Operator (physics)

An operator is a function over a space of physical states onto another space of states. The simplest example of the utility of operators is the study of symmetry (which makes the concept of a group useful in this context). Because of this, they are useful tools in classical mechanics. Operators are even more important in quantum mechanics, where they form an intrinsic part of the formulation of the theory.

Operators in classical mechanics

In classical mechanics, the movement of a particle (or system of particles) is completely determined by the Lagrangian Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L(q, \dot{q}, t)} or equivalently the Hamiltonian Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H(q, p, t)} , a function of the generalized coordinates q, generalized velocities Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot{q} = \mathrm{d} q / \mathrm{d} t} and its conjugate momenta:

p={\frac {\partial L}{\partial {\dot {q}}}}

If either L or H is independent of a generalized coordinate q, meaning the L and H do not change when q is changed, which in turn means the dynamics of the particle are still the same even when q changes, the corresponding momenta conjugate to those coordinates will be conserved (this is part of Noether's theorem, and the invariance of motion with respect to the coordinate q is a symmetry). Operators in classical mechanics are related to these symmetries.

More technically, when H is invariant under the action of a certain group of transformations G:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S\in G, H(S(q,p))=H(q,p)} .

The elements of G are physical operators, which map physical states among themselves.

Table of classical mechanics operators

Transformation	Operator	Position	Momentum
Translational symmetry	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X(\mathbf{a})}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{r}\rightarrow \mathbf{r} + \mathbf{a}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{p}\rightarrow \mathbf{p}}
Time translation symmetry	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(t_0)}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{r}(t)\rightarrow \mathbf{r}(t+t_0)}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{p}(t)\rightarrow \mathbf{p}(t+t_0)}
Rotational invariance	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R(\mathbf{\hat{n}},\theta)}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{r}\rightarrow R(\mathbf{\hat{n}},\theta)\mathbf{r}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{p}\rightarrow R(\mathbf{\hat{n}},\theta)\mathbf{p}}
Galilean transformations	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G(\mathbf{v})}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{r}\rightarrow \mathbf{r} + \mathbf{v}t}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{p}\rightarrow \mathbf{p} + m\mathbf{v}}
Parity	$P$	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{r}\rightarrow -\mathbf{r}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{p}\rightarrow -\mathbf{p}}
T-symmetry	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{r}\rightarrow \mathbf{r}(-t)}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{p}\rightarrow -\mathbf{p}(-t)}

where $R({\hat {\boldsymbol {n}}},\theta )$ is the rotation matrix about an axis defined by the unit vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{\boldsymbol{n}}} and angle θ.

Generators

If the transformation is infinitesimal, the operator action should be of the form

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I + \epsilon A, }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} is the identity operator, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon} is a parameter with a small value, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} will depend on the transformation at hand, and is called a generator of the group. Again, as a simple example, we will derive the generator of the space translations on 1D functions.

As it was stated, $T_{a}f(x)=f(x-a)$ . If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a=\epsilon} is infinitesimal, then we may write

T_{\epsilon }f(x)=f(x-\epsilon )\approx f(x)-\epsilon f'(x).

This formula may be rewritten as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_\epsilon f(x) = (I-\epsilon D) f(x)}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D} is the generator of the translation group, which in this case happens to be the derivative operator. Thus, it is said that the generator of translations is the derivative.

The exponential map

The whole group may be recovered, under normal circumstances, from the generators, via the exponential map. In the case of the translations the idea works like this.

The translation for a finite value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} may be obtained by repeated application of the infinitesimal translation:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_a f(x) = \lim_{N\to\infty} T_{a/N} \cdots T_{a/N} f(x)}

with the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cdots} standing for the application Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} times. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} is large, each of the factors may be considered to be infinitesimal:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_a f(x) = \lim_{N\to\infty} \left(I - \frac{a}{N} D\right)^N f(x).}

But this limit may be rewritten as an exponential:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_a f(x) = \exp(-aD) f(x).}

To be convinced of the validity of this formal expression, we may expand the exponential in a power series:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_a f(x) = \left( I - aD + {a^2 D^2 \over 2!} - {a^3 D^3 \over 3!} + \cdots \right) f(x).}

The right-hand side may be rewritten as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) - af'(x) + \frac{a^2}{2!} f''(x) - \frac{a^3}{3!} f^{(3)}(x) + \cdots}

which is just the Taylor expansion of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x-a)} , which was our original value for $T_{a}f(x)$ .

