# Unitary operator

A **unitary operator** is a linear operator \( U \) on a [Hilbert space](https://www.edgepedia.org/hilbert-space) whose adjoint \( U^{\dagger} \) is also its inverse, so that \( U^{\dagger}U = UU^{\dagger} = I \), where \( I \) is the identity operator<sup>[1](https://interactivetextbooks.tudelft.nl/mqp-v2/operators.html)</sup>. Equivalently, a unitary operator preserves inner products: for any two states \( \ket{\psi} \) and \( \ket{\chi} \), \( \braket{U\psi|U\chi} = \braket{\psi|\chi} \)<sup>[2](https://www.lancaster.ac.uk/staff/schomeru/lecturenotes/Quantum%20Mechanics/S8.html)</sup>. Unitarity is the linear-algebraic expression of preserving lengths and angles in a complex vector space, and it is the property that makes unitary operators central to quantum mechanics, where they represent symmetries, basis changes, and time evolution<sup>[1](https://interactivetextbooks.tudelft.nl/mqp-v2/operators.html)</sup><sup> • </sup><sup>[3](https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/79a8091ef4f18d8e3ff76a097e8db33c_MIT8_05F13_Chap_06.pdf)</sup>.

| Key fact | Detail |
|---|---|
| Definition | \( U^{\dagger}U = UU^{\dagger} = I \); the adjoint is a two-sided inverse<sup>[1](https://interactivetextbooks.tudelft.nl/mqp-v2/operators.html)</sup> |
| Inner products | \( \braket{U\psi|U\chi} = \braket{\psi|\chi} \) for all states; norms and transition probabilities are preserved<sup>[2](https://www.lancaster.ac.uk/staff/schomeru/lecturenotes/Quantum%20Mechanics/S8.html)</sup> |
| Group structure | The identity is unitary, the inverse of a unitary operator is unitary, and products of unitary operators are unitary, so the unitary operators on a Hilbert space form a group<sup>[1](https://interactivetextbooks.tudelft.nl/mqp-v2/operators.html)</sup> |
| Eigenvalues | All eigenvalues \( a_n \) of a unitary operator satisfy \( \lvert a_n \rvert = 1 \)<sup>[2](https://www.lancaster.ac.uk/staff/schomeru/lecturenotes/Quantum%20Mechanics/S8.html)</sup> |
| Exponential form | \( U = \exp(i\tau H) \) with \( H \) Hermitian and \( \tau \) real is unitary, with \( U^{-1} = U^{\dagger} = \exp(-i\tau H) \)<sup>[2](https://www.lancaster.ac.uk/staff/schomeru/lecturenotes/Quantum%20Mechanics/S8.html)</sup> |
| Time evolution | Quantum time evolution is generated by a unitary operator \( U(t,t_0) \); for a time-independent Hamiltonian, \( U(t) = e^{-iHt/\hbar} \)<sup>[3](https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/79a8091ef4f18d8e3ff76a097e8db33c_MIT8_05F13_Chap_06.pdf)</sup><sup> • </sup><sup>[4](https://ocw.mit.edu/courses/22-02-introduction-to-applied-nuclear-physics-spring-2012/b5106a499ae03e36b5a2e002355668f9_MIT22_02S12_lec_ch6.pdf)</sup> |
| Stone's theorem | Every strongly continuous one-parameter unitary group has the form \( U(t) = e^{itA} \) for a unique self-adjoint generator \( A \)<sup>[5](https://bpb-us-e1.wpmucdn.com/sites.psu.edu/dist/1/4020/files/2017/12/analysis-slides-278829v.pdf)</sup> |

## Definition and basic properties

The defining condition \( U^{\dagger}U = UU^{\dagger} = I \) says that the adjoint of \( U \), the operator satisfying \( \braket{\psi|A^{\dagger}\chi} = \braket{A\psi|\chi} \), undoes \( U \) from both sides<sup>[1](https://interactivetextbooks.tudelft.nl/mqp-v2/operators.html)</sup><sup> • </sup><sup>[2](https://www.lancaster.ac.uk/staff/schomeru/lecturenotes/Quantum%20Mechanics/S8.html)</sup>. In an orthonormal basis this becomes a condition on matrix elements: \( \sum_l U_{nl}U^{*}_{ml} = \delta_{nl} \), the statement that the rows of the matrix form an orthonormal set, as do the columns<sup>[2](https://www.lancaster.ac.uk/staff/schomeru/lecturenotes/Quantum%20Mechanics/S8.html)</sup>.

