Expectation Values

Every computed molecular property reported by quantum chemistry software — dipole moments, quadrupole moments, polarizabilities, NMR shielding tensors, spin densities — is an expectation value of the corresponding operator evaluated over a computed wavefunction or electron density, whether that wavefunction comes from Hartree–Fock, coupled cluster, or density functional theory. When a paper reports a computed dipole moment of a drug candidate to compare with a measured Stark-effect splitting, the number quoted is literally $\langle \Psi | \hat{\boldsymbol\mu} | \Psi \rangle$ evaluated on the optimized electronic wavefunction. In NMR spectroscopy, the observed bulk magnetization is $\langle S_z \rangle$ summed over an ensemble of nuclear spins, and the entire vector model of pulsed NMR — on which every modern structural biology and metabolomics measurement depends — is built from tracking how this expectation value precesses and relaxes over time. In electronic structure theory, the variational principle itself is stated as a claim about an expectation value: $\langle \Psi_{\text{trial}} | \hat H | \Psi_{\text{trial}}\rangle \ge E_0$, which is the operational foundation of every basis-set calculation performed in computational chemistry, from small-molecule DFT to large-scale materials screening. In spectroscopy generally, transition intensities are governed by expectation values of the transition dipole operator between different states, $\langle \psi_f | \hat{\boldsymbol\mu} | \psi_i \rangle$, determining which electronic or vibrational transitions are allowed or forbidden — the operational basis of UV-Vis and IR selection rules used daily in structure elucidation. Population analysis schemes (Mulliken, natural bond orbital, Bader's QTAIM) partition expectation values of the number operator or position operator across atomic centers to assign "atomic charges," a concept every organic chemist uses informally but which is, underneath, an expectation-value calculation. Even something as applied as designing a magnetic resonance imaging contrast agent relies on computing expectation values of spin and electronic operators to predict relaxation rates.

Ask the tutor: How does the expectation value formalism change once you move from a pure state $\Psi$ to a mixed statistical ensemble described by a density matrix?

Imagine preparing a thousand identical hydrogen atoms in the same $1s$ state and measuring the electron–nucleus distance $r$ on each one, using some (idealized) instantaneous position probe. You will not get the same number a thousand times. You will get a scatter of distances, weighted by the probability density $|\psi_{1s}|^2$, and if you average all thousand readings you converge on a specific number: $\langle r \rangle = 1.5\,a_0$. That average — not any single reading — is what $\langle r \rangle$ means, and it is exactly what the operator formalism computes without ever needing to "measure" anything. The key conceptual move is that operators act inside an integral sandwiched between $\Psi^*$ and $\Psi$. The wavefunction alone is not observable; $|\Psi|^2 d\tau$ is a probability, and any observable quantity must be built by weighting the operator's action by that probability density. This is precisely why the expectation value is written as a triple structure: bra $\langle\Psi|$, operator $\hat A$, ket $|\Psi\rangle$. The bra–ket notation is not decoration — it encodes "how the operator's effect on $\Psi$ overlaps back with $\Psi$ itself," which is the mathematical definition of a weighted average. A second essential picture: any wavefunction can be expanded in the eigenstates of the operator you care about, $\Psi = \sum_n c_n \psi_n$ with $\hat A \psi_n = a_n \psi_n$. Measuring $A$ on this state can only ever yield one of the eigenvalues $a_n$, with probability $|c_n|^2$ (this is the Measurement Postulate in miniature). The expectation value is then nothing but the ordinary statistical mean of a discrete probability distribution, $\sum_n |c_n|^2 a_n$ — the same arithmetic you'd use to average dice rolls weighted by their probabilities, except the "outcomes" are quantized eigenvalues like energy levels or angular momentum quantum numbers.

A third picture worth holding in mind is the difference between the spread of a distribution and its mean. Two wavefunctions can share the same $\langle x \rangle$ while having wildly different $\langle x^2 \rangle$ — one sharply localized, one smeared over a large region. The mean alone never tells you about the shape; you need the second moment, and the gap between $\langle x^2\rangle$ and $\langle x \rangle^2$ is precisely the variance $\sigma_x^2$ that reappears as the width of a spectral line or the delocalization of an electron in a conjugated system.

The central postulate connects an operator, a state, and a number:

where: $\hat A$ is the Hermitian operator representing the observable, $\Psi$ is normalized ($\int \Psi^*\Psi\, d\tau = 1$), $d\tau$ is the full volume element (all spatial and spin coordinates), and the operator acts on $\Psi$ first, with the result then dotted against $\Psi^*$.

Reading: this is a weighted average, structurally identical to $\bar{x} = \sum_i p_i x_i$ from elementary statistics, except the "outcomes" are eigenvalues of $\hat A$ and the "probabilities" are supplied automatically by $|\Psi|^2$. If $\Psi$ is not normalized, the correct general formula is the ratio

where: the denominator rescales the un-normalized overlap so the answer is independent of any arbitrary constant multiplying $\Psi$.