The mathematical properties of physical operators are a topic of great importance in itself. For further information, see C*-algebra and Gelfand–Naimark theorem.

Operators in quantum mechanics

The mathematical formulation of quantum mechanics (QM) is built upon the concept of an operator.

Physical pure states in quantum mechanics are represented as unit-norm vectors (probabilities are normalized to one) in a special complex Hilbert space. Time evolution in this vector space is given by the application of the evolution operator.

Any observable, i.e., any quantity which can be measured in a physical experiment, should be associated with a self-adjoint linear operator. The operators must yield real eigenvalues, since they are values which may come up as the result of the experiment. Mathematically this means the operators must be Hermitian.^[1] The probability of each eigenvalue is related to the projection of the physical state on the subspace related to that eigenvalue. See below for mathematical details about Hermitian operators.

In the wave mechanics formulation of QM, the wavefunction varies with space and time, or equivalently momentum and time (see position and momentum space for details), so observables are differential operators.

In the matrix mechanics formulation, the norm of the physical state should stay fixed, so the evolution operator should be unitary, and the operators can be represented as matrices. Any other symmetry, mapping a physical state into another, should keep this restriction.

Wavefunction

The wavefunction must be square-integrable (see L^p spaces), meaning:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \iiint_{\R^3} |\psi(\mathbf{r})|^2 \, d^3\mathbf{r} = \iiint_{\R^3} \psi(\mathbf{r})^*\psi(\mathbf{r}) \, d^3\mathbf{r} < \infty }

and normalizable, so that:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \iiint_{\R^3} |\psi(\mathbf{r})|^2 \, d^3\mathbf{r} = 1 }

Two cases of eigenstates (and eigenvalues) are:

for discrete eigenstates Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle | \psi_i \rangle } forming a discrete basis, so any state is a sum Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi\rangle = \sum_i c_i|\phi_i\rangle} where c_i are complex numbers such that |c_i|² = c_i^*c_i is the probability of measuring the state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\phi_i\rangle} , and the corresponding set of eigenvalues a_i is also discrete - either finite or countably infinite. In this case, the inner product of two eigenstates is given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle \phi_i \vert \phi_j\rangle=\delta_{ij}} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta_{mn}} denotes the Kronecker Delta. However,
for a continuum of eigenstates Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle | \psi_i \rangle } forming a continuous basis, any state is an integral Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi\rangle = \int c(\phi) \, d\phi|\phi\rangle } where c(φ) is a complex function such that |c(φ)|² = c(φ)^*c(φ) is the probability of measuring the state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\phi\rangle} , and there is an uncountably infinite set of eigenvalues a. In this case, the inner product of two eigenstates is defined as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle \phi' \vert \phi\rangle=\delta(\phi - \phi')} , where here Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta(x-y)} denotes the Dirac Delta.

Linear operators in wave mechanics

Let $ψ$ be the wavefunction for a quantum system, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{A}} be any linear operator for some observable $A$ (such as position, momentum, energy, angular momentum etc.). If $ψ$ is an eigenfunction of the operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{A}} , then

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{A} \psi = a \psi ,}

where $a$ is the eigenvalue of the operator, corresponding to the measured value of the observable, i.e. observable $A$ has a measured value $a$ .

If $ψ$ is an eigenfunction of a given operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{A}} , then a definite quantity (the eigenvalue $a$ ) will be observed if a measurement of the observable $A$ is made on the state $ψ$ . Conversely, if $ψ$ is not an eigenfunction of ${\hat {A}}$ , then it has no eigenvalue for ${\hat {A}}$ , and the observable does not have a single definite value in that case. Instead, measurements of the observable $A$ will yield each eigenvalue with a certain probability (related to the decomposition of $ψ$ relative to the orthonormal eigenbasis of ${\hat {A}}$ ).

In bra–ket notation the above can be written;

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \hat{A} \psi &= \hat{A} \psi ( \mathbf{r} ) = \hat{A} \left\langle \mathbf{r} \mid \psi \right\rangle = \left\langle \mathbf{r} \left\vert \hat {A} \right\vert \psi \right\rangle \\ a \psi &= a \psi ( \mathbf{r} ) = a \left\langle \mathbf{r} \mid \psi \right\rangle = \left\langle \mathbf{r} \mid a \mid \psi \right\rangle \\ \end{align} }

that are equal if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left| \psi \right\rangle } is an eigenvector, or eigenket of the observable $A$ .

