The language of quantum chemistry — wavefunctions and operators

A particle’s state as a complex function, every observable as a Hermitian operator — the two basic tools of quantum chemistry, followed end to end through one concrete Gaussian wavepacket.

Opening

The book spans 14 chapters from atomic bonding to band structure, but every chapter is written in the same two tools — the wavefunction and the operator. By the end of this chapter the reader should be able to say clearly what “the state of an electron” means, and how the act of “measuring its position” translates into a formula. The destination of this chapter is one concrete wavefunction — the Gaussian wavepacket — touched by hand and by Python, with normalization, expectation values, and uncertainty all checked directly.

Main 1 — Wavefunction and probability

Quantum mechanics describes a particle’s state by a complex-valued function $\psi(\vec r)$ (psi, the wavefunction). In classical mechanics the state was the pair $(\vec r, \vec p)$ of position and momentum; in quantum mechanics the particle does not hold both at once — it holds only a single complex function spread over space.

This function is not directly observable. What is observable is the probability density $|\psi(\vec r)|^2$. The probability of finding the particle in the volume element $d^3 r$ is

$$dP(\vec r) = |\psi(\vec r)|^2\, d^3 r,$$

and the fact that the particle exists somewhere is expressed by the normalization condition

$$\int |\psi(\vec r)|^2\, d^3 r = 1.$$

Even though $\psi$ itself is not observed, its phase does not vanish from the story. When two wavefunctions are combined — as in the double slit or in the overlap of molecular orbitals — the phase determines constructive vs destructive interference. The phase only becomes invisible when one “measures” a single state.

Main 2 — Observables and Hermitian operators

Every measurable quantity is represented by one Hermitian operator. For now you can read “Hermitian” as: a linear transformation engineered so that its eigenvalues are real — because real eigenvalues are what physical measurements actually return.

• Position $\hat x = x$ (just multiplication)
• Momentum $\hat p = -i\hbar\, \partial / \partial x$
• Kinetic energy $\hat T = -\hbar^2/(2m)\, \nabla^2$
• Potential $\hat V = V(\vec r)$ (multiplication by a function of position)

Here $\hbar$ (h-bar) is Planck’s constant divided by $2\pi$, $m$ is the particle mass, and $\nabla^2$ is the Laplacian.

The expectation value of operator $\hat A$ in state $\psi$ — the mean of measurements over infinitely many identically prepared copies of the state — is defined by

$$\langle A \rangle = \int \psi^*(\vec r)\, \hat A\, \psi(\vec r)\, d^3 r,$$

where $\psi^*$ is the complex conjugate of $\psi$. This integral reappears almost everywhere in quantum chemistry — when computing energies, dipole moments, bond orders, anything that has a number attached.

Main 3 — A concrete example: the Gaussian wavepacket

Drop down from abstraction once. Consider the one-dimensional wavefunction

$$\psi(x) = (2\pi\sigma^2)^{-1/4} \exp\!\left( -\frac{x^2}{4\sigma^2} \right),$$

where $\sigma$ (sigma) is a positive width parameter. $\psi$ is real and peaks at the origin. Check normalization first:

$$\int_{-\infty}^{\infty} |\psi(x)|^2\, dx = (2\pi\sigma^2)^{-1/2} \int_{-\infty}^{\infty} e^{-x^2/(2\sigma^2)}\, dx = 1.$$

The last equality uses the Gaussian integral $\int e^{-x^2/(2\sigma^2)}\, dx = \sqrt{2\pi}\,\sigma$. Then the expectation values. Since $|\psi|^2$ is symmetric about the origin,

$$\langle x \rangle = 0,$$

and the variance is

$$\langle x^2 \rangle = \int x^2\, |\psi(x)|^2\, dx = \sigma^2.$$

The position uncertainty is $\Delta x = \sqrt{\langle x^2\rangle - \langle x\rangle^2} = \sigma$. The width parameter $\sigma$ is the position uncertainty.

This is not accidental. The analogous calculation on the momentum side gives $\Delta p = \hbar/(2\sigma)$, and the product is

$$\Delta x\, \Delta p = \frac{\hbar}{2}.$$

This is the state that saturates the equality in Heisenberg’s uncertainty principle $\Delta x\, \Delta p \ge \hbar/2$ — the Gaussian wavepacket is “the narrowest state quantum mechanics allows,” in a precise sense. The full proof is for a later chapter, but this one example already shows the two tools of quantum chemistry meshing together.

In Python

# Numerically verify normalization, expectation values, and uncertainty
# for a Gaussian wavepacket.
import numpy as np
import matplotlib.pyplot as plt

sigma = 1.5
N = 400
x = np.linspace(-10, 10, N)
dx = x[1] - x[0]

# Analytic form, normalization constant included.
psi = (2 * np.pi * sigma**2)**(-0.25) * np.exp(-x**2 / (4 * sigma**2))
rho = psi**2

# Numerical normalization: scale so trapezoidal integral is 1.
norm = np.trapz(rho, x)
rho = rho / norm

# Expectation values and standard deviation.
x_mean = np.trapz(x * rho, x)
x2_mean = np.trapz(x**2 * rho, x)
dx_num = np.sqrt(x2_mean - x_mean**2)

print(f"<x^2> numeric = {x2_mean:.4f}, analytic = {sigma**2:.4f}")
print(f"dx    numeric = {dx_num:.4f}, analytic = {sigma:.4f}")

# Plot |psi|^2 with vertical lines at +/- dx.
plt.plot(x, rho, label=r"$|\psi(x)|^2$")
plt.axvline(+dx_num, color="k", linestyle="--", label=r"$\pm\Delta x$")
plt.axvline(-dx_num, color="k", linestyle="--")
plt.xlabel("x"); plt.ylabel(r"$|\psi|^2$")
plt.legend(); plt.tight_layout()
plt.show()

If the numerical $\langle x^2 \rangle$ matches $\sigma^2 = 2.25$ and the numerical $\Delta x$ matches $\sigma = 1.5$ to two decimal places, the three definitions — normalization, expectation, standard deviation — have all been touched by hand.

To the next chapter

Chapter 2: The Schrödinger equation takes the operators introduced here and ties them into a single equation that governs how $\psi$ evolves in time. With normalization and expectation values already in hand, the Schrödinger equation lands as exactly what it is: the quantum-mechanical statement of energy conservation.