Expectation Values and Orthogonality in the Box

The free-electron (particle-in-a-box) model, built entirely on the expectation values derived here, is still used as a fast qualitative tool for estimating HOMO–LUMO gaps and hence absorption wavelengths of linear conjugated dyes and polyenes, useful in designing chromophores for photovoltaics and biological imaging probes. The orthogonality proof technique — reduce an overlap integral to a product-to-sum trigonometric identity, then show the boundary terms vanish by periodicity — is the direct ancestor of Fourier-analysis arguments used in vibrational spectroscopy, in NMR lineshape theory, and in electronic structure methods where basis-function orthogonality (or its deliberate absence, as in non-orthogonal Gaussian basis sets requiring overlap-matrix corrections) governs the cost and accuracy of a calculation. The trick of computing $\langle p^2\rangle$ via $2m\langle H\rangle$ rather than by brute-force double differentiation reappears constantly in molecular quantum mechanics, wherever the potential is simple over the domain of interest — for instance in estimating zero-point kinetic energy contributions in the harmonic oscillator, and in any perturbative treatment where the unperturbed Hamiltonian's kinetic and potential pieces are cleanly separable. Scanning-tunneling-microscopy quantum corrals, quantum dots, and nanowire heterostructures are all modern physical realizations of confined-particle boxes where these exact expectation-value formulas are used as leading-order estimates before more detailed numerical modeling.

Ask the tutor: How would these same expectation-value integrals change if the box had an asymmetric potential step in the middle instead of flat walls?

The particle-in-a-box eigenstates are standing waves, $\psi_n(x) = \sqrt{2/L}\sin(n\pi x/L)$, confined to $0 \le x \le L$ with $\psi_n = 0$ outside. Once normalization is settled — $\int_0^L |\psi_n|^2\,dx = 1$ — the natural next question is: what does this wavefunction predict? A wavefunction alone is not directly observable; every measurable quantity is extracted by sandwiching an operator between $\psi^*$ and $\psi$ and integrating over all space. This is the defining computational act of quantum chemistry, and the box is the arena where it can be done with pure trigonometry, no special functions, no infinite series. Three features make the box uniquely transparent. First, $|\psi_n(x)|^2 = (2/L)\sin^2(n\pi x/L)$ is a smooth oscillatory density whose peaks and nodes are easy to visualize, so $\langle x\rangle$ has an obvious answer (the midpoint) before any calculation — a good sanity check for the algebra. Second, the momentum operator $\hat p = -i\hbar\,d/dx$ acting on a sine gives a cosine, and integrals of $\sin\cos$ over a full number of half-periods vanish identically, so $\langle p\rangle = 0$ falls out cleanly and can be understood physically: the standing wave is an equal superposition of a right-moving and a left-moving traveling wave, and their momenta cancel. Third, $\langle x^2\rangle$ requires integrating $x^2\sin^2(n\pi x/L)$, which needs one integration by parts (or a table integral) but no more — and the result cleanly interpolates between the quantum ground state and the classical uniform-distribution limit as $n\to\infty$, foreshadowing the correspondence principle used throughout spectroscopy. Orthogonality is the other pillar of this article. Two box eigenstates $\psi_m$ and $\psi_n$ with $m \ne n$ satisfy

This is not a coincidence of the box; it is guaranteed for any two eigenfunctions of a Hermitian operator (here, the box Hamiltonian $\hat H = -\dfrac{\hbar^2}{2m}\dfrac{d^2}{dx^2}$) belonging to different eigenvalues. Verifying it explicitly by direct trigonometric integration in the box — rather than only invoking the general theorem — builds confidence that the general theorem is not a magic trick, and shows exactly which product-to-sum identity does the work. This orthogonality is the reason overlap integrals between different vibrational or electronic states vanish in the absence of perturbation, underlying selection rules in spectroscopy and the vanishing of off-diagonal matrix elements in unperturbed basis sets.

The three central expectation-value integrals for the normalized state $\psi_n(x) = \sqrt{2/L}\sin(n\pi x/L)$, valid for $0 \le x \le L$, are:

where: $n$ is any positive integer quantum number, $L$ is the box length, and the result is independent of $n$.

Reading: the probability density $|\psi_n|^2$ is symmetric about the box's midpoint for every $n$, so the mean position is always exactly $L/2$ regardless of how excited the state is.

where: the first term is the classical (uniform-distribution) mean square position, and the second is a quantum correction that shrinks as $n$ grows.

Reading: as $n \to \infty$, $\langle x^2\rangle_n \to L^2/3$, exactly the mean square position of a classical particle bouncing uniformly across $[0,L]$ — the correspondence principle in action.

where: $\hat p = -i\hbar\, d/dx$ is the momentum operator; the vanishing holds for every $n$.

Reading: the standing wave is an equal mixture of leftward and rightward momentum eigenstates, so their contributions cancel exactly — the particle has no net drift.

where: $E_n = n^2h^2/(8mL^2)$ is the box energy; this identity follows because $\hat H = \hat p^2/(2m)$ inside the box, so $\langle p^2\rangle = 2m\langle E\rangle = 2mE_n$.

Reading: since $\langle p\rangle=0$, all of $\langle p^2\rangle$ is "variance" — the kinetic energy is entirely uncertainty in momentum, never a net drift.

where: $\delta_{mn}$ is the Kronecker delta; this is the orthonormality statement combining normalization ($m=n$) with orthogonality ($m\neq n$).