Due to linearity, vectors can be defined in any number of dimensions, as each component of the vector acts on the function separately. One mathematical example is the del operator, which is itself a vector (useful in momentum-related quantum operators, in the table below).

An operator in n-dimensional space can be written:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{\hat{A}} = \sum_{j=1}^n \mathbf{e}_j \hat{A}_j }

where e_j are basis vectors corresponding to each component operator A_j. Each component will yield a corresponding eigenvalue Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_j} . Acting this on the wave function $ψ$ :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{\hat{A}} \psi = \left( \sum_{j=1}^n \mathbf{e}_j \hat{A}_j \right) \psi = \sum_{j=1}^n \left( \mathbf{e}_j \hat{A}_j \psi \right) = \sum_{j=1}^n \left( \mathbf{e}_j a_j \psi \right) }

in which we have used Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{A}_j \psi = a_j \psi .}

In bra–ket notation:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \mathbf{\hat{A}} \psi = \mathbf{\hat{A}} \psi ( \mathbf{r} ) = \mathbf{\hat{A}} \left\langle \mathbf{r} \mid \psi \right\rangle &= \left\langle \mathbf{r} \left\vert \mathbf{\hat{A}} \right\vert \psi \right\rangle \\ \left ( \sum_{j=1}^n \mathbf{e}_j \hat{A}_j \right ) \psi = \left( \sum_{j=1}^n \mathbf{e}_j \hat{A}_j \right) \psi ( \mathbf{r} ) = \left( \sum_{j=1}^n \mathbf{e}_j \hat{A}_j \right) \left\langle \mathbf{r} \mid \psi \right\rangle &= \left\langle \mathbf{r} \left\vert \sum_{j=1}^n \mathbf{e}_j \hat{A}_j \right\vert \psi \right\rangle \end{align}}

Commutation of operators on Ψ

If two observables A and B have linear operators Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{A} } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{B} } , the commutator is defined by,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[ \hat{A}, \hat{B} \right] = \hat{A} \hat{B} - \hat{B} \hat{A} }

The commutator is itself a (composite) operator. Acting the commutator on ψ gives:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[ \hat{A}, \hat{B} \right] \psi = \hat{A} \hat{B} \psi - \hat{B} \hat{A} \psi . }

If ψ is an eigenfunction with eigenvalues a and b for observables A and B respectively, and if the operators commute:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[ \hat{A}, \hat{B} \right] \psi = 0, }

then the observables A and B can be measured simultaneously with infinite precision, i.e., uncertainties Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta A = 0 } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta B = 0 } simultaneously. ψ is then said to be the simultaneous eigenfunction of A and B. To illustrate this:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \left[ \hat{A}, \hat{B} \right] \psi &= \hat{A} \hat{B} \psi - \hat{B} \hat{A} \psi \\ & = a(b \psi) - b(a \psi) \\ & = 0 . \\ \end{align} }

It shows that measurement of A and B does not cause any shift of state, i.e., initial and final states are same (no disturbance due to measurement). Suppose we measure A to get value a. We then measure B to get the value b. We measure A again. We still get the same value a. Clearly the state (ψ) of the system is not destroyed and so we are able to measure A and B simultaneously with infinite precision.

If the operators do not commute:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[ \hat{A}, \hat{B} \right] \psi \neq 0, }

they cannot be prepared simultaneously to arbitrary precision, and there is an uncertainty relation between the observables

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta A \Delta B \geq \left|\frac{1}{2}\langle[A, B]\rangle\right|}

even if ψ is an eigenfunction the above relation holds. Notable pairs are position-and-momentum and energy-and-time uncertainty relations, and the angular momenta (spin, orbital and total) about any two orthogonal axes (such as L_x and L_y, or s_y and s_z, etc.).^[2]

Expectation values of operators on Ψ

The expectation value (equivalently the average or mean value) is the average measurement of an observable, for particle in region R. The expectation value Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle \hat{A} \right\rangle } of the operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{A} } is calculated from:^[3]

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle \hat{A} \right\rangle = \int_R \psi^{*}\left( \mathbf{r} \right) \hat{A} \psi \left( \mathbf{r} \right) \mathrm{d}^3\mathbf{r} = \left\langle \psi \left| \hat{A} \right| \psi \right\rangle .}

