Three exact solutions and one inequality: quantum chemistry's ground floor, computed

Rebuilt July 2026. The original version of this post was an early draft in lab-report form, and several of its tables reported numbers its own code never produced. This rewrite keeps the subject and the URL, redoes every calculation, and holds each number against the closed-form answer — the same standard the posts that now stand on this one are held to: the Hartree–Fock machinery one floor up, and the correlation gap, measured above that.

Quantum mechanics hands chemistry exactly three systems worth solving with pen and paper: a particle trapped between hard walls, a particle on a spring, and one electron bound to one proton. Everything else — every molecule, every spectrum, every reaction barrier — is approximation layered on top of what these three teach. The box teaches that confinement quantizes energy. The oscillator teaches that motion never stops, even at absolute zero. The hydrogen atom teaches what an orbital actually is. And running through all three is one inequality, Heisenberg’s, which is less a fourth topic than the grammar the other three are written in.

There is a second reason to compute these systems, and it is the working rule of this post. Because the exact answers exist in closed form¹, these are the only problems where a numerical method can be graded against truth rather than against another approximation. Every expectation value below is computed twice — once by numerical integration on a grid, once from the analytic formula — and the two are shown side by side. Where a grid is too short or an integration rule too crude, the failure is displayed rather than hidden. That habit is cheap here and priceless later, where no closed form will be waiting.

All numerics are NumPy/SciPy on a laptop; the hydrogen-atom ladder in §4 uses psi4 1.11². The pattern for every expectation value is three lines:

import numpy as np
x = np.linspace(0, L, 20001)
psi = np.sqrt(2/L) * np.sin(n*np.pi*x/L)
x_mean = np.trapezoid(x * psi**2, x)      # <x> = ∫ x |ψ|² dx

1. The equation and the rule

The time-independent Schrödinger equation in one dimension,

\[ -\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + V(x)\,\psi = E\,\psi , \]

is an eigenvalue problem: the potential \(V(x)\) goes in, and out come the allowed energies \(E_n\) and their wavefunctions \(\psi_n(x)\). The Born rule gives \(|\psi(x)|^2\) its meaning as a probability density, which obliges every wavefunction to be normalized, \(\int |\psi|^2\,dx = 1\), and makes every measurable prediction an integral:

\[ \langle A \rangle = \int \psi^*\hat{A}\,\psi\,dx , \qquad \Delta A = \sqrt{\langle A^2\rangle - \langle A\rangle^2} . \]

Those integrals are where theory meets numerics. On a computer, \(\psi\) is an array, the integral is a quadrature rule, and the domain is finite — three approximations before any physics happens. The three systems below are chosen because each one changes \(V(x)\) in the simplest possible way (walls, a parabola, a Coulomb well) and because for each, every integral above has a closed form to grade the array against.

2. The particle in a box: confinement quantizes

Take \(V = 0\) between hard walls at \(x=0\) and \(x=L\), infinite outside. The wavefunction must vanish at both walls, and that boundary condition alone does all the work: only sine waves that fit an integer number of half-wavelengths into the box survive,

\[ \psi_n(x) = \sqrt{\tfrac{2}{L}}\,\sin\!\Big(\tfrac{n\pi x}{L}\Big), \qquad E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}, \qquad n = 1, 2, 3, \ldots \]

Nothing about this is dynamical. Quantization falls out of geometry — a continuous energy would require a wave that fails to vanish at a wall. For an electron in a box the size of a small conjugated molecule, \(L = 1\) nm, the computed levels are \(E_1 = 0.376\) eV, \(E_2 = 1.504\) eV, \(E_3 = 3.384\) eV:

Figure 1. Wavefunctions of an electron in a 1 nm box, each drawn at the height of its energy level (dashed lines; \(\psi\) scaled by \(0.35\) for visibility). The state \(n\) has \(n-1\) interior nodes, and the energy climbs as \(n^2\) — the gaps widen going up, the opposite of the hydrogen atom’s crowding-together in §4.

The probability densities show where the electron actually is:

Figure 2. Probability densities for the same three states. Every curve encloses unit area; the dashed line marks \(\langle x\rangle = L/2\), the same for all three by symmetry. The \(n\)-th state splits its probability into \(n\) lobes separated by nodes where the electron is never found.

