Expectation Value

Expectation Values in Quantum Mechanics

In quantum mechanics, while the outcome of a single measurement of an observable is unpredictable beyond a set of discrete or continuous eigenvalues, we can still determine the average value we would obtain if we performed the measurement on a large number of identically prepared systems. This average value is known as the expectation value. It is a fundamental concept that connects the abstract quantum state to the results of physical experiments.

Definition of Expectation Value

For a quantum system in a particular state, an observable (a measurable physical quantity like position, momentum, or energy) is represented by a Hermitian operator, let's call it L. The possible values that can be obtained when measuring L are the eigenvalues of the operator L.

If the operator L has a discrete set of eigenvalues {λi} with corresponding probabilities P(λi) for the system in a given state, the expectation value (or mean value) of L, denoted as ⟨L⟩, is defined as the probability-weighted average of these eigenvalues:

Calculating Expectation Values using the State Vector

The Overall Phase Factor of a State-Vector

Time Evolution of Expectation Values (Ehrenfest's Theorem)

How do the expectation values of observables change over time? This is described by Ehrenfest's theorem, which provides a connection between the commutator of an observable with the Hamiltonian and the time derivative of its expectation value.

For an observable L represented by a Hermitian operator that may or may not depend explicitly on time (i.e., L=L(t)), the time evolution of its expectation value in a state ∣Ψ(t)⟩ is given by:

Derivation of Ehrenfest's Theorem

Example: Spin in a Magnetic Field

These equations show that the expectation value of the spin vector ⟨σ⟩=(⟨σx⟩,⟨σy⟩,⟨σz⟩) precesses around the z-axis (the direction of the magnetic field) with angular velocity ω. This classical-like precession of the average behavior is analogous to the Larmor precession of a classical magnetic dipole in a magnetic field. It illustrates how Ehrenfest's theorem provides a link between quantum expectation values and classical equations of motion, particularly for quantities whose operators have simple commutation relations with the Hamiltonian. However, it is crucial to remember that this describes the average behavior; a single measurement of σx, σy, or σz will still yield only their respective eigenvalues (+1 or −1 for the Pauli operators).

The expectation value of an observable in quantum mechanics represents the average outcome of measurements performed on an ensemble of identically prepared systems. It is defined probabilistically as the sum of eigenvalues weighted by their probabilities. Alternatively and equivalently, for a normalized state ∣Ψ⟩, the expectation value of an observable L (represented by a Hermitian operator) is calculated as ⟨L⟩=⟨Ψ∣L∣Ψ⟩.

An overall phase factor in the state vector does not alter any physically measurable quantities like probabilities or expectation values, and thus is physically irrelevant. The time evolution of the expectation value of an observable is governed by Ehrenfest's theorem, (d/dt)⟨L⟩=i/ℏ⟨[H,L]⟩+⟨∂L/∂t⟩. This theorem highlights the role of the commutator of the observable's operator with the Hamiltonian in determining how the average value of the observable changes over time, providing a bridge between quantum dynamics and classical mechanics.

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