Quantum Observable Representation Postulate

Scientific claim

Quantum Observable Representation Postulate

Claim

Let $A$ be an observable physical quantity of a quantum system, such as position, momentum, energy or spin. Quantum mechanics associates $A$ with a self-adjoint linear operator $\hat A$ acting on the system’s complex Hilbert space $\mathcal H$ .

The symbol $\hat A^\dagger$ denotes the adjoint of $\hat A$ . It is defined by the condition

\langle\phi|\hat A\psi\rangle = \langle\hat A^\dagger\phi|\psi\rangle

for all vectors $|\psi\rangle$ in the domain of $\hat A$ and all appropriate vectors $|\phi\rangle$ in the domain of $\hat A^\dagger$ .

The operator representing the observable is self-adjoint, meaning

\hat A=\hat A^\dagger,

including equality of the operators’ domains.

Theoretical: SupportedExperimental: Supported

Assumptions

No assumptions recorded.

Proof

The association between an observable physical quantity $A$ and a self-adjoint operator $\hat A$ is a postulate of quantum mechanics. It is not derived from more fundamental statements within the standard theory.

Let $\mathcal H$ be the complex Hilbert space associated with a quantum system. The operator $\hat A$ representing the observable $A$ is a linear transformation that maps state vectors in its domain to other vectors in the same Hilbert space:

\hat A:D(\hat A)\rightarrow\mathcal H,

where $D(\hat A)\subseteq\mathcal H$ is the domain of $\hat A$ .

Linearity means that for any $|\psi\rangle,|\phi\rangle\in D(\hat A)$ and any $\alpha,\beta\in\mathbb C$ ,

\hat A\left(\alpha|\psi\rangle+\beta|\phi\rangle\right) = \alpha\hat A|\psi\rangle+\beta\hat A|\phi\rangle.

If $\mathcal H$ has finite dimension $n$ and an orthonormal basis

\{|e_1\rangle,\ldots,|e_n\rangle\}

is chosen, then every vector $|\psi\rangle\in\mathcal H$ can be represented by an $n$ -component column vector. In this basis, $\hat A$ is represented by an $n\times n$ square matrix whose entries are

A_{ij}=\langle e_i|\hat A|e_j\rangle.

The matrix is square because $\hat A$ maps vectors from the $n$ -dimensional Hilbert space back into the same $n$ -dimensional Hilbert space.

The symbol $\hat A^\dagger$ denotes the adjoint of $\hat A$ . Abstractly, it is defined by

\langle\phi|\hat A\psi\rangle = \langle\hat A^\dagger\phi|\psi\rangle

for vectors in the appropriate domains.

In a finite-dimensional orthonormal basis, the adjoint is represented by the conjugate transpose of the matrix representing $\hat A$ :

(A^\dagger)_{ij}=A_{ji}^*,

where $A_{ji}^*$ is the complex conjugate of $A_{ji}$ .

The operator is self-adjoint when

\hat A=\hat A^\dagger.

In finite dimensions, this means that its matrix satisfies

A=A^\dagger,

or equivalently,

A_{ij}=A_{ji}^*.

Such a matrix is called a Hermitian matrix.

For example, a spin- $\frac12$ system has Hilbert space $\mathbb C^2$ . An observable acting on this system is therefore represented by a $2\times2$ Hermitian matrix. The spin- $z$ operator is

\hat S_z= \frac{\hbar}{2} \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}.

For an infinite-dimensional Hilbert space, an operator is not generally represented by an ordinary finite square matrix. It may instead appear as a differential operator. For example, the momentum operator in the position representation is

\hat P=-i\hbar\frac{d}{dx}.

It can nevertheless be represented as an infinite matrix after choosing a countable orthonormal basis. In this setting, the operator’s domain must be treated carefully: an unbounded operator is self-adjoint only when both its action and its domain agree with those of its adjoint.

The mathematical suitability of self-adjoint operators can be seen from their expectation values. Let $|\psi\rangle$ be a normalized vector in the domain of $\hat A$ . The expectation value associated with $A$ is

\langle\hat A\rangle_\psi = \langle\psi|\hat A|\psi\rangle.

Taking its complex conjugate gives

\langle\psi|\hat A|\psi\rangle^* = \langle\psi|\hat A^\dagger|\psi\rangle.

Since $\hat A$ is self-adjoint,

\hat A^\dagger=\hat A.

Therefore,

\langle\psi|\hat A|\psi\rangle^* = \langle\psi|\hat A|\psi\rangle,

which means

\langle\hat A\rangle_\psi\in\mathbb R.

Thus, self-adjoint operators are suitable representations of real-valued physical quantities. In finite-dimensional systems they are represented by square Hermitian matrices, while in infinite-dimensional systems they may be represented by differential operators or infinite-dimensional matrices.

These mathematical properties explain the suitability of self-adjoint operators, but they do not prove that physical observables must be represented by them. That association remains a postulate supported by the experimental success of quantum mechanics.

Examples

Position of a particle
Let $X$ be the observable physical quantity describing the position of a particle moving in one dimension. It is represented in the position representation by the self-adjoint position operator $\hat X$ .
Its action on a wavefunction $\psi(x)$ is
$(\hat X\psi)(x)=x\psi(x).$
The operator multiplies the wavefunction’s value at each position $x$ by that position.
Momentum of a particle
Let $P$ be the observable physical quantity describing the momentum of a particle moving in one dimension. In the position representation, it is represented by the momentum operator
$\hat P=-i\hbar\frac{d}{dx},$
where $i$ is the imaginary unit and $\hbar$ is the reduced Planck constant.
Its action on a sufficiently differentiable wavefunction $\psi(x)$ is
$(\hat P\psi)(x) = -i\hbar\frac{d\psi(x)}{dx}.$
The domain and boundary conditions must be specified for $\hat P$ to be self-adjoint.
Energy of a particle
Let $E$ be the observable physical quantity describing the total energy of a nonrelativistic particle of mass $m$ . It is represented by the Hamiltonian operator $\hat H$ .
For a particle moving in a potential $V(x)$ ,
$\hat H = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V(x).$
Its action on a wavefunction is
$(\hat H\psi)(x) = -\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2} + V(x)\psi(x).$
Under appropriate conditions on its domain and on $V(x)$ , the Hamiltonian is self-adjoint.
Spin along the $z$ -axis
Let $S_z$ be the observable physical quantity describing the component of a spin- $\frac12$ particle’s intrinsic angular momentum along the $z$ -axis. It is represented by
$\hat S_z = \frac{\hbar}{2} \begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix}.$
The operator is self-adjoint because its conjugate transpose is equal to itself:
$\hat S_z^\dagger=\hat S_z.$