Unitary Evolution, Observables, and Commutators

A unitary operator $U$ satisfies $U^\dagger U = U U^\dagger = I$. Its defining property is that it preserves inner products: $\langle U\phi|U\psi\rangle = \langle\phi|\psi\rangle$ for all $|\phi\rangle$, $|\psi\rangle$. Taking $|\phi\rangle = |\psi\rangle$ shows that $U$ also preserves norms, so probabilities are unchanged.

Unitary operators are the automorphisms of Hilbert space. They map orthonormal bases to orthonormal bases and can be thought of as generalized rotations or reflections in the space of quantum states. Every unitary transformation is reversible, with inverse $U^\dagger$.

The eigenvalues of a unitary operator lie on the unit circle. If $U|\lambda\rangle = \lambda|\lambda\rangle$, then $\langle\lambda|U^\dagger U|\lambda\rangle = |\lambda|^2 \langle\lambda|\lambda\rangle = \langle\lambda|\lambda\rangle$, so $|\lambda|^2 = 1$ and $\lambda = e^{i\theta}$ for some real $\theta$.

In quantum computing, quantum gates are represented by unitary matrices. The Hadamard gate, phase gates, and CNOT gate are all unitary, ensuring that the total probability of all computational basis outcomes remains 1 after any sequence of gates.

In the Schrödinger picture, the state of an isolated quantum system evolves according to $|\psi(t)\rangle = U(t)|\psi(0)\rangle$, where $U(t)$ is a unitary operator. For a time-independent Hamiltonian $H$, the evolution operator is $U(t) = e^{-iHt/\hbar}$.

The unitarity of $U(t)$ follows because $H$ is Hermitian. The adjoint of $e^{-iHt/\hbar}$ is $e^{+iHt/\hbar}$, and their product is $I$. This ensures that the norm of $|\psi(t)\rangle$ is constant in time, which is the conservation of total probability.

The differential form of unitary evolution is the Schrödinger equation: $i\hbar \frac{d}{dt}|\psi(t)\rangle = H|\psi(t)\rangle$. It is first order in time, so the initial state uniquely determines the future state. The Hamiltonian $H$ is the generator of time translations.

Unitary evolution is deterministic and reversible. If two states start orthogonal, they remain orthogonal for all time. This linearity and reversibility contrast sharply with the probabilistic, irreversible nature of quantum measurement.

Every measurable physical quantity corresponds to a Hermitian operator $A$ on the Hilbert space. The possible outcomes of a measurement are the eigenvalues of $A$, which are guaranteed to be real. The states with definite values are the corresponding eigenvectors.

The expectation value of $A$ in state $|\psi\rangle$ is $\langle A \rangle = \langle\psi|A|\psi\rangle$. It represents the average result obtained when $A$ is measured many times on identically prepared systems. The variance is $\sigma_A^2 = \langle A^2 \rangle - \langle A \rangle^2$.

If $|\psi\rangle$ is an eigenstate of $A$, then $\sigma_A = 0$ and the measurement outcome is certain. More generally, the probability of obtaining eigenvalue $a_i$ is $p_i = |\langle a_i|\psi\rangle|^2$, where $|a_i\rangle$ is the normalized eigenvector. This is the generalized Born rule.

Examples of observables include position $\hat{x}$, momentum $\hat{p}$, angular momentum $\hat{L}_z$, and spin components $\hat{S}_z$. Each has a Hermitian operator and a corresponding set of eigenstates that form a complete basis.

A projective measurement of an observable $A$ with spectral decomposition $A = \sum_i a_i P_i$ yields outcome $a_i$ with probability $p_i = \langle\psi|P_i|\psi\rangle$. After the measurement, the state collapses to $P_i|\psi\rangle / \sqrt{p_i}$, provided $p_i > 0$.

The projectors $P_i$ are orthogonal and complete: $P_i P_j = \delta_{ij} P_i$ and $\sum_i P_i = I$. Orthogonality ensures that distinct outcomes are mutually exclusive, while completeness guarantees that the total probability $\sum_i p_i$ equals 1 for any normalized state.