Now the accountability check. The closed forms for this system are \(\langle x\rangle = L/2\) and \(\langle x^2\rangle = L^2\big(\tfrac{1}{3} - \tfrac{1}{2n^2\pi^2}\big)\); the numerics are trapezoidal integration on a 20 001-point grid:

Table 1. Position uncertainty of an electron in a 1 nm box: numerical integration against the closed form, agreeing to all six digits shown. All states sit at \(\langle x\rangle = 0.5\) nm exactly.

Two physical readings of that last column. First, \(\Delta x\) grows with \(n\) — higher states spread their probability more evenly across the box. Second, it grows toward a ceiling: a uniform distribution on \([0, L]\) has \(\Delta x = L/\sqrt{12} \approx 0.2887\) nm, and \(n=3\) is already within 4% of it. High quantum numbers wash out the lobes and recover the classical answer — the correspondence principle, visible in a table.

The chemistry hiding in this toy: the \(n=1 \to 2\) gap for our 1 nm box is \(1.13\) eV, a photon of about \(1100\) nm. Stretch or shrink the box and the gap scales as \(1/L^2\) — which is, to first crudeness, why lengthening the conjugated backbone of a dye red-shifts its absorption³. One boundary condition, one visible consequence.

3. The harmonic oscillator: motion never stops

Replace the walls with a spring, \(V(x) = \tfrac{1}{2}m\omega^2x^2\) — the universal small-oscillation limit of any potential minimum, which is why this model underwrites every vibrational spectrum. The solutions are Gaussians dressed with Hermite polynomials,

\[ \psi_n(x) = N_n\,H_n(\alpha x)\,e^{-\alpha^2x^2/2}, \qquad \alpha = \sqrt{\tfrac{m\omega}{\hbar}}, \qquad E_n = \hbar\omega\big(n + \tfrac{1}{2}\big) . \]

The spectrum is a ladder with even rungs \(\hbar\omega\) apart, and the bottom rung is not at zero: \(E_0 = \tfrac{1}{2}\hbar\omega\). For the numbers below the particle is an electron and \(\omega = 5\times10^{14}\ \mathrm{s^{-1}}\), giving \(\hbar\omega = 0.329\) eV — about \(2650\ \mathrm{cm^{-1}}\), so this artificial oscillator happens to sit right in the vibrational infrared, between a C–H and an O–H stretch.

Figure 3. Harmonic-oscillator wavefunctions (scaled) drawn at their energy levels inside the parabolic potential (dashed). The ground state is a pure Gaussian sitting \(\tfrac{1}{2}\hbar\omega = 0.165\) eV above the bottom of the well — the zero-point energy. Note the tails leaking past the classical turning points, where the potential exceeds the state’s total energy.

That non-negotiable \(0.165\) eV floor is not a technicality. Zero-point energy is why helium never freezes at atmospheric pressure, and its dependence on mass (\(\omega \propto 1/\sqrt{\mu}\)) is why deuteration slows C–H bond cleavage — the kinetic isotope effect is zero-point arithmetic.

Where the old draft went wrong — and how to catch it. The previous version of this post tabulated \(\Delta x\cdot\Delta p = \hbar/2\) for every oscillator state and offered it as a computational verification of the uncertainty principle. It was nothing of the sort: the code had computed \(\Delta p := \hbar/(2\Delta x)\) — momentum uncertainty defined by the relation it claimed to verify. The correct procedure computes \(\langle p^2\rangle\) independently from the wavefunction’s curvature, \(\langle p^2 \rangle = \hbar^2\!\int |d\psi/dx|^2\,dx\), and the exact answer for eigenstates is

\[ \Delta x\,\Delta p = \big(n + \tfrac{1}{2}\big)\hbar , \]

which touches the Heisenberg floor only at \(n=0\). Both computed columns now match their closed forms:

Table 2. Position and momentum uncertainties for the oscillator, with \(\langle p^2\rangle\) computed from the numerical derivative of \(\psi\) on a 40 001-point grid. The product climbs the ladder as \((n+\tfrac{1}{2})\hbar\); only the ground state is a minimum-uncertainty state. Both \(\Delta x\) and \(\Delta p\) match \(\sqrt{(n+\tfrac{1}{2})\hbar/m\omega}\) and \(\sqrt{(n+\tfrac{1}{2})\hbar m\omega}\) to the digits shown.