Three consequences follow directly from the definition. The identity operator is unitary. If \( T \) is unitary, its inverse \( T^{-1} \), which equals \( T^{\dagger} \), is also unitary. If \( T \) and \( U \) are unitary, so is the product \( TU \). The collection of all unitary operators on a Hilbert space therefore carries the structure of a group<sup>[1](https://interactivetextbooks.tudelft.nl/mqp-v2/operators.html)</sup>.

Because a unitary operator preserves inner products, it preserves the norm of every vector and the overlap between any pair of vectors. In quantum mechanics the squared overlap \( \lvert\braket{\psi|\chi}\rvert^2 \) is a transition probability, so unitary transformations leave all physical probabilities unchanged<sup>[2](https://www.lancaster.ac.uk/staff/schomeru/lecturenotes/Quantum%20Mechanics/S8.html)</sup><sup> • </sup><sup>[6](https://www.sciencedirect.com/science/article/pii/S0375960108014709)</sup>.

**Spectral properties.** In finite dimensions, Hermitian and unitary operators have orthonormal bases of eigenvectors; in infinite-dimensional Hilbert spaces, they need not<sup>[13](https://math.hodapp.com/lectures/higher/functional-analysis/spectral-theorem.html)</sup>. The two classes differ in their eigenvalues: Hermitian operators have real eigenvalues, while the eigenvalues of a unitary operator all have modulus one<sup>[2](https://www.lancaster.ac.uk/staff/schomeru/lecturenotes/Quantum%20Mechanics/S8.html)</sup>. A unitary operator on a finite-dimensional space can be diagonalized by a unitary transformation, whereas an infinite-dimensional unitary operator need not have an orthonormal basis of eigenvectors; two operators related by a unitary similarity transformation share the same eigenvalues even though their matrices differ<sup>[7](https://www.physics.unlv.edu/~bernard/phy721_99/tex_notes/node9.html)</sup>.

## Relation to Hermitian operators and basis changes

Unitary operators are the complex analogs of orthogonal matrices that rotate a coordinate system. Any change of basis from one orthonormal basis \( \{\ket{n}\} \) to another \( \{\ket{\tilde{n}}\} \) can be written as \( \ket{\tilde{n}} = U\ket{n} \) for a suitable unitary \( U \)<sup>[2](https://www.lancaster.ac.uk/staff/schomeru/lecturenotes/Quantum%20Mechanics/S8.html)</sup>.

A large and important class of unitary operators arises by exponentiating Hermitian operators. For a Hermitian \( H \) and a real constant \( \tau \), the operator \( U = \exp(i\tau H) \), defined by the [Taylor series](https://www.edgepedia.org/taylor-series) \( \exp(A) = \sum_{n=0}^{\infty} A^n/n! \), is unitary, with \( U^{-1} = U^{\dagger} = \exp(-i\tau H) \)<sup>[2](https://www.lancaster.ac.uk/staff/schomeru/lecturenotes/Quantum%20Mechanics/S8.html)</sup>. Conversely, any symmetry transformation of space that leaves the volume element invariant, such as a spatial translation, has a unitary operator representation<sup>[8](https://oer.physics.manchester.ac.uk/AQM2/Notes/Notes-3.4.html)</sup>.

A concrete finite-dimensional example: for any angle \( \phi \), the matrix

\[ \begin{pmatrix} \cos\phi & \sin\phi \\ i\sin\phi & -i\cos\phi \end{pmatrix} \]

is unitary<sup>[1](https://interactivetextbooks.tudelft.nl/mqp-v2/operators.html)</sup>.

## Time evolution in quantum mechanics

The postulate of unitary time evolution states that for any quantum system there is a unitary operator \( U(t,t_0) \) such that the state at time \( t \) is obtained from the state at time \( t_0 \) by \( \ket{\Psi,t} = U(t,t_0)\ket{\Psi,t_0} \)<sup>[3](https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/79a8091ef4f18d8e3ff76a097e8db33c_MIT8_05F13_Chap_06.pdf)</sup>. This operator is unique and does not depend on which state it acts on; a single operator generates the evolution of all possible states<sup>[3](https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/79a8091ef4f18d8e3ff76a097e8db33c_MIT8_05F13_Chap_06.pdf)</sup>.

