# Operator (physics)
In physics, an operator is a mathematical object that acts on the state of a system to produce another state or a number; in quantum mechanics, every measurable physical quantity, or observable, is represented by a linear operator acting on the wavefunction or state vector[1](https://www.damtp.cam.ac.uk/user/tong/qm/qmhtml/S3.html). Operators are written with a hat, such as \( \hat{x} \) for position or \( \hat{p} \) for momentum, although the hat convention is often dropped in later courses[1](https://www.damtp.cam.ac.uk/user/tong/qm/qmhtml/S3.html).
| Key fact | Detail |
|---|---|
| Definition | A linear operator \( \widehat{O} \) maps a state to another state, obeying \( \widehat{O}[\alpha\psi_1+\psi_2] = \alpha\widehat{O}\psi_1 + \widehat{O}\psi_2 \); on an N-dimensional basis it is represented by an N×N matrix[1](https://www.damtp.cam.ac.uk/user/tong/qm/qmhtml/S3.html) • [2](https://ocw.mit.edu/courses/22-51-quantum-theory-of-radiation-interactions-fall-2012/774a0d883fca9c660912cc029e58669e_MIT22_51F12_Ch2.pdf) |
| Observables | Each observable is a Hermitian (self-adjoint) operator; its eigenvalues are real and are the only possible measurement outcomes[1](https://www.damtp.cam.ac.uk/user/tong/qm/qmhtml/S3.html) |
| Canonical commutation relation | \( [\hat{x},\hat{p}] = i\hbar \); in three dimensions \( [\hat{r}_a,\hat{p}_b] = i\hbar\,\delta_{a,b} \), so different components commute[3](https://ocw.mit.edu/courses/22-02-introduction-to-applied-nuclear-physics-spring-2012/0456d26b0767e6aab3bace1e6f86d78b_MIT22_02S12_lec_ch2.pdf) |
| Consequence of non-commutation | Operators that do not commute correspond to incompatible observables, which cannot have simultaneously definite values; commuting Hermitian operators share a common set of eigenvectors[4](https://paulskrzypczyk.github.io/qm-lecture-notes/comm-up) |
| Time evolution | The state evolves as \( |\psi(t)\rangle = \widehat{U}(t,t_0)|\psi(t_0)\rangle \) with a unitary operator satisfying \( i\hbar\, d\widehat{U}/dt = \widehat{H}\widehat{U} \); for time-independent \( \widehat{H} \), \( \widehat{U} = \exp[-i(t-t_0)\widehat{H}/\hbar] \)[5](https://www.lancaster.ac.uk/staff/schomeru/lecturenotes/Quantum%20Information%20Processing/S1.html) |
| Classical analogy | Commutators correspond to \( i\hbar \) times Poisson brackets: \( [x,p] = i\hbar \) while \( \{x,p\}_{pb} = 1 \)[6](https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/79a8091ef4f18d8e3ff76a097e8db33c_MIT8_05F13_Chap_06.pdf) |
## Operators and observables
In classical mechanics the state of a particle is its position \( \mathbf{x} \) and momentum \( \mathbf{p} \), and any observable is a function \( f(\mathbf{x},\mathbf{p}) \), such as angular momentum \( \mathbf{L} = \mathbf{x}\times\mathbf{p} \) or energy \( E = \mathbf{p}^2/2m + V(\mathbf{x}) \). In quantum mechanics the state is a wavefunction, viewed abstractly as a vector in a [Hilbert space](https://www.edgepedia.org/hilbert-space), and all observables are represented by linear operators on that space[1](https://www.damtp.cam.ac.uk/user/tong/qm/qmhtml/S3.html).
The connection to measurement runs through eigenvalues. Given an operator \( \widehat{O} \), its eigenvalue equation is \( \widehat{O}\psi = \lambda\psi \), and the collection of all eigenvalues is called the spectrum. The outcome of any measurement of \( \widehat{O} \) lies in the spectrum of \( \widehat{O} \); the time-independent [Schrödinger equation](https://www.edgepedia.org/schrodinger-equation) is simply the eigenvalue equation for the Hamiltonian, whose spectrum gives the possible energy levels[1](https://www.damtp.cam.ac.uk/user/tong/qm/qmhtml/S3.html).
