Expectation Values

Prerequisites: Statistics, Canonical Quantum Mechanics. Full path (18 pages)

Intuitive
Concrete
Abstract

Intuitive

Roll a fair die many times and average the results. The average gets closer and closer to 3.5, even though no single roll ever shows 3.5. This long-run average is the expectation value. It does not tell us what happens in one trial, but it summarizes what happens on average in many.

In quantum mechanics the expectation value plays a central role because a single measurement on a quantum system generally gives a random result. Prepare many copies of an electron in the same state, measure the position of each one, and the results scatter. The theory does not predict the individual outcomes, only their probabilities. The expectation value of the position is the mean of all these measured positions, and it is one of the quantities the theory predicts sharply. The spread of the results around this mean is described by the expectation value of the squared deviation, which is how the uncertainty relations are formulated.

The same idea appears in statistical physics. A gas contains far too many molecules to follow each one, so we describe it by the probabilities of its microscopic configurations. Quantities such as pressure or magnetization are then expectation values over these probabilities, and correlation functions, like the one in the quote below, are expectation values of products of quantities at different points.

Concrete

To that end, consider a locally defined microscopic variable which I will denote $\psi(r)$ . In a ferromagnet this might well be the local magnetization, M(r), or spin vector, S(r), at point r in ordinary d-dimensional (Euclidean) space; in a fluid it might be the deviation $\delta p(r)$ , of the fluctuating density at r from the mean density. In QFT the local variables $\psi(r)$ are the basic quantum fields which are 'operator valued.' For a magnetic system in which quantum mechanics was important, M(r) and S(r) would, likewise, be operators. However, the distinction is of relatively minor importance so that we may, for ease, suppose $\psi(r)$ is a simple classical variable. It will be most interesting when $\psi$ is closely related to the order parameter for the phase transition and critical behavior of concern.

By means of a scattering experiment (using light, x rays, neutrons, electrons, etc.) one can often observe the corresponding pair correlation function (or basic 'two-point function') $G(r)=\langle \psi(0) \psi(r)\rangle$ where the angular brackets $\langle \cdot \rangle$ denote a statistical average over the thermal fluctuations that characterize all equilibrium systems at nonzero temperature. (Also understood, when $\psi(r)$ is an operator, are the corresponding quantum-mechanical expectation values.) Physically, $G(r)$ is important since it provides a direct measure of the influence of the leading microscopic fluctuations at the origin $0$ on the behavior at a point distance $r = |r|$ away. But, almost by definition, in the vicinity of an appropriate critical point - for example the Curie point of a ferromagnet when $\psi = \vec M$ or the gas-liquid critical point when $\psi = \delta p$ - a strong "ordering" influence or correlation spreads out over, essentially, macroscopic distances. As a consequence, precisely at criticality one rather generally finds a power-law decay, namely, $G(r) \approx D/d^{d-2+\eta}$ as $r \to \infty$ which is characterized by the Critical Exponents (or critical index) $d - 2 + \eta$ .

Michael Fisher in Conceptual Foundations of Quantum Field Theory, Edited by Cao

No-Nonsense Quantum Mechanics by Jakob Schwichtenberg, section "Statistical Tools in Quantum Mechanics"

Abstract

Probability theory. For a random variable $X$ on a probability space $(\Omega,\mathcal{F},P)$ the expectation value is the Lebesgue integral $\mathbb{E}[X]=\int_\Omega X\,dP$ , defined when $\int|X|\,dP<\infty$ . It is a linear functional, positive ( $X\ge0\Rightarrow\mathbb{E}[X]\ge0$ ) and normalized ( $\mathbb{E}[1]=1$ ).

Quantum mechanics. For a self-adjoint operator $A$ with spectral measure $E_A$ and a normalized vector $\psi$ in a Hilbert space, the Borel measure $\mu_\psi(B)=\langle\psi|E_A(B)|\psi\rangle$ is a probability measure on the spectrum of $A$ : the distribution of measurement outcomes. Its mean is $\langle A\rangle_\psi=\int\lambda\,d\mu_\psi(\lambda)=\langle\psi|A|\psi\rangle,$ where the last expression is valid for $\psi$ in the domain of $A$ . A mixed state is a density operator $\rho$ (positive, trace class, $\operatorname{tr}\rho=1$ ), and for bounded $A$ $\langle A\rangle_\rho=\operatorname{tr}(\rho A).$ The map $A\mapsto\operatorname{tr}(\rho A)$ is again a positive, normalized linear functional on the bounded operators $\mathcal{B}(\mathcal{H})$ , and it is normal (continuous for monotone limits). Every normal state on $\mathcal{B}(\mathcal{H})$ is of this form. Gleason's theorem gives the converse direction from probabilities: for $\dim\mathcal{H}\ge3$ , every countably additive probability assignment to the closed subspaces (projections) of $\mathcal{H}$ is given by a density operator.

Algebraic formulation. This motivates defining a state on a unital C-algebra $\mathfrak{A}$ of observables as a linear functional $\omega:\mathfrak{A}\to\mathbb{C}$ with $\omega(a^*a)\ge0$ and $\omega(1)=1$ ; $\omega(a)$ is the expectation value of $a$ . Commutative C-algebras are algebras $C(X)$ of continuous functions on a compact Hausdorff space $X$ , and by the Riesz-Markov theorem their states are probability measures on $X$ , so this definition contains the classical case. For a noncommutative algebra the GNS construction produces from $\omega$ a Hilbert space $\mathcal{H}_\omega$ , a representation $\pi_\omega$ and a cyclic vector $\Omega_\omega$ with $\omega(a)=\langle\Omega_\omega|\pi_\omega(a)|\Omega_\omega\rangle .$ In quantum field theory the vacuum expectation values $\langle0|\phi(x_1)\cdots\phi(x_n)|0\rangle$ are the basic data: by the Wightman reconstruction theorem they determine the Hilbert space and the field operators.

Expectation Values

Intuitive

Concrete

Abstract

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