Functions of Operators and the Spectral Calculus

The Core Idea: Acting on Eigenvalues

Imagine an operator \( \hat{A} \) as a machine that stretches space along specific axes (eigenvectors) by specific factors (eigenvalues). If you want to apply a function \( f \) to this machine, you do not need to understand the complex geometry of the whole space. Instead, you simply apply \( f \) to the stretching factors. For example, if \( \hat{A} \) stretches a vector by a factor of 2 along a specific axis, then \( \hat{A}^2 \) stretches it by \( 2^2 = 4 \), and \( \sqrt{\hat{A}} \) stretches it by \( \sqrt{2} \). The direction of the stretch (the eigenvector) remains unchanged; only the magnitude is modified by the function. This principle holds for any well-behaved function \( f \), provided \( \hat{A} \) is self-adjoint, because the eigenvectors form a stable, orthogonal basis that does not mix under the operation.

Formal Definition via Spectral Resolution

Let \( \hat{A} \) be a self-adjoint operator with distinct eigenvalues \( \lambda_1, \dots, \lambda_n \) and corresponding orthogonal projectors \( P_k \). According to the spectral resolution, we can write \( \hat{A} = \sum_{k=1}^{n} \lambda_k P_k \). For any function \( f: \mathbb{R} \to \mathbb{C} \) defined on the spectrum of \( \hat{A} \), the function of the operator is defined as:

This definition is consistent with polynomial functions. If \( f(x) = x^m \), then \( f(\hat{A}) = \hat{A}^m \). The power of this definition lies in its generality; it extends naturally to transcendental functions like exponentials and logarithms. Because the projectors \( P_k \) are mutually orthogonal and idempotent (\( P_j P_k = \delta_{jk} P_k \)), the algebraic properties of \( f(\hat{A}) \) are determined entirely by the scalar values \( f(\lambda_k) \).

Commutativity and Functional Calculus

A crucial consequence of the spectral calculus is that any two functions of the same self-adjoint operator commute. Let \( f(\hat{A}) \) and \( g(\hat{A}) \) be two such functions. Their product is:

Using the orthogonality condition \( P_j P_k = \delta_{jk} P_k \), the double sum collapses to a single sum:

This result is fundamental in quantum mechanics. It implies that if two observables are functions of the same self-adjoint operator (or commute with it), they share a common eigenbasis. This is the mathematical underpinning of the concept of compatible observables, where measurements of one quantity do not disturb the state defined by the other.

Physical Applications: Time Evolution

The most prominent application of the spectral calculus is in the time evolution of quantum states. The time-evolution operator is given by \( \hat{U}(t) = e^{-i\hat{H}t/\hbar} \), where \( \hat{H} \) is the Hamiltonian. Since \( \hat{H} \) is self-adjoint, we can apply the spectral definition:

Here, \( \lambda_k \) are the energy eigenvalues. If a system is in an energy eigenstate \( |e_k\rangle \), the time evolution is simply a phase factor \( e^{-i\lambda_k t/\hbar} \). For a general state \( |\psi\rangle = \sum_{k} c_k |e_k\rangle \), the evolved state is \( |\psi(t)\rangle = \sum_{k} c_k e^{-i\lambda_k t/\hbar} |e_k\rangle \). This shows that the spectral calculus reduces the complex differential equation \( i\hbar \frac{d}{dt}|\psi\rangle = \hat{H}|\psi\rangle \) to a set of independent scalar phase rotations.