**Why Hermitian.** Not every linear operator qualifies as a physical observable. The adjoint \( \widehat{O}^{\dagger} \) is defined by the inner-product relation \( \langle\psi|\widehat{O}\varphi\rangle = \langle\widehat{O}^{\dagger}\psi|\varphi\rangle \), and an operator is Hermitian, or self-adjoint, if \( \widehat{O} = \widehat{O}^{\dagger} \)[1](https://www.damtp.cam.ac.uk/user/tong/qm/qmhtml/S3.html). Hermitian operators have real eigenvalues, and eigenfunctions belonging to distinct eigenvalues are orthogonal; on the operator's domain the eigenvectors form a complete orthogonal basis[2](https://ocw.mit.edu/courses/22-51-quantum-theory-of-radiation-interactions-fall-2012/774a0d883fca9c660912cc029e58669e_MIT22_51F12_Ch2.pdf). Real eigenvalues matter because laboratory instruments measure real numbers, never complex ones[1](https://www.damtp.cam.ac.uk/user/tong/qm/qmhtml/S3.html).
The position operator is trivially Hermitian, since multiplication by \( \mathbf{x} \) passes through the complex conjugate unchanged. The momentum operator \( \widehat{\mathbf{p}} = -i\hbar\nabla \) is Hermitian by an integration by parts: the two minus signs, one from integration by parts and one from conjugating the factor of \( i \), cancel. This is one way to see why the momentum operator necessarily carries a factor of \( i \)[1](https://www.damtp.cam.ac.uk/user/tong/qm/qmhtml/S3.html). A fine point of terminology: "Hermitian" and "self-adjoint" are synonymous for finite-dimensional Hilbert spaces, but in infinite-dimensional spaces of functions they differ subtly through the domain on which the operator acts[1](https://www.damtp.cam.ac.uk/user/tong/qm/qmhtml/S3.html) • [7](https://www.cambridge.org/core/books/abs/path-integrals-and-hamiltonians/operators/A84B99E6BA1AF0DE3B8870DC38B265C4).
## Commutators and incompatible observables
Operators do not in general commute: applying \( \hat{X} \) then \( \hat{P} \) to a wavefunction gives a different result from applying \( \hat{P} \) then \( \hat{X} \). The commutator \( [\hat{X},\hat{P}] = \hat{X}\hat{P} - \hat{P}\hat{X} \) measures this difference. Acting on a wavefunction \( \Psi(x) \), with \( \hat{P} = -i\hbar\,\partial/\partial x \), the product rule gives \( [\hat{X},\hat{P}]\Psi(x) = i\hbar\,\Psi(x) \), and since this holds for all wavefunctions it is written, as an abuse of notation, \( [\hat{X},\hat{P}] = i\hbar \)[4](https://paulskrzypczyk.github.io/qm-lecture-notes/comm-up). This is the canonical commutation relation, and it shows that the operators for the two fundamental properties of a particle, position and momentum, do not commute[4](https://paulskrzypczyk.github.io/qm-lecture-notes/comm-up). The same result follows by applying the operators directly to \( \psi(x) \) and using the product rule on \( x\psi(x) \)[3](https://ocw.mit.edu/courses/22-02-introduction-to-applied-nuclear-physics-spring-2012/0456d26b0767e6aab3bace1e6f86d78b_MIT22_02S12_lec_ch2.pdf).
In three dimensions the relation generalizes to \( [\hat{r}_a, \hat{p}_b] = i\hbar\,\delta_{a,b} \), where \( \delta_{a,b} \) is the [Kronecker delta](https://www.edgepedia.org/kronecker-delta): vector components in different directions commute, with commutator zero[3](https://ocw.mit.edu/courses/22-02-introduction-to-applied-nuclear-physics-spring-2012/0456d26b0767e6aab3bace1e6f86d78b_MIT22_02S12_lec_ch2.pdf).
