Commutators and the Uncertainty Principle

The commutator structure underlying orbital angular momentum ($[\hat L_x,\hat L_y]=i\hbar\hat L_z$, and its generalization to electron and nuclear spin operators with identical algebra) is the direct basis for NMR and EPR spectroscopy: the fact that $\hat S_x$, $\hat S_y$, $\hat S_z$ do not commute is why spin precesses about an applied field rather than aligning instantaneously, and this precession is the Larmor frequency detected in every NMR experiment used in structure elucidation, MRI, and protein folding studies. Selection rules in UV-Vis and IR spectroscopy — which transitions are allowed — depend on whether the transition dipole operator commutes or fails to commute with parity and angular-momentum operators of the initial and final states, governed by the same commutator algebra. In quantum computing and molecular qubits, non-commuting spin operators are the entire resource that makes superposition-based logic gates possible; commuting operators would make quantum computers behave classically. In spectroscopic line-shape theory, the energy–time uncertainty relation (a looser cousin of the operator relation, since time is not represented by a Hermitian operator in standard non-relativistic quantum mechanics) explains natural linewidths: short-lived excited states (fast radiative or non-radiative decay, small $\Delta t$) have correspondingly broad spectral linewidths ($\Delta E$), a phenomenon routinely used to extract excited-state lifetimes from fluorescence and femtosecond transient-absorption spectra. In computational chemistry, the choice of basis functions and operators that commute with the molecular point-group symmetry operations (a generalized commutator condition, $[\hat H,\hat R]=0$ for symmetry operation $\hat R$) is exactly what allows block-diagonalization of the Hamiltonian by irreducible representation, the computational trick that makes electronic-structure calculations on symmetric molecules tractable — this connects directly to how The Measurement Postulate guarantees that simultaneous eigenstates of $\hat H$ and symmetry operators exist whenever they commute, letting spectroscopists assign vibrational and electronic transitions by symmetry label alone.

:::quiz[If [Â,B̂] = 0 for two Hermitian operators, what can be said about their eigenstates?]

• No two states can ever be eigenstates of both
• A complete set of simultaneous eigenfunctions of both operators exists ✓
• Only the ground state can be a simultaneous eigenstate
• The two operators must be identical

:::quiz[What is the value of [x̂, p̂ₓ] in one dimension?]

• Zero
• ħ
• iħ ✓
• -iħ²

Ask the tutor: Can you show how the same Cauchy–Schwarz argument changes if the state is not normalized, or if $\hat A$ and $\hat B$ act on a finite-dimensional spin space instead of continuous position/momentum?

Every observable quantity in quantum mechanics — position, momentum, energy, angular momentum — is represented by a Hermitian operator acting on the wavefunction. A measurement of that observable can yield a perfectly sharp, uncertainty-free value if and only if the wavefunction is an eigenfunction of the corresponding operator. The central question this article answers is: given two observables $A$ and $B$, can a single wavefunction be an eigenfunction of both operators at once, so that both quantities are simultaneously sharp? The answer is controlled entirely by the commutator,

which is itself an operator (it acts on a function, producing another function). If $[\hat A,\hat B]f = 0$ for every well-behaved function $f$, the operators commute, and a fundamental theorem guarantees the existence of a complete set of functions that are simultaneous eigenfunctions of both $\hat A$ and $\hat B$. Chemists exploit this constantly: the hydrogen-atom orbitals $\psi_{n\ell m_\ell}$ are simultaneous eigenfunctions of $\hat H$, $\hat L^2$, and $\hat L_z$, because these three operators pairwise commute. That is why "quantum numbers" $n$, $\ell$, $m_\ell$ exist as a good labeling scheme — it is not a convention, it is forced by commutation. If instead $[\hat A,\hat B]\neq 0$, no complete set of simultaneous eigenfunctions exists (barring isolated special states). The classic example is $\hat L_x$ and $\hat L_y$: an electron with sharp $L_z = m_\ell\hbar$ has, in general, completely undetermined $L_x$ and $L_y$ — this is why the "orbital angular momentum vector" pictured in textbooks is always drawn precessing on a cone rather than pointing in a fixed direction. The same non-commutation, applied to $\hat x$ and $\hat p_x$, produces the famous position–momentum uncertainty relation, but the relation is completely general: any pair of non-commuting Hermitian operators obeys an analogous inequality.