The lesson generalizes beyond this bug. A verification that assumes its conclusion will pass with flying colors on any input, which is exactly what makes it worse than no verification at all. The fix was not better physics — it was refusing to let one uncertainty be derived from the other.

Why is only \(n=0\) minimum-uncertainty? Because the Gaussian is the unique shape that saturates Heisenberg’s inequality, and the oscillator’s ground state is a Gaussian — a fact §5 returns to from the other direction.

4. The hydrogen atom: the one atom solved exactly

Swap the parabola for the Coulomb well, \(V(r) = -e^2/4\pi\epsilon_0 r\), and the problem becomes three-dimensional and real: this is an actual atom, and the last one for which the Schrödinger equation surrenders a closed form. The ground state is

\[ \psi_{1s}(r) = \frac{1}{\sqrt{\pi a_0^3}}\,e^{-r/a_0}, \qquad E_1 = -\tfrac{1}{2}\,E_h = -13.606\ \mathrm{eV}, \]

with \(a_0 = 0.5292\) Å the Bohr radius. The density \(|\psi_{1s}|^2\) peaks at the nucleus, but the electron is not most likely to be found at the nucleus: the probability of a radius \(r\) carries the volume of the shell at that radius, \(P(r) = 4\pi r^2|\psi_{1s}|^2\), and the shell volume vanishes at \(r=0\). The competition between the growing \(r^2\) and the dying exponential puts the peak exactly at \(a_0\):

Figure 4. The radial probability density of the hydrogen 1s state. The most probable radius is exactly the Bohr radius (red dashed), but the mean radius sits at \(1.5\,a_0 = 0.794\) Å (grey dashed) — the long exponential tail drags the average outward. Two different questions, two different answers, one distribution.

Numerics held accountable. The closed forms are \(\langle r\rangle = \tfrac{3}{2}a_0 = 0.7938\) Šand \(\langle r^2\rangle = 3a_0^2 = 0.8401\) Ų. Here is what a grid does to them, from careless to careful:

Table 3. Grid convergence of the 1s radial moments. Truncating at \(5a_0\) quietly discards 0.3% of the electron and 2.7% of \(\langle r^2\rangle\) — the \(r^2\) weight amplifies exactly the tail the truncation removed. The integration rule matters far less than the domain: the worst error here is a boundary decision, not a quadrature decision. (Renormalizing a truncated grid, as the old draft did, hides the missing norm but not the biased moments.)

Now point research-grade machinery at the same atom. Psi4 solving hydrogen with unrestricted Hartree–Fock is a deliberately overqualified tool — and that is the point, because for a one-electron system Hartree–Fock’s mean-field approximation is no approximation at all. There is no second electron to average over: the correlation energy is identically zero, and the only thing separating the computed energy from the exact \(-\tfrac{1}{2}\,E_h\) is the finite basis set. Hydrogen is the one atom where the basis-set error is the whole error, in its purest laboratory form:

Table 4. UHF/psi4 energies for the hydrogen atom up the basis ladder⁴, converging monotonically — as the variational principle guarantees — toward the exact nonrelativistic, clamped-nucleus value of \(-0.5\,E_h\).

Figure 5. Basis-set error of the hydrogen-atom UHF energy against basis size, both axes logarithmic. Four orders of magnitude of error retired in six rungs. With no correlation energy in the problem, this entire descent is the one-electron basis doing its only job: shaping a single orbital.

Three readings of this ladder. First, the minimal basis misses by chemistry-sized amounts. STO-3G’s error is \(0.91\) eV — about \(88\) kJ/mol, bond-strength territory, on the simplest system in existence. Second, most of a big basis is spectator here. cc-pV5Z carries 55 functions, but a \(^2S\) atom can only use the \(s\)-type ones — the \(p\), \(d\), \(f\), \(g\) polarization functions cannot mix into a spherically symmetric state, and the ladder narrows only because each rung also deepens the \(s\) set (2, 3, 4, then 5 \(s\)-functions from cc-pVDZ to cc-pV5Z). Third, the last few micro-hartrees are the cusp. The exact 1s orbital has a kink at the nucleus — \(e^{-r/a_0}\) comes to a point at \(r=0\) — while every Gaussian is smooth there, so a finite sum of Gaussians can chase the cusp but never land it⁵. Readers of the correlation-gap post have met this enemy before: the electron–electron cusp of the Coulomb hole is what makes correlation energies crawl toward the basis limit. Same mathematical villain, different pair of particles.