This can be generalized to any function F of an operator:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle F \left( \hat{A} \right) \right\rangle = \int_R \psi(\mathbf{r})^{*} \left[ F \left( \hat{A} \right) \psi(\mathbf{r}) \right] \mathrm{d}^3 \mathbf{r} = \left\langle \psi \left| F \left( \hat{A} \right) \right| \psi \right\rangle , }

An example of F is the 2-fold action of A on ψ, i.e. squaring an operator or doing it twice:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} F\left(\hat{A}\right) &= \hat{A}^2 \\ \Rightarrow \left\langle \hat{A}^2 \right\rangle &= \int_R \psi^{*} \left( \mathbf{r} \right) \hat{A}^2 \psi \left( \mathbf{r} \right) \mathrm{d}^3\mathbf{r} = \left\langle \psi \left\vert \hat{A}^2 \right\vert \psi \right\rangle \\ \end{align}\,\!}

Hermitian operators

The definition of a Hermitian operator is:^[1]

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{A} = \hat{A}^\dagger}

Following from this, in bra–ket notation:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle \phi_i \left| \hat{A} \right| \phi_j \right\rangle = \left\langle \phi_j \left| \hat{A} \right| \phi_i \right\rangle^*.}

Important properties of Hermitian operators include:

real eigenvalues,
eigenvectors with different eigenvalues are orthogonal,
eigenvectors can be chosen to be a complete orthonormal basis,

Operators in matrix mechanics

An operator can be written in matrix form to map one basis vector to another. Since the operators are linear, the matrix is a linear transformation (aka transition matrix) between bases. Each basis element Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_j } can be connected to another,^[3] by the expression:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_{ij} = \left\langle \phi_i \left| \hat{A} \right| \phi_j \right\rangle,}

which is a matrix element:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{A} = \begin{pmatrix} A_{11} & A_{12} & \cdots & A_{1n} \\ A_{21} & A_{22} & \cdots & A_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ A_{n1} & A_{n2} & \cdots & A_{nn} \\ \end{pmatrix} }

A further property of a Hermitian operator is that eigenfunctions corresponding to different eigenvalues are orthogonal.^[1] In matrix form, operators allow real eigenvalues to be found, corresponding to measurements. Orthogonality allows a suitable basis set of vectors to represent the state of the quantum system. The eigenvalues of the operator are also evaluated in the same way as for the square matrix, by solving the characteristic polynomial:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \det\left( \hat{A} - a \hat{I} \right) = 0 ,}

where I is the n × n identity matrix, as an operator it corresponds to the identity operator. For a discrete basis:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{I} = \sum_i |\phi_i\rangle\langle\phi_i|}

while for a continuous basis:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{I} = \int |\phi\rangle\langle\phi| \mathrm{d}\phi}

Inverse of an operator

A non-singular operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{A}} has an inverse Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{A}^{-1} } defined by:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{A}\hat{A}^{-1} = \hat{A}^{-1}\hat{A} = \hat{I} }

If an operator has no inverse, it is a singular operator. In a finite-dimensional space, an operator is non-singular if and only if its determinant is nonzero:

\det \left({\hat {A}}\right)\neq 0

and hence the determinant is zero for a singular operator.

Table of QM operators

The operators used in quantum mechanics are collected in the table below (see for example^[1]^[4]). The bold-face vectors with circumflexes are not unit vectors, they are 3-vector operators; all three spatial components taken together.

