The hydrogen atom

Result

The hydrogen atom

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Statement

Atomic orbitals as exact solutions of the Schrodinger equation.

Why it matters

particle-in-a-box showed how confinement alone quantises energy in the simplest possible geometry; the hydrogen atom is the first genuinely realistic three-dimensional system for which the Schrödinger equation can still be solved exactly, and its solution is the direct origin of the atomic orbitals (\(s\), \(p\), \(d\), \ldots) and quantum numbers that organise the entire periodic table and every subsequent bonding theory. Every result in this network that refers to an orbital, an electron configuration, or a quantum number ultimately traces back to this exact solution as its foundational reference case, since no other real atom or molecule admits an exact analytical solution at all.

Hypotheses

The electron moves in the exact Coulomb potential of a single, fixed (infinitely heavy or externally clamped) point-charge nucleus.This single-particle, central-field problem is what makes an exact analytical solution possible at all; treating the nucleus as genuinely fixed (rather than allowing for the small correction from its finite mass and the two-body reduced-mass motion) introduces only a very small, well-understood correction, but any system with more than one electron immediately loses exact solvability, since electron–electron repulsion is not a central-field, one-body potential (hartree-fock exists specifically to approximate this harder, many-electron case). The problem is treated non-relativistically, and electron spin is added afterward as an extra, independent quantum number rather than emerging from the equation itself.A fully relativistic treatment (the Dirac equation) is required to obtain electron spin and fine-structure splitting as a genuine consequence of the underlying equation rather than as a separately appended label; for most chemical purposes the non-relativistic treatment developed here, with spin appended by hand, is entirely sufficient.

Proof

\hat H = -\frac{\hbar^2}{2m}\nabla^2 - \frac{e^2}{4\pi\epsilon_0 r}

The full three-dimensional time-independent Schrödinger equation for a single electron in the exact Coulomb potential of a nucleus of charge \(+e\) is written down; the potential energy term depends only on \(r\), the electron–nucleus distance, making this a genuine central-field problem. B

\psi(r,\theta,\phi) = R(r)\,\Theta(\theta)\,\Phi(\phi)

Because the potential depends only on \(r\) and not on the angular coordinates, the equation separates cleanly in spherical coordinates into independent radial and angular parts, a standard technique (separation of variables) applicable to any central-field problem; the angular parts, common to every central-field problem, are the spherical harmonics. B

n=1,2,3,\ldots; \quad l=0,1,\ldots,n-1; \quad m_l=-l,\ldots,+l

Requiring the separated radial and angular solutions to be well-behaved (finite, single-valued, and normalisable everywhere, including at the origin and as \(r\to\infty\)) forces three independent quantum numbers to take only specific, discrete, mutually constrained integer values — the exact same physical requirement (boundary-condition-enforced quantisation) that produced discrete energies in particle-in-a-box, now generalised to three dimensions with an added angular-momentum structure. B

E_n = -\frac{m e^4}{8\epsilon_0^2h^2n^2} = -\frac{13.6\,\text{eV}}{n^2}

Solving the radial equation explicitly gives energies depending only on the principal quantum number \(n\), not on \(l\) or \(m_l\) — a special, "accidental" degeneracy specific to the exact \(1/r\) Coulomb potential, reproducing exactly the same energy-level formula the Bohr model had already found empirically-successfully, but now derived rigorously from the full Schrödinger equation. A

Result

E_n = -\frac{13.6\,\text{eV}}{n^2}, \qquad \text{degeneracy} = n^2\ \text{(before spin)}

Reading. The hydrogen atom's allowed energies depend only on the principal quantum number \(n\); for each \(n\), the angular momentum quantum number \(l\) can take \(n\) different values (\(0\) to \(n-1\)), and each \(l\) has \(2l+1\) possible orientations \(m_l\), giving \(n^2\) total degenerate spatial orbitals (\(2n^2\) states once spin is included) at each energy level.

Scope. Exact for a genuine one-electron, one-nucleus system (hydrogen itself, or any hydrogen-like ion such as \(\text{He}^+\) or \(\text{Li}^{2+}\) with \(Z\) substituted appropriately); the \(n\)-only energy degeneracy is broken immediately in any many-electron atom, where electron–electron repulsion makes energy depend on \(l\) as well (the basis of the building-up principle's orbital energy ordering).

Corollaries & converses

huckel-theory's and lcao-molecular-orbitals' entire framework of building molecular orbitals from atomic orbitals presupposes that atomic orbitals (the \(s,p,d\) shapes derived exactly here) are a meaningful, well-defined starting basis — this result is what actually supplies that basis, rather than it being an assumption made for convenience.
variational-principle offers the standard alternative route to approximate energies for systems (essentially every atom or molecule besides hydrogen-like ions) where an exact solution of this kind is not available.
Converse: the close numerical agreement between this exact quantum-mechanical energy formula and the earlier, semi-classical Bohr model's energy formula is not a coincidence; it reflects a genuine, historically important correspondence between the two theories for this specific, exactly solvable one-electron problem.

Fails without

Apply the exact \(n\)-only degeneracy pattern of Step 4 to a many-electron atom: once a second electron is present, electron–electron repulsion breaks the central-field, exactly-solvable structure the derivation depends on (Hypotheses), and orbital energies split apart by \(l\) as well as \(n\) — the "accidental" \(n\)-only degeneracy found here is a special feature of the pure \(1/r\) one-electron potential, not a general atomic property.
Ignore the non-relativistic hypothesis and expect the formula to capture fine-structure splitting: spectroscopically observed fine structure (small energy splittings within a given \(n,l\) level) arises from relativistic and spin–orbit effects entirely absent from this non-relativistic treatment, which predicts a false degeneracy among states that a fully relativistic treatment splits apart.

