Expectation value (quantum mechanics)

In quantum mechanics, the expectation value is the probabilistic expected value of the result (measurement) of an experiment. It is not the most probable value of a measurement; indeed the expectation value may have zero probability of occurring. It is a fundamental concept in all areas of quantum physics.

Operational definition

Consider an operator A. So the expectation value is

${\displaystyle \langle A\rangle =\langle \psi |A|\psi \rangle }$ in Dirac notation

Formalism in quantum mechanics

In quantum theory, an experimental setup is described by the observable $A$ to be measured, and the state $\sigma$ of the system. The expectation value of $A$ in the state $\sigma$ is denoted as $\langle A \rangle_\sigma$.

Mathematically, $A$ is a self-adjoint operator on a Hilbert space. In the most commonly used case in quantum mechanics, $\sigma$ is a pure state, described by a normalized^{[lower-alpha 1]} vector $\psi$ in the Hilbert space. The expectation value of $A$ in the state $\psi$ is defined as

(1)      $\langle A \rangle_\psi = \langle \psi | A | \psi \rangle$.

If dynamics is considered, either the vector $\psi$ or the operator $A$ is taken to be time-dependent, depending on whether the Schrödinger picture or Heisenberg picture is used. The evolution of the expectation value does not depend on this choice, however.

If $A$ has a complete set of eigenvectors $\phi _{j}$, with eigenvalues $a_{j}$, then (1) can be expressed as

(2)      $\langle A \rangle_\psi = \sum_j a_j |\langle \psi | \phi_j \rangle|^2$.

This expression is similar to the arithmetic mean, and illustrates the physical meaning of the mathematical formalism: The eigenvalues $a_{j}$ are the possible outcomes of the experiment,^{[lower-alpha 2]} and their corresponding coefficient $|\langle \psi | \phi_j \rangle|^2$ is the probability that this outcome will occur; it is often called the transition probability.

A particularly simple case arises when $A$ is a projection, and thus has only the eigenvalues 0 and 1. This physically corresponds to a "yes-no" type of experiment. In this case, the expectation value is the probability that the experiment results in "1", and it can be computed as

(3)      $\langle A \rangle_\psi = \| A \psi \|^2$.

In quantum theory, also operators with non-discrete spectrum are in use, such as the position operator $Q$ in quantum mechanics. This operator does not have eigenvalues, but has a completely continuous spectrum. In this case, the vector $\psi$ can be written as a complex-valued function $\psi (x)$ on the spectrum of $Q$ (usually the real line). For the expectation value of the position operator, one then has the formula

(4)      $\langle Q \rangle_\psi = \int_{-\infty}^{\infty} \, x \, |\psi(x)|^2 \, dx$.

A similar formula holds for the momentum operator $P$, in systems where it has continuous spectrum.

All the above formulas are valid for pure states $\sigma$ only. Prominently in thermodynamics, also mixed states are of importance; these are described by a positive trace-class operator $\rho = \sum_i \rho_i | \psi_i \rangle \langle \psi_i |$, the statistical operator or density matrix. The expectation value then can be obtained as

(5)      $\langle A \rangle_\rho = \mathrm{Trace} (\rho A) = \sum_i \rho_i \langle \psi_i | A | \psi_i \rangle = \sum_i \rho_i \langle A \rangle_{\psi_i}$.

General formulation

In general, quantum states $\sigma$ are described by positive normalized linear functionals on the set of observables, mathematically often taken to be a C* algebra. The expectation value of an observable $A$ is then given by

(6)      $\langle A \rangle_\sigma = \sigma(A)$.

If the algebra of observables acts irreducibly on a Hilbert space, and if $\sigma$ is a normal functional, that is, it is continuous in the ultraweak topology, then it can be written as

$\sigma (\cdot) = \mathrm{Trace} (\rho \; \cdot)$

with a positive trace-class operator $\rho$ of trace 1. This gives formula (5) above. In the case of a pure state, $\rho= |\psi\rangle\langle\psi|$ is a projection onto a unit vector $\psi$. Then $\sigma = \langle \psi |\cdot \; \psi\rangle$, which gives formula (1) above.

