Superposition of States

Superposition of stationary states is not a textbook abstraction; it is the operational description behind entire experimental fields. In femtosecond pump–probe spectroscopy, a short laser pulse creates exactly the kind of vibrational wavepacket worked out in Example 2 — a coherent superposition of vibrational eigenstates on an electronically excited potential energy surface — and the resulting oscillation of the transient absorption signal at the vibrational beat frequency is a direct, real-time observation of the $\cos(\omega_{nm}t)$ terms derived above. This underlies time-resolved studies of photosynthetic energy transfer, photoisomerization (e.g., the retinal chromophore in vision), and coherent control schemes that use shaped laser pulses to steer chemical reactions by engineering the relative phases $\Delta\alpha$ between populated states. In NMR and EPR spectroscopy, superpositions of nuclear or electron spin eigenstates ("coherences") are the entire basis of pulsed multidimensional experiments: a $\pi/2$ pulse creates precisely a superposition of spin-up and spin-down eigenstates, whose time evolution at the Larmor frequency is detected as the free-induction decay signal, then Fourier-transformed back into an NMR spectrum. Rotational wavepackets — superpositions of molecular rotational eigenstates created by short laser pulses — are used to align and orient molecules in gas-phase reaction dynamics experiments and in strong-field physics for high-harmonic generation. Quantum computing platforms built from trapped ions or superconducting qubits manipulate exactly this same superposition-coefficient machinery, engineering $c_0|0\rangle + c_1|1\rangle$ states and tracking their relative phase as the physical resource for quantum algorithms. Even the simple particle-in-a-box superposition of this article is the pedagogical seed for understanding electron wavepacket dynamics in quantum dots and conjugated polyene chromophores used in dye chemistry.

Ask the tutor: How does the relative phase between superposition coefficients determine whether two vibrational wavepackets interfere constructively or destructively when they later overlap?

Consider an electron confined to a one-dimensional box, with orthonormal energy eigenfunctions $\psi_n(x) = \sqrt{2/L}\,\sin(n\pi x/L)$. Suppose the electron is prepared, say by an ultrafast excitation pulse, in some wavefunction $\Psi(x)$ that is not one of these sine functions — perhaps a wavepacket localized near one wall. Quantum mechanics insists this $\Psi$ is nevertheless expressible exactly as an infinite sum $\sum_n c_n \psi_n(x)$. The eigenfunctions $\{\psi_n\}$ form a complete orthonormal set, exactly as sines and cosines do in a Fourier series, and any well-behaved function on the same domain can be built from them. The crucial conceptual leap is that $\Psi$ being "in superposition" does not mean the electron is secretly in one of the $\psi_n$ and we simply don't know which — that would be a classical statistical mixture, describable by a probability distribution alone. Instead $\Psi$ is a genuinely new quantum state, a coherent sum in which the complex phases of the $c_n$ matter physically. Interference between terms shows up whenever you compute the expectation value of an operator that doesn't share the $\psi_n$ eigenbasis (such as position, in the energy eigenbasis), and it shows up dramatically once you let the state evolve in time, since each term acquires its own energy-dependent phase. The first figure below shows a two-level superposition of box eigenfunctions and how its probability density is not simply the average of the two individual densities — cross terms appear. The second figure shows the associated time evolution: the same superposition, evaluated at different times, sloshes back and forth in space at the Bohr frequency $(E_2-E_1)/\hbar$, a direct signature of coherence.

The mathematical backbone is completeness combined with orthonormality. If $\{\psi_n\}$ are the (orthonormal, complete) eigenfunctions of a Hermitian operator $\hat{A}$ with eigenvalues $a_n$,

where: $\Psi$ is an arbitrary physically acceptable wavefunction on the same domain, $\psi_n$ are eigenfunctions of $\hat A$ satisfying $\hat A \psi_n = a_n \psi_n$, and $c_n$ are complex expansion coefficients. Reading: this says every legitimate state is reachable from the eigenbasis of any observable — the eigenfunctions of a Hermitian operator span the whole space of admissible wavefunctions, so nothing is left out. The coefficients are extracted by projection, exploiting orthonormality $\langle \psi_m|\psi_n\rangle = \delta_{mn}$:

where: the integral runs over the full domain of the coordinate, $\psi_n^{*}$ is the complex conjugate of the $n$-th eigenfunction, and $\langle\psi_n|\Psi\rangle$ is Dirac bracket notation for the same overlap integral. Reading: $c_n$ measures "how much $\psi_n$ is contained in $\Psi$," exactly as a Fourier coefficient measures how much of a given sine wave is present in a signal. Normalization of $\Psi$ forces the coefficients to satisfy a sum rule:

where: $|c_n|^2$ is the modulus-squared of each (generally complex) coefficient. Reading: this is the statement that the probabilities of all possible measurement outcomes must add to one — it is the Pythagorean theorem of Hilbert space, since the $\psi_n$ are orthogonal unit vectors. If $\hat A$ is chosen to be the Hamiltonian $\hat H$, with $\hat H \psi_n = E_n \psi_n$, then the full time-dependent wavefunction built from this superposition evolves as

where: $c_n$ are the same time-independent coefficients fixed at $t=0$, and each term picks up its own oscillating phase factor governed by its energy $E_n$. Reading: individual stationary states just spin in phase and never change their probability density; superpositions of different energies do change in time, because the relative phase between terms, $e^{-i(E_n-E_m)t/\hbar}$, sweeps through a full cycle at the Bohr frequency $\omega_{nm} = (E_n-E_m)/\hbar$. The expectation value of any operator $\hat B$ in a superposition state picks up interference (cross) terms between different basis components:

where: the double sum includes diagonal terms $n=m$ (weighted by $|c_n|^2$, the "classical" probabilities) and off-diagonal terms $n\neq m$ (the coherences, weighted by $c_n^* c_m$). Reading: it is precisely these off-diagonal cross terms that make a superposition fundamentally richer than a statistical mixture — they vanish identically for a classical ensemble but survive here, encoding genuine quantum interference.

Example 1 — Foundational: projecting a triangular wavefunction onto particle-in-a-box eigenstates. Take an electron in a 1-D box of length $L$, with energy eigenfunctions $\psi_n(x)=\sqrt{2/L}\sin(n\pi x/L)$ and eigenvalues $E_n = n^2h^2/8mL^2$. Suppose at $t=0$ the (normalized) wavefunction is the simple asymmetric superposition

chosen with unequal weights (0.36 and 0.64, deliberately non-round and asymmetric so the arithmetic can't hide a wrong sign or a swapped index) and zero relative phase for now. Check normalization first. $|c_1|^2 + |c_3|^2 = 0.36+0.64=1.00$. Good — this is a properly normalized state, consistent with the sum rule derived above. Interpretation of coefficients. A measurement of energy on this state yields $E_1$ with probability $|c_1|^2=0.36$ (36%) and $E_3$ with probability $|c_3|^2=0.64$ (64%); no other energy is ever obtained, because $c_2=c_4=\dots=0$. Expectation value of energy. Using $E_n=n^2h^2/8mL^2$, so $E_1 = h^2/8mL^2$ and $E_3=9h^2/8mL^2$:

Note $\langle E\rangle$ here is a simple weighted average — because $\hat H$ is diagonal in its own eigenbasis, there are no cross terms in $\langle \hat H\rangle$, only in operators (like $x$ or $p$) that mix different $n$. Time evolution. At later times,

The probability density $|\Psi(x,t)|^2$ contains a cross term oscillating at $\omega_{31}=(E_3-E_1)/\hbar = 8h^2/(8mL^2\hbar) = h^2/(mL^2\hbar)$, causing the electron's spatial distribution to slosh periodically between favoring the left/right lobes — exactly as in the second figure, but now for the $n=1,3$ pair rather than $n=1,2$. Limiting case. If instead $c_3\to 0$ and $c_1\to 1$ (all weight on one eigenstate), $\Psi$ reduces to a pure stationary state: $|\Psi(x,t)|^2$ becomes time-independent, and all oscillatory cross terms vanish identically, confirming that superposition dynamics require at least two populated eigenstates with different energies.

Example 2 — Applied: a vibrational wavepacket coefficient from a Franck–Condon-like overlap. In ultrafast spectroscopy, a short laser pulse promotes a molecule's nuclear wavefunction from a ground-state vibrational level onto an excited electronic potential energy surface, creating a vibrational wavepacket — literally a superposition of the excited-state harmonic oscillator eigenfunctions $\{\chi_v\}$. Suppose (in a simplified model) the transferred nuclear wavefunction $\Phi(Q)$ on the excited surface is well approximated by a truncated three-term expansion,

and a Franck–Condon overlap calculation (projection of the shifted ground-state Gaussian onto the excited harmonic-oscillator eigenfunctions) gives unnormalized coefficients $c_0' = 0.55$, $c_1' = 0.71$, $c_2' = 0.29$ (deliberately unequal, non-special values representing a modest displacement of the excited-state potential minimum). Step 1: normalize. Compute the sum of squares:

Divide each by $\sqrt{0.8907}=0.9438$ to enforce $\sum|c_n|^2=1$:

Check: $0.583^2+0.752^2+0.307^2 = 0.340+0.565+0.094=0.999\approx1$ (rounding). This normalization step is not bookkeeping fluff — it is what makes $|c_n|^2$ a genuine probability, per the sum rule derived above. Step 2: read off populations. A measurement of vibrational quantum number on the excited surface would give $v=0$ with probability $34.0\%$, $v=1$ with probability $56.5\%$, $v=2$ with probability $9.4\%$ — the wavepacket is dominated by $v=1$ character, consistent with a moderately (not extremely) displaced excited-state minimum. Step 3: predict the beat pattern. The wavepacket subsequently oscillates back and forth in $Q$ (this is the vibrational coherence probed in pump–probe spectroscopy) with the dominant beat frequency set by adjacent-level spacing, $\omega_{10}=(E_1-E_0)/\hbar = \omega_{vib}$ for a harmonic oscillator (equally spaced levels), plus a weaker second harmonic from the $v=0\leftrightarrow v=2$ coherence at $2\omega_{vib}$. This is precisely the physical origin of the vibrational quantum beats recorded in femtosecond transient-absorption experiments — a direct experimental readout of superposition coefficients and their relative phases.

Example 3 — Exam/competition level: extracting coefficients from a piecewise wavefunction and predicting a measurement outcome. A particle in an infinite box of length $L=1.00\,\text{nm}$ is prepared at $t=0$ in the (normalized, real) wavefunction

using the box eigenfunctions $\psi_n(x)=\sqrt{2/L}\sin(n\pi x/L)$, where $A$ is a normalization constant to be found, chosen with unequal integer coefficients (3 and $-2$) and a non-adjacent pair of quantum numbers (1 and 4) to keep the arithmetic asymmetric and the answer unambiguous. Step 1: find $A$ by orthonormality. Since $\psi_1,\psi_4$ are orthonormal,

using $\langle\psi_1|\psi_4\rangle=0$ to kill the cross terms. So

Step 2: identify the true coefficients. Comparing to $\Psi = c_1\psi_1 + c_4\psi_4$,

Check the sum rule: $c_1^2 + c_4^2 = 9/13 + 4/13 = 13/13 = 1$. Exactly one, as required — note that the negative sign on $c_4$ contributes to $|c_4|^2$ exactly as a positive value would, since only the magnitude enters the probability. Step 3: predict a single energy measurement. $E_n = n^2h^2/8mL^2$. Numerically, with $m=9.109\times10^{-31}\,\text{kg}$, $h=6.626\times10^{-34}\,\text{J s}$, $L=1.00\times10^{-9}\,\text{m}$:

So $E_1 = 6.025\times10^{-20}\,\text{J}$ (0.3762 eV) and $E_4 = 16 E_1 = 9.640\times10^{-19}\,\text{J}$ (6.018 eV). A single measurement of energy gives $E_1$ with probability $9/13=69.23\%$ or $E_4$ with probability $4/13=30.77\%$ — never any intermediate value. Step 4: expectation value and its limiting check.

Limiting-case check: if the coefficient of $\psi_4$ were sent to zero, $c_4\to0$ forces (via the sum rule) $c_1\to \pm1$, recovering a pure $\psi_1$ stationary state with $\langle E\rangle \to E_1$ exactly and zero variance — the superposition machinery degenerates smoothly to the single-eigenstate limit, as it must, since $\langle E^2\rangle - \langle E\rangle^2 \to E_1^2-E_1^2=0$ directly foreshadowing the energy–time uncertainty relation for superposed states with nonzero energy spread.

The mathematics of expanding an arbitrary function in an orthonormal basis predates quantum mechanics by a century — it is Fourier's method, generalized. But the physical interpretation of expansion coefficients as probability amplitudes for discrete quantum states is due above all to Max Born, who in 1926 proposed that $|\psi|^2$ (and, by direct extension, $|c_n|^2$) gives a probability density/probability, resolving the ambiguity of what Schrödinger's wavefunction actually represented. Born's insight, developed while working out scattering theory using Schrödinger's newly published wave mechanics, turned an elegant differential equation into a genuinely predictive statistical theory — and it is this statistical reading of expansion coefficients that underlies every application discussed here, from vibrational wavepackets to NMR coherences.