Reading: eigenfunctions belonging to different energy eigenvalues of a Hermitian Hamiltonian never overlap — a structural fact used every time a wavefunction is expanded in a basis set.

Example 1 — Foundational: ground and first excited state, $L = 1.00\ \text{nm}$. Take $L = 1.00\times10^{-9}$ m, and evaluate $\langle x\rangle$, $\langle x^2\rangle$, $\Delta x$ for $n=1$ and $n=2$, plus confirm orthogonality between them. For both states, $\langle x\rangle_n = L/2 = 0.500$ nm — no calculation needed beyond the derived formula. For $\langle x^2\rangle$, use

With $L^2 = 1.00\ \text{nm}^2$: For $n=1$:

Then

For $n=2$:

Then

So $\Delta x$ grows from $n=1$ to $n=2$ (0.181 nm → 0.266 nm), approaching the classical limit $\Delta x_\infty = L/\sqrt{12} = 0.2887$ nm as $n$ increases — consistent with the correspondence principle. Orthogonality check: $S_{12} = \dfrac{2}{L}\displaystyle\int_0^L \sin(\pi x/L)\sin(2\pi x/L)\,dx$. Using the Step 3 result with $k = m-n = -1$ and $k=m+n=3$, both nonzero integers, both cosine integrals vanish, giving $S_{12}=0$ exactly — as it must, since $E_1 \neq E_2$.

Example 2 — Applied: momentum uncertainty as a bond-order estimate in a conjugated chain. The free-electron (particle-in-a-box) model treats a conjugated polyene's π-electrons as particles confined along the chain length. Consider a chain of effective length $L = 0.729$ nm (a specific asymmetric value, representative of a short polyene with several conjugated double bonds), and ask: for an electron in the $n=3$ state, what is $\Delta p$, and what velocity scale does it represent? From the derivation,

With $n=3$, $\hbar = 1.0546\times10^{-34}\ \text{J s}$, $L = 0.729\times10^{-9}$ m:

Dividing by the electron mass $m_e = 9.109\times10^{-31}$ kg gives a characteristic speed scale (not a literal velocity, since $\langle p\rangle = 0$ and this is a spread, not a drift):

This is about 0.5% of the speed of light — safely non-relativistic, confirming that a non-relativistic Schrödinger treatment is self-consistent for this system, and giving a rough sense of how "fast" the π-electron's momentum content is spread even though its average drift is exactly zero. This is the kind of order-of-magnitude cross-check that justifies applying the simple box model to real π-systems before doing any more sophisticated Hückel-level calculation.

Example 3 — Competition-level: verifying orthogonality and computing a joint expectation across mixed states. Consider the (unnormalized, for pedagogical purposes) superposition

built from two orthonormal box eigenstates, in a box of length $L$. Find the normalization constant needed to make $\Phi$ a proper wavefunction, and compute $\langle x\rangle_\Phi$ in terms of $\langle x\rangle_2 = \langle x\rangle_5 = L/2$ and any cross terms.

Normalization. Because $\psi_2$ and $\psi_5$ are orthonormal (proved above: $m=2,n=5$ gives nonzero integers $m-n=-3$ and $m+n=7$, so $S_{25}=0$; and each is separately normalized, $S_{22}=S_{55}=1$),

So the normalized state is $\Phi_N = \Phi/2$, i.e., coefficients $c_2 = 1/2$ and $c_5 = \sqrt3/2$, satisfying $|c_2|^2+|c_5|^2 = 1/4+3/4=1$ as required.

Expectation value of $x$ in the mixed state. In general, for $\Phi_N = c_2\psi_2 + c_5\psi_5$ with real orthonormal $\psi_2,\psi_5$,

The first two (diagonal) terms use the already-derived $\langle x\rangle_n = L/2$ for both $n=2$ and $n=5$:

The cross term $\int_0^L \psi_2\,x\,\psi_5\,dx$ is generally nonzero — unlike the overlap integral $S_{25}$, the operator $x$ does not commute with the symmetry that forced $S_{25}=0$, so this off-diagonal position matrix element survives. Explicitly,

which by the product-to-sum identity becomes $\frac1L\int_0^L x\big[\cos(3\pi x/L) - \cos(7\pi x/L)\big]dx$, each piece handled by the same integration-by-parts machine used for $\langle x\rangle_n$ above (with odd $k=3,7$ giving nonzero boundary contributions since $\cos(k\pi)=-1$ for odd $k$, unlike the $k$ even case in Step 2 of the first derivation). Carrying this out gives a nonzero numerical coefficient of $L$; the point of the example is structural — that off-diagonal position matrix elements between orthogonal states need not vanish, and this is precisely the mechanism behind allowed optical transitions (nonzero transition dipole moments) between different box/orbital states, foreshadowing selection rules in spectroscopy.

Though the specific problem of a particle in a box predates Dirac by several years (it appears in early treatments by Schrödinger and in pedagogical form soon after), the machinery used throughout this article — writing observables as Hermitian operators, expressing predictions as sandwich integrals, and deriving orthogonality from Hermiticity rather than from case-by-case trigonometric coincidence — is Dirac's systematic contribution. Every time a chemist writes $\langle x\rangle = \int\psi^*x\psi\,dx$ or invokes "orthogonal basis functions," they are using notation and logic Dirac built to make quantum mechanics computable and interpretable across every subsequent exactly solvable model, including the ones the box directly foreshadows.