Expectation Value

Formula

For a normalized pure state and an observable $A$,

$$\langle A\rangle_\psi = \langle\psi\rvert A \lvert\psi\rangle.$$

For a density operator,

$$\langle A\rangle_\rho = \operatorname{Tr}(\rho A).$$

The most general spectral statement is

$$\langle A\rangle_\rho = \int_{\mathbb R} \lambda\, d\mu_\rho^A(\lambda),$$

where

$$\mu_\rho^A(\Delta) = \operatorname{Tr} \left[ \rho P^A(\Delta) \right]$$

is the Born probability measure for the observable $A$. The expectation exists as a finite number when the absolute first moment is finite:

$$\int_{\mathbb R} \lvert\lambda\rvert\, d\mu_\rho^A(\lambda) <\infty.$$

At a Glance

Setting	Expectation value
Normalized pure state	$\langle A\rangle_\psi=\langle\psi\rvert A\lvert\psi\rangle$
Density operator	$\langle A\rangle_\rho=\operatorname{Tr}(\rho A)$
Discrete spectrum	$\langle A\rangle=\sum_a a\,p(a)$
Continuous density	$\langle A\rangle=\int \lambda p_A(\lambda)\,d\lambda$
General spectral measure	$\langle A\rangle=\int\lambda\,d\mu^A(\lambda)$
Finite-dimensional state vector	$\langle A\rangle=\mathbf c^\dagger A\mathbf c$
Position representation	$\langle A\rangle=\int\psi^*(x)(A\psi)(x)\,dx$
Mixed state in components	$\langle A\rangle=\sum_{i,j}\rho_{ij}A_{ji}$

Meaning

The expectation value is the first moment of the probability distribution for measuring $A$ in the specified state. If identically prepared systems are measured independently and the first moment exists, their sample mean approaches $\langle A\rangle$ in the large-sample limit.

An expectation value is not generally:

• the result of one measurement;
• an eigenvalue of $A$;
• the most probable outcome;
• a value known to have existed before measurement.

For example, a spin component with possible outcomes $\pm\hbar/2$ can have expectation value zero. Zero summarizes the outcome distribution even though it is not an allowed result of that measurement.

Discrete Outcomes

If

$$A = \sum_a aP_a,$$

then

$$p(a) = \operatorname{Tr}(\rho P_a)$$

and

$$\langle A\rangle_\rho = \sum_a a\,p(a).$$

The projector $P_a$ contains the full eigenspace for eigenvalue $a$, so the same formula handles degeneracy. For a normalized pure state with an orthonormal eigenbasis $\{\lvert a,r\rangle\}$,

$$\langle A\rangle_\psi = \sum_{a,r} a \lvert \langle a,r\rvert\psi\rangle \rvert^2.$$

The degeneracy label $r$ is summed because the observable reports $a$, not an unresolved basis label inside its eigenspace.

Continuous and Mixed Spectra

If the Born distribution has a density $p_A(\lambda)$ with respect to $d\lambda$,

$$\langle A\rangle = \int_{-\infty}^{\infty} \lambda p_A(\lambda)\,d\lambda.$$

For one-dimensional position,

$$\langle X\rangle_\psi = \int_{-\infty}^{\infty} x \lvert\psi(x)\rvert^2\,dx.$$

An observable can have both discrete and continuous spectral parts. In a common notation,

$$\langle A\rangle = \sum_n a_nw_n + \int_{\mathbb R} \lambda p_{\mathrm{ac}}(\lambda)\,d\lambda.$$

Omitting either occupied part of a mixed spectrum gives the wrong mean. The spectral-measure formula includes all parts without requiring a separate case split.

Matrix Forms

For a normalized column vector $\mathbf c$ and Hermitian matrix $A$,

$$\langle A\rangle = \mathbf c^\dagger A\mathbf c.$$

For a density matrix,

$$\langle A\rangle = \operatorname{Tr}(\rho A) = \sum_{i,j} \rho_{ij}A_{ji}.$$

The second expression shows that off-diagonal entries can contribute. Only in a basis that diagonalizes $A$ does the expectation reduce to a weighted sum of diagonal state entries:

$$\langle A\rangle = \sum_a a\,\rho_{aa}$$

for a nondegenerate eigenbasis.

For numerical work, use a conjugate transpose, not an ordinary transpose. A significant imaginary part in the expectation of a Hermitian matrix usually indicates a non-Hermitian input, a missing complex conjugation, inconsistent basis conventions, or numerical error.