**Commutation and measurement.** Non-commutation is a signature of incompatibility. When two quantities' operators fail to commute, a particle cannot simultaneously possess definite values of both, so the quantities cannot be measured at the same time; this is the basis of Heisenberg's uncertainty principle, which says in rough terms that greater precision in a particle's position implies greater imprecision in its momentum, and the other way around. Conversely, if two operators commute, a particle can have definite values of both quantities at once[4](https://paulskrzypczyk.github.io/qm-lecture-notes/comm-up). The formal statement is a theorem: if two Hermitian operators commute, they share a common set of eigenvectors[2](https://ocw.mit.edu/courses/22-51-quantum-theory-of-radiation-interactions-fall-2012/774a0d883fca9c660912cc029e58669e_MIT22_51F12_Ch2.pdf). A worked example is the free particle, whose Hamiltonian \( H = \hat{p}^2/2m \) commutes with \( \hat{p} \); the momentum eigenfunctions \( e^{ikx} \) are also energy eigenfunctions with \( E = \hbar^2 k^2/2m \)[3](https://ocw.mit.edu/courses/22-02-introduction-to-applied-nuclear-physics-spring-2012/0456d26b0767e6aab3bace1e6f86d78b_MIT22_02S12_lec_ch2.pdf).
Commutator algebra follows familiar rules: any operator commutes with scalars and with any function of itself, \( [A, f(A)] = 0 \); products obey \( [A, BC] = [A,B]C + B[A,C] \). From these, \( [x, p^2] = 2i\hbar p \)[3](https://ocw.mit.edu/courses/22-02-introduction-to-applied-nuclear-physics-spring-2012/0456d26b0767e6aab3bace1e6f86d78b_MIT22_02S12_lec_ch2.pdf).
## Time evolution and unitary operators
A second important class of operators governs dynamics. The time evolution of quantum states is governed by the Schrödinger equation \( i\hbar\, d|\psi(t)\rangle/dt = \widehat{H}(t)|\psi(t)\rangle \), where \( \widehat{H} \) is a Hermitian operator called the Hamiltonian. The general solution is \( |\psi(t)\rangle = \widehat{U}(t,t_0)|\psi(t_0)\rangle \), where \( \widehat{U} \) is a unitary operator, meaning \( \widehat{U}^{\dagger}\widehat{U} = \mathbb{I} \), which preserves inner products and hence probabilities. The evolution operator itself satisfies \( i\hbar\, d\widehat{U}/dt = \widehat{H}\widehat{U} \) with \( \widehat{U}(t_0,t_0) = \mathbb{I} \)[5](https://www.lancaster.ac.uk/staff/schomeru/lecturenotes/Quantum%20Information%20Processing/S1.html).
When the Hamiltonian is time independent, the equation is solved by an operator exponential,
\[ \widehat{U}(t,t_0) = \exp\left[-i(t-t_0)\widehat{H}/\hbar\right], \]
where the exponential of an operator is defined by its power series \( \exp\widehat{A} = \sum_{k=0}^{\infty} \widehat{A}^k/k! \). Using the eigenrepresentation \( \widehat{H} = \sum_n E_n |n\rangle\langle n| \), this becomes \( \widehat{U} = \sum_n \exp[-i(t-t_0)E_n/\hbar]\,|n\rangle\langle n| \)[5](https://www.lancaster.ac.uk/staff/schomeru/lecturenotes/Quantum%20Information%20Processing/S1.html). Unitarity follows because \( \widehat{H} \) is Hermitian[8](https://etneil.github.io/grad_qm_lec_notes/time_evo.html).
**Pictures.** The same evolution can be formulated in equivalent ways. In the usual [Schrödinger picture](https://www.edgepedia.org/schrodinger-picture) the states evolve and the operators stay constant; in the [Heisenberg picture](https://www.edgepedia.org/heisenberg-picture) the wavefunctions are fixed and the operators carry the time development, governed by their commutator with the Hamiltonian[9](https://quantummechanics.ucsd.edu/ph130a/130_notes/node191.html). A third formulation expresses evolution entirely through the unitary propagator \( \widehat{U}(t) \)[10](https://ocw.mit.edu/courses/22-02-introduction-to-applied-nuclear-physics-spring-2012/b5106a499ae03e36b5a2e002355668f9_MIT22_02S12_lec_ch6.pdf).