Reading: multiplying $\Psi$ by any nonzero constant $c$ must not change a physical prediction; the ratio form guarantees this because both numerator and denominator scale as $|c|^2$ and cancel. When $\Psi$ is expanded in the orthonormal eigenbasis of $\hat A$, with $\hat A \psi_n = a_n\psi_n$ and $\Psi = \sum_n c_n \psi_n$, the expectation value collapses to a discrete sum:

where: $c_n = \langle \psi_n | \Psi \rangle$ are expansion coefficients and $|c_n|^2$ is the probability of measuring eigenvalue $a_n$, satisfying $\sum_n |c_n|^2 = 1$.

Reading: this is the discrete-spectrum version of the integral definition and makes explicit that $\langle A \rangle$ is a genuine statistical mean over quantized outcomes, never itself an eigenvalue unless the state happens to be a pure eigenstate. Finally, the variance — how far a single measurement typically strays from the mean — is defined through the second moment:

where: $\langle A^2\rangle = \langle \Psi|\hat A^2|\Psi\rangle$ is the expectation value of the squared operator, and $\sigma_A$ (the standard deviation) is the root-mean-square spread of measurement outcomes.

Reading: this quantity is never negative, vanishes only if $\Psi$ is an eigenstate of $\hat A$, and is the direct precursor to the Uncertainty Principle, where $\sigma_x \sigma_p \geq \hbar/2$.

Example 1 — Foundational: average position in an asymmetric excited state of a particle in a box. Consider an electron confined to a one-dimensional box of length $L = 1.20\ \text{nm}$ (a crude but standard model for a conjugated polyene chromophore), in the $n=2$ stationary state

This state is already normalized (a standard particle-in-a-box result), so $\langle x \rangle = \int_0^L \psi_2^* \, x\, \psi_2 \, dx$ with no extra normalization ratio needed.

Using $\sin^2\theta = \tfrac12(1-\cos 2\theta)$ and the standard box integral $\int_0^L x\sin^2(n\pi x/L)\,dx = L^2/4$ (valid for any integer $n \ge 1$, including $n=2$ here), this evaluates cleanly to

For $L = 1.20\ \text{nm}$, $\langle x \rangle = 0.600\ \text{nm}$ — the electron is on average at the geometric center of the box, exactly as symmetry demands (the density $|\psi_2|^2$ is symmetric about $x=L/2$ regardless of quantum number $n$). This is the sanity check: any asymmetric-looking intermediate integral must still deliver a midpoint mean because of that reflection symmetry. Now the physically more informative quantity, the spread. For $n=2$ the box-integral formula for the second moment is

With $n=2$:

Numerically, $1/3 = 0.33333$ and $1/(8\pi^2) = 0.012665$, so $\langle x^2\rangle = 0.32067\, L^2$. With $L = 1.20\ \text{nm}$, $L^2 = 1.440\ \text{nm}^2$, giving

The variance is

so $\sigma_x = 0.319\ \text{nm}$ — the electron's position genuinely fluctuates over roughly a quarter of the box length even though its average sits exactly at the center. This is the first concrete illustration that $\langle x^2\rangle \ne \langle x\rangle^2$: here $0.4618 \ne 0.3600$, and the 0.102 nm² gap is exactly the physically meaningful spread.

Limiting case check. As $n \to \infty$, $\langle x^2\rangle \to L^2/3$ (the classical uniform-distribution value), because the oscillatory term $1/(2n^2\pi^2)$ vanishes — recovering the correspondence-principle limit where quantum averages match a classical particle bouncing uniformly between the walls.

Example 2 — Applied: average electron–nucleus distance in the hydrogen 1s and 2p$_z$ orbitals. The radial expectation value $\langle r \rangle$ sets the effective "size" chemists use to rationalize periodic trends and orbital overlap. For the hydrogen ground state,

already normalized, with $a_0 = 0.529\ \text{Å}$ the Bohr radius. The position operator here is simply multiplication by $r$, and the volume element is $d\tau = r^2\sin\theta\, dr\, d\theta\, d\phi$. Because $\psi_{1s}$ has no angular dependence, the angular integral gives the full solid angle $4\pi$, leaving

Using the standard Gamma-function integral $\int_0^\infty r^3 e^{-\alpha r}\,dr = 6/\alpha^4$ with $\alpha = 2/a_0$:

