Expectation value (quantum mechanics) explained

In quantum mechanics, the expectation value is the probabilistic expected value of the result (measurement) of an experiment. It can be thought of as an average of all the possible outcomes of a measurement as weighted by their likelihood, and as such it is not the most probable value of a measurement; indeed the expectation value may have zero probability of occurring (e.g. measurements which can only yield integer values may have a non-integer mean). It is a fundamental concept in all areas of quantum physics.

Operational definition

A . The expectation value is then \langle A \rangle = \langle \psi | A | \psi \rangle in Dirac notation with |\psi \rangle a normalized state vector.

Formalism in quantum mechanics

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 separable complex Hilbert space. In the most commonly used case in quantum mechanics, \sigma is a pure state, described by a normalized vector \psi in the Hilbert space. The expectation value of A in the state \psi is defined as

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} so that A = \sum _{j} a _{j} | \phi _{j\rangle} \langle \phi _{j} |, then can be expressed as^[1]

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, 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 asX in quantum mechanics. This operator has a completely continuous spectrum, with eigenvalues and eigenvectors depending on a continuous parameter, x . Specifically, the operator X acts on a spatial vector | x \rangle as X | x \rangle = x |x\rangle .^[2] In this case, the vector \psi can be written as a complex-valued function \psi(x) on the spectrum of X (usually the real line). This is formally achieved by projecting the state vector | \psi \rangle onto the eigenvalues of the operator, as in the discrete case $\backslash psi(x)\; \backslash equiv\; \backslash langle\; x\; |\; \backslash psi\; \backslash rangle$. It happens that the eigenvectors of the position operator form a complete basis for the vector space of states, and therefore obey a completeness relation in quantum mechanics:$$\backslash int\; |x\; \backslash rangle\; \backslash langle\; x|\; \backslash ,\; dx\; \backslash equiv\; \backslash mathbb$$

The above may be used to derive the common, integral expression for the expected value, by inserting identities into the vector expression of expected value, then expanding in the position basis:

\begin{align} \langle X \rangle _{\psi} & = \langle \psi | X | \psi \rangle = \langle \psi | I X I | \psi \rangle \\ & = \iint \langle \psi | x \rangle \langle x | X | x' \rangle \langle x' | \psi \rangle dx dx' \\ & = \iint \langle x | \psi \rangle ^{*} x' \langle x | x' \rangle \langle x' | \psi \rangle dx dx' \\ & = \iint \langle x | \psi \rangle ^{*} x' \delta(x - x') \langle x' | \psi \rangle dx dx' \\ & = \int \psi(x) ^{*} x \psi(x) dx = \int x \psi(x) ^{*} \psi(x) dx = \int x |\psi(x)| ^{2} dx \end{align}

Where the orthonormality relation of the position basis vectors

\langle x | x' \rangle = \delta(x - x') , reduces the double integral to a single integral. The last line uses the modulus of a complex valued function to replace \psi ^{*\psi} with |\psi| ^{2} , which is a common substitution in quantum-mechanical integrals.

The expectation value may then be stated, where is unbounded, as the formula

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

All the above formulas are valid for pure states

\sigma only. Prominently in thermodynamics and quantum optics, also mixed states are of importance; these are described by a positive trace-class operator $\backslash rho\; =\; \backslash sum\_i\; p\_i\; |\; \backslash psi\_i\; \backslash rangle\; \backslash langle\; \backslash psi\_i\; |$, the statistical operator or density matrix. The expectation value then can be obtained as

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

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$$\backslash sigma\; (\backslash cdot)\; =\; \backslash operatorname\; (\backslash rho\; \backslash ;\; \backslash cdot)$$with a positive trace-class operator \rho of trace 1. This gives formula 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 | ⋅ \psi\rangle , which gives formula 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,$$A\; =\; \backslash int\; a\; \backslash ,\; dP(a)$$with a projection-valued measure P . For the expectation value of A in a pure state \sigma = \langle \psi | ⋅ \psi \rangle , this means$$\backslash langle\; A\; \backslash rangle\_\backslash sigma\; =\; \backslash int\; a\; \backslash ;\; d\; \backslash langle\; \backslash psi\; |\; P(a)\; \backslash psi\backslash rangle,$$which may be seen as a common generalization of formulas and 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,^[3] and as charged states in quantum field theory.^[4] In these cases, the expectation value is determined only by the more general formula .

Example in configuration space

As an example, consider a quantum mechanical particle in one spatial dimension, in the configuration space representation. Here the Hilbert space is

l{H} = L ^{2(R)} , the space of square-integrable functions on the real line. Vectors \psi\in l{H} are represented by functions \psi(x) , called wave functions. The scalar product is given by $\backslash langle\; \backslash psi\_1\; |\; \backslash psi\_2\; \backslash rangle\; =\; \backslash int\; \backslash psi\_1^\backslash ast\; (x)\; \backslash psi\_2(x)\; \backslash ,\; dx$. The wave functions have a direct interpretation as a probability distribution:\rho(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$$(Q\; \backslash psi)\; (x)\; =\; x\; \backslash psi(x)\; .$$

The expectation value, or mean value of measurements, of

Q performed on a very large number of identical independent systems will be given by$$\backslash langle\; Q\; \backslash rangle\_\backslash psi=\; \backslash langle\; \backslash psi\; |\; Q\; |\; \backslash psi\; \backslash rangle=\; \backslash int\_^\; \backslash psi^\backslash ast(x)\; \backslash ,\; x\; \backslash ,\; \backslash psi(x)\; \backslash ,\; dx=\; \backslash int\_^\; x\; \backslash ,\; \backslash rho(x)\; \backslash ,\; dx\; .$$

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, $\backslash mathbf\; =\; -i\; \backslash hbar\; \backslash ,\; \backslash frac$. Explicitly, its expectation value is\langle p \rangle _{\psi} = - i\hbar \int ^{inf ty} _{-inf ty} \psi ^{\ast(x)} (d\psi(x))/(dx) dx.

Not all operators in general provide a measurable 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

• Rayleigh quotient
• Uncertainty principle
• Virial theorem

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:

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

Notes and References

1. http://physics.mq.edu.au/~jcresser/Phys301/Chapters/Chapter14.pdf Probability, Expectation Value and Uncertainty
2. Book: Cohen-Tannoudji, Claude, 1933-. Quantum mechanics. Volume 2. Diu, Bernard,, Laloë, Franck, 1940-, Hemley, Susan Reid,, Ostrowsky, Nicole, 1943-, Ostrowsky, D. B.. June 2020. 978-3-527-82272-0. Weinheim. 1159410161.
3. Book: Bratteli . Ola . Robinson . Derek W . 1987 . Springer . 978-3-540-17093-8 . 2nd edition . Ola Bratteli . Derek W. Robinson.
4. Book: Haag, Rudolf . Rudolf Haag

   . Rudolf Haag . Local Quantum Physics . Springer . 1996 . Chapter IV . 3-540-61451-6.