The post-measurement state is an eigenstate of $A$ corresponding to the observed eigenvalue. Repeating the measurement immediately yields the same outcome with certainty. This repeatability is a defining feature of projective measurements.

The measurement process is not unitary on the system alone; it is an interaction between the system and an apparatus. In a fully unitary description of system plus apparatus, the apparent collapse emerges from entanglement between the two.

The commutator of two operators $A$ and $B$ is $[A,B] = AB - BA$. It measures the failure of $A$ and $B$ to commute. If $[A,B] = 0$, the operators commute and are called compatible; otherwise they are incompatible.

Commutators satisfy several algebraic identities. They are antisymmetric: $[A,B] = -[B,A]$. They obey the Jacobi identity $[A,[B,C]] + [B,[C,A]] + [C,[A,B]] = 0$. They also satisfy the Leibniz rule $[A,BC] = [A,B]C + B[A,C]$.

The commutator of two Hermitian operators is anti-Hermitian: $[A,B]^\dagger = -[A,B]$. Therefore $i[A,B]$ is Hermitian. This is why uncertainty relations involve the expectation value of $i[A,B]$, which is real.

The most famous commutator in quantum mechanics is $[\hat{x}, \hat{p}] = i\hbar I$. It encodes the canonical commutation relation and directly implies the Heisenberg uncertainty principle. All quantum commutators are descendants of this fundamental relation.

Two observables $A$ and $B$ are compatible if they commute: $[A,B] = 0$. In finite dimensions, commuting Hermitian operators can be simultaneously diagonalized. This means there exists an orthonormal basis $\{|a_i,b_j\rangle\}$ in which both $A$ and $B$ are diagonal.

When $A$ and $B$ are compatible, a state can have definite values for both observables at the same time. Measuring $A$ does not disturb a subsequent measurement of $B$, and the order of measurements does not matter. Classical physics is the limiting case in which all relevant observables commute.

Incompatible observables do not commute. They cannot share a complete set of eigenvectors. A state that is an eigenstate of one is generally a superposition of eigenstates of the other. Measuring one observable randomizes the value of the other.

Spin components along different axes are incompatible: $[\sigma_x, \sigma_y] = 2i\sigma_z \neq 0$. A particle with definite $S_z$ is in a superposition of $S_x$ eigenstates, so a measurement of $S_x$ yields $\pm \hbar/2$ with equal probability.

For any two observables $A$ and $B$, the Robertson uncertainty relation states $\sigma_A \sigma_B \ge \frac{1}{2}|\langle [A,B] \rangle|$. The right-hand side involves the expectation value of the commutator in the state $|\psi\rangle$. If $A$ and $B$ commute, the lower bound is zero and both can be precisely known.

The proof uses the Cauchy-Schwarz inequality applied to the kets $(A - \langle A\rangle)|\psi\rangle$ and $(B - \langle B\rangle)|\psi\rangle$. The imaginary part of their inner product is half the commutator expectation, which gives the uncertainty bound.

The standard Heisenberg relation $\sigma_x \sigma_p \ge \hbar/2$ follows directly from $[\hat{x}, \hat{p}] = i\hbar I$, since $|\langle [\hat{x},\hat{p}] \rangle| = \hbar$. The generalized relation applies to any pair of observables and any state.

A state saturates the bound when the two deviation kets are proportional with a purely imaginary constant. For position and momentum, the minimum-uncertainty states are Gaussians. For spin, special spin-coherent states can saturate angular-momentum uncertainty relations.

The Pauli matrices are $\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$, $\sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}$, and $\sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$. Compute the commutator $[\sigma_x, \sigma_y]$.

Step 1 -- Multiply $\sigma_x \sigma_y$. $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} = \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix} = i\sigma_z$.

Step 2 -- Multiply $\sigma_y \sigma_x$. $\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} -i & 0 \\ 0 & i \end{pmatrix} = -i\sigma_z$.

Step 3 -- Subtract. $[\sigma_x, \sigma_y] = \sigma_x \sigma_y - \sigma_y \sigma_x = i\sigma_z - (-i\sigma_z) = 2i\sigma_z$. This is the fundamental spin commutation relation (without factors of $\hbar/2$).