## Relation to classical mechanics
The operator formalism mirrors the Hamiltonian formulation of classical mechanics. In classical mechanics the [Poisson bracket](https://www.edgepedia.org/poisson-bracket) \( \{A,B\}_{pb} \) gives the time derivative of any observable function of \( x \) and \( p \); the quantum commutator plays the same role, and the correspondence is \( [A,B] \Leftrightarrow i\hbar\{A,B\}_{pb} \). The check case is \( [x,p] = i\hbar \) against \( \{x,p\}_{pb} = 1 \)[6](https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/79a8091ef4f18d8e3ff76a097e8db33c_MIT8_05F13_Chap_06.pdf). This analogy justifies calling \( H \) in the Schrödinger equation the Hamiltonian, although any Hermitian operator with units of energy can serve as a Hamiltonian regardless of any connection to a classical theory[6](https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/79a8091ef4f18d8e3ff76a097e8db33c_MIT8_05F13_Chap_06.pdf).
## References
1. [3 The Formalism of Quantum Mechanics, Quantum Mechanics by David Tong, University of Cambridge](https://www.damtp.cam.ac.uk/user/tong/qm/qmhtml/S3.html)
2. [22.51 Course Notes, Chapter 2: Mathematical Formalism of Quantum Mechanics, MIT OpenCourseWare](https://ocw.mit.edu/courses/22-51-quantum-theory-of-radiation-interactions-fall-2012/774a0d883fca9c660912cc029e58669e_MIT22_51F12_Ch2.pdf)
3. [Lecture notes, Chapter 2: Introduction to Quantum Mechanics, MIT 22.02, MIT OpenCourseWare](https://ocw.mit.edu/courses/22-02-introduction-to-applied-nuclear-physics-spring-2012/0456d26b0767e6aab3bace1e6f86d78b_MIT22_02S12_lec_ch2.pdf)
4. [Chapter 7: Commutation and the Uncertainty Principle, Quantum Mechanics I lecture notes, Paul Skrzypczyk](https://paulskrzypczyk.github.io/qm-lecture-notes/comm-up)
5. [I Quantum mechanics, PHYS483: Quantum information processing lecture notes, Lancaster University](https://www.lancaster.ac.uk/staff/schomeru/lecturenotes/Quantum%20Information%20Processing/S1.html)
6. [Quantum Physics II, Lecture Notes 6, MIT 8.05, MIT OpenCourseWare](https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/79a8091ef4f18d8e3ff76a097e8db33c_MIT8_05F13_Chap_06.pdf)
7. [Operators (Chapter 3), Path Integrals and Hamiltonians, Cambridge University Press](https://www.cambridge.org/core/books/abs/path-integrals-and-hamiltonians/operators/A84B99E6BA1AF0DE3B8870DC38B265C4)
8. [4 Time evolution in quantum mechanics, Graduate Quantum Mechanics Lecture Notes](https://etneil.github.io/grad_qm_lec_notes/time_evo.html)
9. [The Heisenberg Picture, Physics 130a notes, UC San Diego](https://quantummechanics.ucsd.edu/ph130a/130_notes/node191.html)
10. [Lecture notes, Chapter 6: Time Evolution in Quantum Mechanics, MIT 22.02, MIT OpenCourseWare](https://ocw.mit.edu/courses/22-02-introduction-to-applied-nuclear-physics-spring-2012/b5106a499ae03e36b5a2e002355668f9_MIT22_02S12_lec_ch6.pdf)
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*Topic: Encyclopedia › Physical world and mathematics › Physics › Quantum physics › Quantum mechanics › Quantum formalism and states › Operators, observables, and angular momentum › Quantum operators and observables (overview)*
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