Operator (common name/s)	Cartesian component	General definition	SI unit	Dimension
Position	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \hat{x} &= x, & \hat{y} &= y, & \hat{z} &= z \end{align}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{\hat{r}} = \mathbf{r} \,\!}	m	[L]
Momentum	General Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \hat{p}_x & = -i \hbar \frac{\partial}{\partial x}, & \hat{p}_y & = -i \hbar \frac{\partial}{\partial y}, & \hat{p}_z & = -i \hbar \frac{\partial}{\partial z} \end{align}}	General Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{\hat{p}} = -i \hbar \nabla \,\!}	J s m⁻¹ = N s	[M] [L] [T]⁻¹
Momentum	Electromagnetic field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \hat{p}_x = -i \hbar \frac{\partial}{\partial x} - qA_x \\ \hat{p}_y = -i \hbar \frac{\partial}{\partial y} - qA_y \\ \hat{p}_z = -i \hbar \frac{\partial}{\partial z} - qA_z \end{align}}	Electromagnetic field (uses kinetic momentum; A, vector potential) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \mathbf{\hat{p}} & = \mathbf{\hat{P}} - q\mathbf{A} \\ & = -i \hbar \nabla - q\mathbf{A} \\ \end{align}\,\!}	J s m⁻¹ = N s	[M] [L] [T]⁻¹
Kinetic energy	Translation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \hat{T}_x & = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} \\[2pt] \hat{T}_y & = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial y^2} \\[2pt] \hat{T}_z & = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial z^2} \\ \end{align} }	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \hat{T} & = \frac{1}{2m}\mathbf{\hat{p}}\cdot\mathbf{\hat{p}} \\ & = \frac{1}{2m}(-i \hbar \nabla)\cdot(-i \hbar \nabla) \\ & = \frac{-\hbar^2 }{2m}\nabla^2 \end{align}\,\!}	J	[M] [L]² [T]⁻²
	Electromagnetic field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \hat{T}_x & = \frac{1}{2m}\left(-i \hbar \frac{\partial}{\partial x} - q A_x \right)^2 \\ \hat{T}_y & = \frac{1}{2m}\left(-i \hbar \frac{\partial}{\partial y} - q A_y \right)^2 \\ \hat{T}_z & = \frac{1}{2m}\left(-i \hbar \frac{\partial}{\partial z} - q A_z \right)^2 \end{align}\,\!}	Electromagnetic field (A, vector potential) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \hat{T} & = \frac{1}{2m}\mathbf{\hat{p}}\cdot\mathbf{\hat{p}} \\ & = \frac{1}{2m}(-i \hbar \nabla - q\mathbf{A})\cdot(-i \hbar \nabla - q\mathbf{A}) \\ & = \frac{1}{2m}(-i \hbar \nabla - q\mathbf{A})^2 \end{align}\,\!}	J	[M] [L]² [T]⁻²
	Rotation (I, moment of inertia) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \hat{T}_{xx} & = \frac{\hat{J}_x^2}{2I_{xx}} \\ \hat{T}_{yy} & = \frac{\hat{J}_y^2}{2I_{yy}} \\ \hat{T}_{zz} & = \frac{\hat{J}_z^2}{2I_{zz}} \\ \end{align}\,\!}	Rotation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{T} = \frac{\mathbf{\hat{J}}\cdot\mathbf{\hat{J}}}{2I} \,\!} ^{[citation needed]}	J	[M] [L]² [T]⁻²
Potential energy	N/A	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{V} = V\left( \mathbf{r}, t \right) = V \,\!}	J	[M] [L]² [T]⁻²
Total energy	N/A	Time-dependent potential: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{E} = i \hbar \frac{\partial}{\partial t} \,\!} Time-independent: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{E} = E \,\!}	J	[M] [L]² [T]⁻²
Hamiltonian		Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \hat{H} & = \hat{T} + \hat{V} \\ & = \frac{1}{2m}\mathbf{\hat{p}}\cdot\mathbf{\hat{p}} + V \\ & = \frac{1}{2m}\hat{p}^2 + V \\ \end{align} \,\!}	J	[M] [L]² [T]⁻²
Angular momentum operator	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \hat{L}_x & = -i\hbar \left(y {\partial \over \partial z} - z {\partial \over \partial y}\right) \\ \hat{L}_y & = -i\hbar \left(z {\partial \over \partial x} - x {\partial \over \partial z}\right) \\ \hat{L}_z & = -i\hbar \left(x {\partial \over \partial y} - y {\partial \over \partial x}\right) \end{align}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{\hat{L}} = \mathbf{r} \times -i\hbar \nabla }	J s = N s m	[M] [L]² [T]⁻¹
Spin angular momentum	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \hat{S}_x &= {\hbar \over 2} \sigma_x & \hat{S}_y &= {\hbar \over 2} \sigma_y & \hat{S}_z &= {\hbar \over 2} \sigma_z \end{align}} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \sigma_x &= \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \\ \sigma_y &= \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} \\ \sigma_z &= \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \end{align}} are the Pauli matrices for spin-1/2 particles.	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{\hat{S}} = {\hbar \over 2} \boldsymbol{\sigma} \,\!} where σ is the vector whose components are the Pauli matrices.	J s = N s m	[M] [L]² [T]⁻¹
Total angular momentum	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \hat{J}_x & = \hat{L}_x + \hat{S}_x \\ \hat{J}_y & = \hat{L}_y + \hat{S}_y \\ \hat{J}_z & = \hat{L}_z + \hat{S}_z \end{align}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \mathbf{\hat{J}} & = \mathbf{\hat{L}} + \mathbf{\hat{S}} \\ & = -i\hbar \mathbf{r}\times\nabla + \frac{\hbar}{2}\boldsymbol{\sigma} \end{align}}	J s = N s m	[M] [L]² [T]⁻¹
Transition dipole moment (electric)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \hat{d}_x & = q\hat{x}, & \hat{d}_y & = q\hat{y}, & \hat{d}_z & = q\hat{z} \end{align}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{\hat{d}} = q \mathbf{\hat{r}} }	C m	[I] [T] [L]