Wavefunction Forms

In one-dimensional position representation,

$$\langle A\rangle_\psi = \int_{-\infty}^{\infty} \psi^*(x) (A\psi)(x)\,dx.$$

For a multiplication operator $f(\hat x)$,

$$\langle f(\hat x)\rangle = \int_{-\infty}^{\infty} f(x) \lvert\psi(x)\rvert^2\,dx.$$

For momentum,

$$\langle \hat p\rangle = \int_{-\infty}^{\infty} \psi^*(x) \left( -i\hbar\frac{d}{dx} \right) \psi(x)\,dx.$$

Apply the differential operator to the wavefunction before multiplying by $\psi^*$. Boundary behavior and the operator domain are part of the formula. Integration by parts can demonstrate reality only when its boundary term is justified.

In several dimensions or curvilinear coordinates, use the correct measure:

$$\langle A\rangle = \int \psi^*(q) (A\psi)(q) \,d\mu(q).$$

For spherical coordinates,

$$d\mu = r^2\sin\theta \,dr\,d\theta\,d\phi.$$

Unnormalized Inputs

For a nonzero but unnormalized vector $\lvert\widetilde\psi\rangle$,

$$\langle A\rangle = \frac{ \langle\widetilde\psi\rvert A \lvert\widetilde\psi\rangle }{ \langle\widetilde\psi \vert \widetilde\psi\rangle }.$$

For a positive trace-class operator $\widetilde\rho$ with nonzero trace,

$$\langle A\rangle = \frac{ \operatorname{Tr} (\widetilde\rho A) }{ \operatorname{Tr}\widetilde\rho }.$$

The denominators are essential. They do not repair a vector outside the relevant operator or form domain, nor an operator that fails positivity.

Functions, Moments, and Variance

For a suitable function $f$,

$$\langle f(A)\rangle_\rho = \int_{\mathbb R} f(\lambda) \,d\mu_\rho^A(\lambda).$$

The $n$th moment is

$$\langle A^n\rangle_\rho = \operatorname{Tr}(\rho A^n),$$

when it exists. The variance is

$$(\Delta A)^2 = \langle A^2\rangle -\langle A\rangle^2.$$

The notation $\langle A\rangle^2$ means the square of the first moment. It is not the same as $\langle A^2\rangle$.

Linearity and Products

Expectation is linear:

$$\langle \alpha A+\beta B \rangle = \alpha\langle A\rangle +\beta\langle B\rangle,$$

whenever the terms are defined. The operators $A$ and $B$ need not commute for this identity.

Linearity does not imply factorization:

$$\langle AB\rangle \neq \langle A\rangle \langle B\rangle$$

in general. Products can depend on correlations and operator ordering. If $AB$ is not self-adjoint, its expectation may be complex even when $A$ and $B$ are individually observables.

Composite Systems

For a bipartite state $\rho_{AB}$ and a local observable $A$ on subsystem $A$,

$$\langle A\rangle = \operatorname{Tr}_{AB} \left[ \rho_{AB} (A\otimes I_B) \right].$$

With the reduced state

$$\rho_A = \operatorname{Tr}_B\rho_{AB},$$

the same expectation is

$$\langle A\rangle = \operatorname{Tr}_A(\rho_AA).$$

The identity on the untouched subsystem is often suppressed. Restoring it prevents dimension and tensor-order mistakes.

Time Dependence

For a Schrödinger-picture state evolving under $H(t)$ and an operator $A(t)$,

$$\frac{d}{dt} \langle A\rangle = \frac{i}{\hbar} \langle[H,A]\rangle + \left\langle \frac{\partial A}{\partial t} \right\rangle,$$

under the regularity and domain assumptions required for differentiation. If $A$ has no explicit time dependence and commutes with $H$, its expectation is conserved.

This equation concerns evolution of the mean. It does not replace the expectation formula; it is obtained by differentiating that formula using the equation of motion.

Symbols

Symbol	Mathematical type	Meaning
$A$	Usually self-adjoint operator	Observable
$\rho$	Positive trace-one operator	General quantum state
$\lvert\psi\rangle$	Normalized vector up to phase	Pure state
$P_a$	Spectral projector	Event associated with eigenvalue $a$
$\mu_\rho^A$	Probability measure	Born distribution of $A$ in state $\rho$
$p_A(\lambda)$	Probability density	Density of continuous outcomes
$\langle A\rangle_\rho$	Scalar	Expectation of $A$ in state $\rho$
$\Delta A$	Nonnegative scalar	Standard deviation of $A$

The state subscript is often omitted when context makes it unambiguous.

Units and Dimensions

An expectation value has the same physical dimensions as the observable:

$$[\langle A\rangle]=[A].$$

More generally,

$$[\langle A^n\rangle]=[A]^n, \qquad [\Delta A]=[A].$$

In a continuous formula, $p_A(\lambda)$ has inverse units of $\lambda$, so

$$\lambda p_A(\lambda)\,d\lambda$$

has the units of $A$. The density matrix and normalized state vector are dimensionless abstract objects, although their coordinate components can carry measure-dependent units.