The uncertainty principle is often mis-stated as a statement about clumsy measuring instruments disturbing a system. In fact it is a statement about the wavefunction itself: even with a perfect, non-disturbing instrument, a quantum state that has a sharp value of $A$ necessarily has a spread of possible values for any $B$ that does not commute with $A$. The spread $\Delta A$ is defined exactly as in statistics,

the root-mean-square deviation of repeated measurements of $A$ on identically prepared systems — this is the same object introduced when discussing Expectation Values. The uncertainty principle bounds the product $\Delta A\,\Delta B$ from below by the expectation value of the commutator. A vivid chemical picture: consider the particle-in-a-box or a harmonic oscillator ground state. The wavefunction is spread over a finite region of $x$ (nonzero $\Delta x$) and correspondingly must contain a spread of momentum components (nonzero $\Delta p$) — this spread in momentum is exactly what supplies the zero-point kinetic energy that cannot be removed even at $T=0$. A vibrating C–H bond, no matter how cold, retains vibrational energy $\tfrac12\hbar\omega$ precisely because localizing the bond length (finite $\Delta x$) forces a nonzero $\Delta p$, and nonzero $\Delta p$ means nonzero $\langle p^2\rangle$, which means nonzero kinetic energy.

Definition of the commutator. For any two operators,

where: $\hat A\hat B$ means "apply $\hat B$ first, then $\hat A$," to some test function; the commutator is itself a new operator, evaluated by acting on an arbitrary function $f(x)$ and simplifying.

Reading: if $[\hat A,\hat B]f = 0$ for all $f$, the order of measurement/operation does not matter — $A$ and $B$ are compatible. If not, order matters, and this failure is quantified.

The canonical commutator.

where: $\hat x = x$ (multiplication), $\hat p_x = -i\hbar\,d/dx$, and $\hbar = h/2\pi = 1.0546\times10^{-34}\ \text{J s}$.

Reading: this single nonzero commutator is the ultimate algebraic source of the position–momentum uncertainty principle, and by extension (via the correspondence in angular-momentum operators built from $\hat x,\hat p$) of orbital angular momentum's non-commutation.

Angular momentum commutators.

where: $\hat L_x,\hat L_y,\hat L_z$ are the Cartesian components of orbital angular momentum, cyclic in $x\to y\to z\to x$.

Reading: no two different Cartesian components of angular momentum can be simultaneously sharp (except in the trivial case $\ell=0$). This is why only $L^2$ and one component (conventionally $L_z$) are used to label orbitals.

Compatibility with $\hat L^2$.

where: $\hat L^2 = \hat L_x^2+\hat L_y^2+\hat L_z^2$ is the total angular momentum magnitude-squared operator.

Reading: $L^2$ and any one component commute, which is exactly why $\ell$ and $m_\ell$ form a valid joint quantum-number pair, but $m_\ell$ alone (not $m_{\ell,x}$ and $m_{\ell,y}$ together) labels the state.

The generalized uncertainty principle.

where: $\Delta A,\Delta B$ are the standard deviations of $A,B$ in state $\psi$, and the right side is evaluated in that same state.

Reading: the product of uncertainties in two observables can never be smaller than half the magnitude of the expectation value of their commutator; if the commutator vanishes in that state, there is no lower bound at all — both can (in principle) be made simultaneously sharp.

The position–momentum uncertainty relation (specific case).

where: this follows by substituting $[\hat x,\hat p_x]=i\hbar$ (a state-independent constant) directly into the generalized relation.

Reading: this is the tightest possible bound; it is achieved with equality only by Gaussian wavepackets (including the harmonic oscillator ground state), and exceeded by every other physically realizable state.

Example 1 — Foundational: computing $[\hat x,\hat p_x]$ and verifying $\Delta x\Delta p \geq \hbar/2$ for a Gaussian. Compute the canonical commutator explicitly by acting on an arbitrary test function $f(x)$:

Subtracting,

So indeed $[\hat x,\hat p_x] = i\hbar$ as an operator identity, independent of the specific $f$ chosen. Now check the bound numerically for a normalized Gaussian wavefunction with an asymmetric, non-round width parameter $\alpha = 0.73\ \text{Å}^{-2}$:

For this standard Gaussian form, the known moments are $\langle x^2\rangle = 1/(4\alpha)$ and $\langle p_x^2\rangle = \alpha\hbar^2$, with $\langle x\rangle=\langle p_x\rangle=0$ by symmetry, so

Multiplying,

Comparing to the bound $\hbar/2 = 0.5000\,\hbar$: the product equals the minimum-uncertainty value (the small numerical discrepancy above is rounding in the intermediate arithmetic; algebraically $\Delta x\Delta p_x = \tfrac12\hbar\sqrt{1/(4\alpha)}\cdot\sqrt{4\alpha}=\hbar/2$ exactly for any $\alpha$). This confirms Gaussians saturate the bound regardless of the specific width chosen — the equality is a structural property of the Gaussian shape, not a coincidence of a particular $\alpha$.

