales as $1/\mu$, so muonic hydrogen's Bohr radius is about $1/186$ of ordinary hydrogen's, roughly $2.85\times10^{-13}$ m, small enough that the muon samples the finite size of the proton itself (this sensitivity is exactly what the muonic-hydrogen Lamb shift measurements exploit to determine the proton's charge radius). Limiting case check: setting $m_\mu \to m_e$ recovers $\mu_\mu \to \mu_e$ from Example 1, confirming the formula reduces correctly to ordinary hydrogen when the orbiting particle's mass is dialed back to the electron mass.

Example 3 — Exam/competition level: verifying that the cross-term cancellation in the kinetic energy separation genuinely requires the specific center-of-mass weighting, by testing an incorrect (unweighted) choice. Suppose a student, in a hurry, mistakenly defines the "center of mass" using an equal-weighted average instead of the correct mass-weighted average:

Show that this choice does not eliminate the cross term, using asymmetric masses $m_e$ and $m_N = 10\,m_e$ (chosen arbitrarily unequal to expose the failure clearly; the real hydrogen mass ratio is even more extreme but this suffices to show the mechanism). Inverting: $\mathbf r_e = \mathbf R_{\text{wrong}} + \mathbf r/2$, $\mathbf r_N = \mathbf R_{\text{wrong}} - \mathbf r/2$. By the chain rule (as in Step 3 of the first derivation, but now with coefficient $1/2$ replacing $m_e/M$ and $m_N/M$):

Now divide by the true masses when assembling kinetic energy (mass does not change just because the coordinate definition changed):

Adding, the cross-term coefficient is

which is manifestly nonzero whenever $m_e \neq m_N$. The cross term survives — the naive unweighted average fails to decouple the kinetic energy. This confirms, by direct counterexample, that the mass-weighted definition $\mathbf R = (m_e\mathbf r_e + m_N\mathbf r_N)/M$ used in the derivation is not an arbitrary convenience but the unique linear combination (up to trivial rescaling) that cancels the cross term for unequal masses; only when $m_e = m_N$ would the naive average happen to work by symmetry, which is precisely why a symmetric test case would have hidden this error.