Common errors

Assuming energy depends on \(l\) as well as \(n\) for hydrogen itself, carrying over intuition from many-electron atoms where this is true (building-up principle) but which does not hold for the exact one-electron hydrogen problem (Result, Scope).
Forgetting the constraint \(l\leq n-1\), and proposing invalid combinations such as \(n=2,\ l=2\), which are not physically allowed solutions of Step 3.
Treating the \(n^2\) spatial degeneracy as the total degeneracy without doubling for the two independent spin states, undercounting the true number of degenerate quantum states at each energy level by a factor of two.
Confusing an orbital (a specific mathematical solution \(\psi\), a probability-amplitude function) with a literal, sharply bounded orbit or trajectory the electron physically follows, an outdated conceptual holdover from the earlier Bohr model that this fully quantum-mechanical treatment specifically supersedes.

Discussion

Erwin Schrödinger solved the hydrogen atom exactly in 1926, in one of the founding papers of wave mechanics, obtaining energies numerically identical to Niels Bohr's earlier (1913) semi-classical model but from a fundamentally different, and far more generally applicable, theoretical foundation; the exact match on energies, despite the very different underlying physical pictures, is one of the striking early triumphs establishing quantum mechanics as the correct successor theory.

The hydrogen atom remains, to this day, essentially the only genuinely exactly solvable atom or small molecule in all of quantum chemistry (alongside a handful of closely related one-electron ions); every approximate method developed later in this unit — variational-principle, hartree-fock, density-functional-theory — exists precisely because no other system of chemical interest admits a comparable exact, closed-form solution, making hydrogen's exact result an indispensable benchmark and conceptual anchor throughout the rest of quantum chemistry.

Common misconception: that an atomic orbital represents a literal, well-defined path or orbit the electron travels along, in analogy with the earlier, now-superseded Bohr model. An orbital is properly a probability-amplitude function, \(|\psi|^2\) giving a probability density for the electron's position rather than a trajectory; the electron does not follow any definite path at all within the fully quantum-mechanical picture developed here.

Worked examples

n=2:\quad l=0\ (\text{one } 2s\text{ orbital}),\quad l=1\ (\text{three } 2p \text{ orbitals: } m_l=-1,0,+1)

Applying Step 3's quantum-number rules at \(n=2\) gives exactly \(1+3=4=n^2\) degenerate spatial orbitals, all sharing the identical energy \(E_2=-13.6/4=-3.4\,\text{eV}\) by Step 4's \(n\)-only dependence. A

\Delta E_{2\to1} = E_2-E_1 = -3.4-(-13.6) = 10.2\,\text{eV}

The energy released on transition from \(n=2\) to \(n=1\) is computed directly from Step 4's formula; converting to wavelength via \(\Delta E=hc/\lambda\) reproduces the observed Lyman-\(\alpha\) spectral line, a direct, historically important experimental confirmation of the energy-level formula. A

E_2-E_1 = 10.2\,\text{eV}\ (\text{Lyman-}\alpha)

Reading. The exact energy formula, derived purely from solving the Schrödinger equation, reproduces the observed hydrogen emission spectrum quantitatively and exactly, the same spectral lines the earlier Bohr model had already matched empirically.

Scope. The identical formula, with \(Z^2\) inserted for a hydrogen-like ion of nuclear charge \(Z\), predicts the analogous spectral series for \(\text{He}^+\), \(\text{Li}^{2+}\), and other one-electron ions.

Problems

List all allowed \((l,m_l)\) combinations for \(n=3\), and confirm the total spatial degeneracy equals \(9=n^2\).
Solution
\(l=0\): \(m_l=0\) (1 orbital, \(3s\)). \(l=1\): \(m_l=-1,0,+1\) (3 orbitals, \(3p\)). \(l=2\): \(m_l=-2,-1,0,+1,+2\) (5 orbitals, \(3d\)). Total: \(1+3+5=9=3^2\), confirming Step 3/4's degeneracy formula.
Compute the energy, in eV, required to ionise a hydrogen atom from its \(n=1\) ground state, and explain why this is also called the atom's first ionisation energy.
Solution
Ionisation corresponds to removing the electron to \(n\to\infty\), where \(E_\infty=0\) by Step 4. The required energy is \(E_\infty-E_1 = 0-(-13.6) = 13.6\,\text{eV}\), matching the well-known hydrogen ionisation energy. This is called the first ionisation energy because it is the energy needed to remove the single, first (and only) electron from the neutral atom.
Explain why the exact energy-level formula of Step 4 depends only on \(n\) and not on \(l\), while in a many-electron atom (as encountered in the building-up principle) orbital energy depends strongly on \(l\) as well.
Solution
The \(n\)-only dependence is a special, "accidental" degeneracy specific to the exact \(1/r\) Coulomb potential of a single electron around a single nucleus (Hypotheses); it arises from a deeper mathematical symmetry unique to this precise potential form. In a many-electron atom, each electron experiences not just the bare nuclear attraction but also repulsion from every other electron, and this repulsion depends on the detailed shape of each orbital (how much an electron in a given \(l\) penetrates close to the nucleus past the other electrons' shielding), breaking the clean \(1/r\) form and, with it, the special \(n\)-only degeneracy found here.