That is also the bridge out of this post. Add one more electron — helium — and the closed forms stop. Hartree–Fock becomes an approximation with a gap below it, and everything in the two posts upstairs becomes necessary. The hydrogen atom is where “orbital” is a fact; everywhere else it is a very good idea.

5. The inequality: localization has a price

The uncertainty principle,

\[ \Delta x \cdot \Delta p \geq \frac{\hbar}{2}, \]

has been present in every table so far — the box states, the oscillator ladder, the hydrogen moments all obey it with room to spare. The clean way to see it saturated is the Gaussian wavepacket \(\psi(x) \propto e^{-x^2/4\sigma^2}\), for which the computed uncertainties (momentum again via the derivative, \(\hbar = 1\)) come out \(\Delta x = \sigma\) and \(\Delta p = 1/2\sigma\) exactly — product \(\tfrac{1}{2}\hbar\), for every \(\sigma\) on the grid, to six digits. Narrow the packet and its momentum spread widens in exact compensation:

Figure 6. Two Gaussian wavepackets in position space (solid) and their momentum-space densities (dashed), same color per packet. The narrow position packet (blue) owns the broad momentum distribution; the wide one (red) owns the sharp one. The product \(\Delta x\,\Delta p\) is \(\hbar/2\) for both — the inequality’s floor, reachable only by this shape.

Where does the trade come from? Not from measurement disturbing anything — from Fourier analysis. A wavefunction sharply peaked in position must be built by superposing many momentum components; the interference that localizes it is the momentum spread. Averaging plane waves whose wavenumbers span a width \(2s\) around a carrier \(k_0\) gives \(\overline{\cos kx} = \cos(k_0x)\,\mathrm{sinc}(sx)\) — a wavetrain confined to an envelope of width \(\sim 1/s\):

Figure 7. Superpositions of plane waves with the same carrier wavenumber but different spreads \(s\) (curves offset vertically). The wide spread of momentum components (red) buys a tightly localized packet; the narrow spread (blue) leaves the wave extended. It is the spread of contributing wavenumbers that localizes, not their count — the packet width scales as \(1/s\), Fourier’s version of \(\Delta x\,\Delta p \sim \hbar\).

And this inequality is why the hydrogen atom of §4 exists at all. Classically the electron should spiral into the nucleus; quantum mechanically, squeezing it into a region \(\sigma\) costs kinetic energy \(\sim \hbar^2/2m\sigma^2\) while the Coulomb well only pays back \(\sim e^2/4\pi\epsilon_0\sigma\). Minimizing the sum lands, up to a factor of order one, on \(\sigma \approx a_0\) and \(E \approx -\tfrac{1}{2}E_h\) — the Bohr radius as the equilibrium point of the uncertainty trade. The atom is a standoff between Heisenberg and Coulomb, and the 1s orbital is the treaty.

6. What the ground floor carries

Each toy in this post is a load-bearing column under the rest of the series. The box’s boundary-condition quantization and node-counting reappear whenever a wavefunction is confined — and its sine functions are the simplest example of a basis in which harder problems get expanded. The oscillator’s ladder is the working model for every vibrational spectrum and every zero-point correction, and its ground state introduced the Gaussian that basis-set builders would later adopt wholesale. The hydrogen 1s is the template for every atomic orbital in every basis set — Slater functions imitate its cusp, Gaussians approximate it in bulk, and the residual few micro-hartrees in Table 4 are the honest price of that convenience. The inequality is the reason atoms have size.

The rule of the post carries further than the physics. Every number here had a closed form waiting to grade it, and the grading caught real errors — a circular momentum “verification,” a truncated grid renormalized into false agreement. One floor up, in the Hartree–Fock machinery, the exact answers stop coming, and the habits formed where truth was available — convergence tables, independent cross-checks, suspicion of too-perfect agreement — are the only part of this post that survives the climb.

References