Examples of applying quantum operators

The procedure for extracting information from a wave function is as follows. Consider the momentum p of a particle as an example. The momentum operator in position basis in one dimension is:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{p} = -i\hbar\frac{\partial }{\partial x}}

Letting this act on ψ we obtain:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{p} \psi = -i\hbar\frac{\partial }{\partial x} \psi ,}

if ψ is an eigenfunction of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{p}} , then the momentum eigenvalue p is the value of the particle's momentum, found by:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -i\hbar\frac{\partial }{\partial x} \psi = p \psi.}

For three dimensions the momentum operator uses the nabla operator to become:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{\hat{p}} = -i\hbar\nabla .}

In Cartesian coordinates (using the standard Cartesian basis vectors e_x, e_y, e_z) this can be written;

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{e}_\mathrm{x}\hat{p}_x + \mathbf{e}_\mathrm{y}\hat{p}_y + \mathbf{e}_\mathrm{z}\hat{p}_z = -i\hbar\left ( \mathbf{e}_\mathrm{x} \frac{\partial }{\partial x} + \mathbf{e}_\mathrm{y} \frac{\partial }{\partial y} + \mathbf{e}_\mathrm{z} \frac{\partial }{\partial z} \right ),}

that is:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{p}_x = -i\hbar \frac{\partial}{\partial x}, \quad \hat{p}_y = -i\hbar \frac{\partial}{\partial y} , \quad \hat{p}_z = -i\hbar \frac{\partial}{\partial z} \,\!}

The process of finding eigenvalues is the same. Since this is a vector and operator equation, if ψ is an eigenfunction, then each component of the momentum operator will have an eigenvalue corresponding to that component of momentum. Acting Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{\hat{p}} } on ψ obtains:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \hat{p}_x \psi & = -i\hbar \frac{\partial}{\partial x} \psi = p_x \psi \\ \hat{p}_y \psi & = -i\hbar \frac{\partial}{\partial y} \psi = p_y \psi \\ \hat{p}_z \psi & = -i\hbar \frac{\partial}{\partial z} \psi = p_z \psi \\ \end{align} \,\!}

References

^ ^1.0 ^1.1 ^1.2 ^1.3 Molecular Quantum Mechanics Parts I and II: An Introduction to Quantum Chemistry (Volume 1), P.W. Atkins, Oxford University Press, 1977, ISBN 0-19-855129-0
^ Ballentine, L. E. (1970), "The Statistical Interpretation of Quantum Mechanics", Reviews of Modern Physics, 42 (4): 358–381, Bibcode:1970RvMP...42..358B, doi:10.1103/RevModPhys.42.358
^ ^3.0 ^3.1 Quantum Mechanics Demystified, D. McMahon, Mc Graw Hill (USA), 2006, ISBN 0-07-145546-9
^ Operators - The Feynman Lectures on Physics

de:Operator (Mathematik)#Operatoren der Physik

[QUANTUM_CHEMISRTY_1977-1] 1.0 ^1.1 ^1.2 ^1.3 Molecular Quantum Mechanics Parts I and II: An Introduction to Quantum Chemistry (Volume 1), P.W. Atkins, Oxford University Press, 1977, ISBN 0-19-855129-0

[Ballentine1970-2] Ballentine, L. E. (1970), "The Statistical Interpretation of Quantum Mechanics", Reviews of Modern Physics, 42 (4): 358–381, Bibcode:1970RvMP...42..358B, doi:10.1103/RevModPhys.42.358

[Quantum_Mechanics_Demystified_2006-3] 3.0 ^3.1 Quantum Mechanics Demystified, D. McMahon, Mc Graw Hill (USA), 2006, ISBN 0-07-145546-9

[4] Operators - The Feynman Lectures on Physics

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[2]

[3]

[4]