Example 2 — Applied: zero-point energy of an HCl-like oscillator from the uncertainty principle. Estimate the minimum vibrational energy of a diatomic oscillator using only the uncertainty principle, then compare to the exact quantum-mechanical zero-point energy, for a bond with force constant $k = 480\ \text{N/m}$ and reduced mass $\mu = 1.628\times10^{-27}\ \text{kg}$ (close to HCl values). Model the total energy classically as a function of position spread and momentum spread, using the harmonic potential and treating $\langle x^2\rangle\approx(\Delta x)^2$, $\langle p^2\rangle\approx(\Delta p)^2$ (valid since $\langle x\rangle=\langle p\rangle=0$ for the ground state):

Impose the minimum-uncertainty condition $\Delta p = \hbar/(2\Delta x)$ (using equality, the tightest possible bound) and substitute:

Minimize with respect to $\Delta x$ by differentiating and setting to zero:

Substituting back into $E$,

Recognizing $\omega = \sqrt{k/\mu}$, this gives $E_{\min} = \tfrac12\hbar\omega$ — exactly the known quantum zero-point energy, derived purely from the uncertainty principle without solving the Schrödinger equation. Plugging in numbers:

Converting to wavenumbers via $\tilde\nu = E/(hc)$:

This is close to the real HCl fundamental of about $2886\ \text{cm}^{-1}$ divided by 2 (since $\tilde\nu_{\text{fundamental}} = 2\times$ zero-point wavenumber for a harmonic oscillator, as $E_1 - E_0 = \hbar\omega$ while $E_0=\tfrac12\hbar\omega$), confirming the estimate is physically sound and the uncertainty principle alone predicts real vibrational spectroscopy correctly in order of magnitude and structure.

Example 3 — Exam/competition level: commutator of $\hat L_x$ and $\hat L_y$, and the resulting bound on simultaneous angular momentum components. Derive $[\hat L_x,\hat L_y] = i\hbar\hat L_z$ from the canonical position–momentum commutators, then use it to compute the minimum uncertainty product for a specific hydrogen-like state with $\ell=2$, $m_\ell = 1$ (a $d$-orbital component). Starting definitions: $\hat L_x = \hat y\hat p_z - \hat z\hat p_y$, $\hat L_y = \hat z\hat p_x - \hat x\hat p_z$. Using $[\hat x_i,\hat p_j]=i\hbar\delta_{ij}$ (each coordinate commutes with momenta along other axes, and $[\hat x_i,\hat p_i]=i\hbar$), expand:

Expanding into four terms and using that operators referring to different, independent Cartesian degrees of freedom commute (e.g., $\hat y$ commutes with $\hat z, \hat p_x, \hat p_z$ since these act on different coordinates):

The second and third terms vanish (all constituent operators mutually commute within each). The first term:

The fourth term:

Adding these surviving pieces:

confirming the stated relation. Now apply the generalized uncertainty principle to a state with definite $\ell=2$, $m_\ell=1$ (one specific $d$-orbital wavefunction, eigenstate of $\hat L_z$ with eigenvalue $m_\ell\hbar = \hbar$):

For $m_\ell = 1$ specifically, $\langle L_z\rangle = \hbar = 1.0546\times10^{-34}\ \text{J s}$, so

This nonzero lower bound is why an electron in this $d$-orbital component, despite having a perfectly sharp $L_z=\hbar$ and sharp $L^2 = \ell(\ell+1)\hbar^2 = 6\hbar^2$, can never simultaneously have sharp $L_x$ and $L_y$: the product of their uncertainties is bounded away from zero exactly by half of $\langle L_z\rangle$. As a limiting-case check, for the $s$-orbital case $\ell=0$ (so necessarily $m_\ell=0$), the bound collapses to $\Delta L_x\Delta L_y \geq 0$ — consistent with the fact that an $s$-orbital has zero angular momentum entirely, all three components sharply zero at once, with no contradiction, since the commutator's expectation value itself vanishes when $\langle L_z\rangle=0$.