So

With $a_0 = 0.529\ \text{Å}$, this gives $\langle r \rangle_{1s} = 0.7935\ \text{Å}$ — a textbook number confirming the formula. Now the non-trivial comparison, chosen with an asymmetric quantum number to show the trend clearly rather than just re-deriving the $1s$ case: the hydrogen $2p_z$ orbital,

already normalized. Squaring and separating radial/angular parts,

where the angular factor $\int_0^{2\pi}\!d\phi \int_0^\pi \cos^2\theta \sin\theta\, d\theta = 2\pi \cdot \tfrac{2}{3} = \tfrac{4\pi}{3}$ is already absorbed into the normalization constant $1/\sqrt{32\pi a_0^3}$, so the clean radial-only formula (a standard hydrogenic result) applies directly:

For $2p$: $n=2$, $\ell=1$, so

With $a_0=0.529\ \text{Å}$: $\langle r \rangle_{2p} = 2.645\ \text{Å}$ — more than three times larger than the $1s$ value ($0.7935\ \text{Å}$), quantitatively capturing the qualitative rule that higher-$n$ orbitals are larger and that, at fixed $n$, higher $\ell$ pulls the electron slightly outward less than lower $\ell$ (compare $2s$, $\ell=0$: $\langle r\rangle_{2s} = \tfrac{a_0}{2}(12-0)=6a_0$, larger than $2p$'s $5a_0$, because the $2s$ orbital's inner lobe is offset by even more density piled up far out).

Limiting case check. As $n \to \infty$ at fixed $\ell$, $\langle r \rangle_{n\ell} \to \tfrac{3}{2}a_0 n^2$, reproducing the $n^2$ scaling of Rydberg-state orbital radii used in Rydberg-atom spectroscopy.

Example 3 — Exam/competition level: expectation value of energy in a non-stationary superposition. A particle in a box of length $L$ is prepared (say, by a sudden perturbation) in the un-normalized superposition

where $\psi_1, \psi_3$ are the (real, normalized, mutually orthogonal) particle-in-a-box eigenfunctions with energies

This tests two things simultaneously: handling an un-normalized state and handling complex coefficients correctly.

Step 1 — Normalize. Compute $\langle \Phi | \Phi \rangle$ using orthonormality of $\psi_1,\psi_3$:

(Cross terms vanish by orthogonality; note $(3\psi_1 - 4i\psi_3)^* = 3\psi_1 + 4i\psi_3$ since $\psi_n$ are real.) So $\Phi$ is 25 times "too big"; the normalized state is $\Psi = \Phi/5$.

Step 2 — Identify the probabilities. In the normalized state, $c_1 = 3/5$ and $c_3 = -4i/5$, so

which correctly sum to 1 — the imaginary unit in $c_3$ contributes nothing extra to the probability, only a phase, illustrating that phases never affect $\langle A\rangle$ when $\hat A$ is diagonal in this basis (energy).

Step 3 — Apply the weighted-average formula. Since $\psi_1,\psi_3$ are eigenstates of the Hamiltonian,

For a specific numerical case, take an electron ($m = 9.109\times10^{-31}$ kg) in $L = 0.500\ \text{nm}$:

Numerator: $(6.626\times10^{-34})^2 = 4.392\times10^{-67}$. Denominator: $8 \times 9.109\times10^{-31} \times 2.500\times10^{-19} = 1.822\times10^{-48}$. So

Then

Verification via the direct ratio form. Using $\langle E \rangle = \langle \Phi|\hat H|\Phi\rangle / \langle \Phi|\Phi\rangle$ directly on the un-normalized $\Phi$: $\langle \Phi|\hat H|\Phi\rangle = 9E_1 + 16E_3 = 9E_1 + 144E_1 = 153E_1$, and dividing by $\langle \Phi|\Phi\rangle = 25$ gives $\langle E \rangle = 153E_1/25 = 6.12 E_1$ — identical to Step 3, confirming that normalizing first versus dividing at the end are equivalent, exactly as the ratio formula guarantees.

Limiting case check. If instead $c_3 \to 0$ (pure $\psi_1$), $\langle E \rangle \to E_1$ exactly, with zero variance — the single-eigenstate limit recovers a sharp, dispersion-free energy, as any correctly built superposition formula must.

The mathematical apparatus of expectation values belongs to the founders of matrix and wave mechanics, but the physicist whose interpretive framework is most directly responsible for how chemists use $\langle A \rangle$ today is Max Born, who supplied the probabilistic reading of $|\Psi|^2$ that makes the entire weighted-average picture meaningful in the first place. Born, trained in Göttingen alongside Heisenberg and Jordan during the frantic years of matrix mechanics' construction, initially worked on scattering theory — computing how an electron beam's wavefunction after collision with a potential related to the probability of finding the electron deflected into a given direction. It was in that scattering context, in 1926, that he proposed treating $|\psi|^2$ as a probability density rather than a literal charge or mass density, a move that took years to be accepted (Schrödinger himself resisted it) but that became the load-bearing statistical foundation beneath every expectation value calculated since.