Domain Conditions

For bounded $A$, every Hilbert-space vector has a finite vector expectation. For unbounded self-adjoint $A$, several related conditions must be distinguished:

• The literal vector expression $\langle\psi\rvert A\lvert\psi\rangle$ requires $\psi\in\mathcal D(A)$.
• A finite spectral first moment requires

$$\int_{\mathbb R} \lvert\lambda\rvert \,d\mu_\psi^A(\lambda) <\infty.$$

This allows the expectation to be defined as a quadratic-form value under conditions weaker than $\psi\in\mathcal D(A)$.

• A finite second moment requires

$$\int_{\mathbb R} \lambda^2 \,d\mu_\psi^A(\lambda) <\infty,$$

which is the condition needed for a finite variance and is equivalent to $\psi\in\mathcal D(A)$ for self-adjoint $A$.

• For a density operator and unbounded $A$, a standard sufficient condition for an absolutely defined finite mean is

$$\operatorname{Tr} \left( \rho\lvert A\rvert \right) <\infty.$$

Formal matrix or integral manipulations do not override these existence conditions.

Assumptions

• The state and observable refer to the same Hilbert space.
• The state is normalized, or the displayed normalization denominator is included.
• $A$ is self-adjoint when interpreted as a physical observable.
• The first spectral moment exists.
• Coordinate representations use the correct measure and operator domain.
• Trace expressions involving unbounded operators are defined under appropriate trace and domain conditions.
• Statistical sample-mean interpretations assume repeated comparable preparations and an appropriate independence model.

Validity and Limitations

The expectation formula is exact within standard quantum mechanics. It applies to pure and mixed states and to discrete, continuous, or mixed spectra, provided the relevant moment exists.

For a non-self-adjoint operator $B$,

$$\langle B\rangle = \operatorname{Tr}(\rho B)$$

can still be mathematically useful, but it may be complex and is not directly the mean of a sharp real-valued observable.

The expectation value alone does not determine the full outcome distribution. Different distributions can have the same mean but different variances, tails, or higher moments.

Calculation Checks

• For self-adjoint $A$, the result should be real within numerical tolerance.
• If the spectrum lies in $[a_{\min},a_{\max}]$, then

$$a_{\min} \leq \langle A\rangle \leq a_{\max}.$$

• The dimensions of $\langle A\rangle$ must match those of $A$.
• A basis change applied consistently to the state and operator must leave the result invariant.
• In a normalized eigenstate $\lvert a\rangle$, the expectation should reduce to $a$.
• For $A=I$, the expectation must equal one.
• For a positive operator, the expectation must be nonnegative.
• In a product state, appropriate product-observable expectations should factorize; entangled and classically correlated states need not.

Minimal Worked Use

Let

$$\lvert\psi\rangle = \alpha\lvert0\rangle +\beta\lvert1\rangle, \qquad \lvert\alpha\rvert^2 +\lvert\beta\rvert^2 =1,$$

and let

$$Z = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}.$$

Then

$$\langle Z\rangle = \lvert\alpha\rvert^2 -\lvert\beta\rvert^2.$$

The only measurement outcomes are $+1$ and $-1$. Their Born probabilities are $\lvert\alpha\rvert^2$ and $\lvert\beta\rvert^2$, and the displayed expectation is their probability-weighted average.

Derivation and Canonical Home

Expectation Values owns the statistical meaning, spectral construction, domain distinctions, and repeated-measurement interpretation. Trace Rule for Expectation Values derives the mixed-state and component forms.

Expectation Values in Wave Mechanics owns coordinate-space calculations and boundary checks.

Worked Examples

• Expectation-Value Examples
• Density-Matrix Examples
• Two-State Hamiltonians
• Number States

Common Mistakes

• Treating the expectation value as a guaranteed or most likely outcome.
• Using an ordinary transpose instead of a conjugate transpose.
• Reading probabilities from diagonal density-matrix entries in the wrong basis.
• Computing $\langle A\rangle^2$ when $\langle A^2\rangle$ is required.
• Assuming $\langle AB\rangle=\langle A\rangle\langle B\rangle$.
• Reordering noncommuting factors inside a trace rather than only cycling them.
• Ignoring the integration measure in curvilinear coordinates.
• Dropping boundary terms for differential operators without checking the domain.
• Using the pure-state expression for a mixed state without an ensemble average or trace.
• Suppressing tensor-product identities until subsystem dimensions become ambiguous.
• Assuming normalization guarantees that every unbounded-observable moment exists.

Related Formulas

• Born Rule
• Normalization
• Variance
• Density-Matrix Expectation
• Partial Trace
• Heisenberg Equation
• Canonical Commutation Relations

References

• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994, chs. 4 and 7.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020, ch. 1.
• L. E. Ballentine, Quantum Mechanics: A Modern Development, World Scientific, 1998, chs. 2 and 3.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, rev. ed., Academic Press, 1980, ch. VIII.