$A$ is assumed to be a self-adjoint operator. In the general case, its spectrum will neither be entirely discrete nor entirely continuous. Still, one can write $A$ in a spectral decomposition,

$A = \int a \, \mathrm{d}P(a)$

with a projector-valued measure $P$. For the expectation value of $A$ in a pure state $\sigma=\langle\psi | \cdot \, \psi \rangle$, this means

$\langle A \rangle_\sigma = \int a \; \mathrm{d} \langle \psi | P(a) \psi\rangle$,

which may be seen as a common generalization of formulas (2) and (4) above.

In non-relativistic theories of finitely many particles (quantum mechanics, in the strict sense), the states considered are generally normal. However, in other areas of quantum theory, also non-normal states are in use: They appear, for example. in the form of KMS states in quantum statistical mechanics of infinitely extended media,^[1] and as charged states in quantum field theory.^[2] In these cases, the expectation value is determined only by the more general formula (6).

Example in configuration space

As an example, let us consider a quantum mechanical particle in one spatial dimension, in the configuration space representation. Here the Hilbert space is $\mathcal{H} = L^2(\mathbb{R})$, the space of square-integrable functions on the real line. Vectors $\psi\in\mathcal{H}$ are represented by functions $\psi (x)$, called wave functions. The scalar product is given by $\langle \psi _{1}|\psi _{2}\rangle =\int \psi _{1}^{\ast }(x)\psi _{2}(x)\,{\mathrm {d}}x$. The wave functions have a direct interpretation as a probability distribution:

$p(x) dx = \psi^*(x)\psi(x) dx$

gives the probability of finding the particle in an infinitesimal interval of length $dx$ about some point $x$.

As an observable, consider the position operator $Q$, which acts on wavefunctions $\psi$ by

$(Q \psi) (x) = x \psi(x)$.

The expectation value, or mean value of measurements, of $Q$ performed on a very large number of identical independent systems will be given by

$\langle Q \rangle_\psi = \langle \psi | Q | \psi \rangle =\int_{-\infty}^{\infty} \psi^\ast(x) \, x \, \psi(x) \, \mathrm{d}x = \int_{-\infty}^{\infty} x \, p(x) \, \mathrm{d}x$.

The expectation value only exists if the integral converges, which is not the case for all vectors $\psi$. This is because the position operator is unbounded, and $\psi$ has to be chosen from its domain of definition.

In general, the expectation of any observable can be calculated by replacing $Q$ with the appropriate operator. For example, to calculate the average momentum, one uses the momentum operator in configuration space, $P=-i\hbar \,d/dx$. Explicitly, its expectation value is

$\langle P\rangle _{\psi }=-i\hbar \int _{{-\infty }}^{{\infty }}\psi ^{\ast }(x)\,{\frac {d\psi (x)}{dx}}\,{\mathrm {d}}x$.

Not all operators in general provide a measureable value. An operator that has a pure real expectation value is called an observable and its value can be directly measured in experiment.

See also

• Uncertainty principle
• Virial theorem

Notes

1. This article always takes $\psi$ to be of norm 1. For non-normalized vectors, $\psi$ has to be replaced with $\psi / \|\psi\|$ in all formulas.
2. It is assumed here that the eigenvalues are non-degenerate.

References

1. Bratteli, Ola; Robinson, Derek W (1987). Operator Algebras and Quantum Statistical Mechanics 1. Springer. ISBN 978-3-540-17093-8. 2nd edition.
2. Haag, Rudolf (1996). Local Quantum Physics. Springer. pp. Chapter IV. ISBN 3-540-61451-6.

Further reading

The expectation value, in particular as presented in the section "Formalism in quantum mechanics", is covered in most elementary textbooks on quantum mechanics.

For a discussion of conceptual aspects, see:

• Isham, Chris J (1995). Lectures on Quantum Theory: Mathematical and Structural Foundations. Imperial College Press. ISBN 978-1-86094-001-9.