Unitarity here has a direct physical meaning: it conserves the norm of the state, \( \braket{\Psi,t|\Psi,t} = \braket{\Psi,t_0|\Psi,t_0} \), which is the conservation of total probability<sup>[3](https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/79a8091ef4f18d8e3ff76a097e8db33c_MIT8_05F13_Chap_06.pdf)</sup>. The evolution operator also satisfies composition and inversion rules: \( U(t_3,t_2)U(t_2,t_1) = U(t_3,t_1) \), \( U(t_1,t_1) = I \), and \( U^{\dagger}(t_2,t_1) = U(t_1,t_2) \), so evolution backward in time is given by the adjoint<sup>[9](https://www.thphys.uni-heidelberg.de/~wetterich/Vorlesungen/Foundations%20of%20quantum%20mechanics%202021/Foundations_of_Quantum_Mechanics.pdf)</sup>.

For a time-independent Hamiltonian \( H \), differentiating the evolution equation \( i\hbar\, \partial_t U(t,t_0) = H\,U(t,t_0) \) with \( U(t_0,t_0) = I \) gives the closed form

\[ U(t) = e^{-iHt/\hbar}, \]

which is unitary precisely because \( H \) is Hermitian: \( U^{\dagger}U = e^{iHt/\hbar}e^{-iHt/\hbar} = I \)<sup>[4](https://ocw.mit.edu/courses/22-02-introduction-to-applied-nuclear-physics-spring-2012/b5106a499ae03e36b5a2e002355668f9_MIT22_02S12_lec_ch6.pdf)</sup><sup> • </sup><sup>[9](https://www.thphys.uni-heidelberg.de/~wetterich/Vorlesungen/Foundations%20of%20quantum%20mechanics%202021/Foundations_of_Quantum_Mechanics.pdf)</sup>. This unitary-operator formulation is a third presentation of quantum evolution, equivalent to the Schrödinger and Heisenberg equations<sup>[4](https://ocw.mit.edu/courses/22-02-introduction-to-applied-nuclear-physics-spring-2012/b5106a499ae03e36b5a2e002355668f9_MIT22_02S12_lec_ch6.pdf)</sup>.

## Stone's theorem

A family \( \{U(t)\}_{t \in \mathbb{R}} \) of unitary operators is a **strongly continuous one-parameter unitary group** if \( U(t) \) is a group homomorphism from \( \mathbb{R} \) to the unitary operators and, for each fixed vector \( \xi \), the map \( t \mapsto U(t)\xi \) is continuous<sup>[5](https://bpb-us-e1.wpmucdn.com/sites.psu.edu/dist/1/4020/files/2017/12/analysis-slides-278829v.pdf)</sup>. Stone's theorem states that every such group has the form \( U(t) = e^{itA} \) for a unique self-adjoint operator \( A \), the infinitesimal generator, obtained as the limit

\[ A\xi = \lim_{t \to 0} \frac{U(t)\xi - \xi}{it} \]

on its domain<sup>[5](https://bpb-us-e1.wpmucdn.com/sites.psu.edu/dist/1/4020/files/2017/12/analysis-slides-278829v.pdf)</sup>. The correspondence is one-to-one: self-adjoint operators and strongly continuous one-parameter unitary groups determine each other<sup>[10](https://math-terek.github.io/texts/GQT-companion.pdf)</sup>.

In nonrelativistic quantum mechanics, time evolution is exactly such a one-parameter unitary group, and its infinitesimal generator is the Hamiltonian operator that quantizes the total energy of the system<sup>[5](https://bpb-us-e1.wpmucdn.com/sites.psu.edu/dist/1/4020/files/2017/12/analysis-slides-278829v.pdf)</sup>. Stone's theorem is thus the precise statement behind the exponential formula \( U(t) = e^{-iHt/\hbar} \): the Hermiticity of \( H \) and the unitarity of \( U(t) \) are two sides of the same correspondence<sup>[11](https://www.physics2u.com/vault/matrix-exponential-generator-flow.html)</sup>.

## Symmetries and Wigner's theorem

The role of unitary operators as carriers of symmetry is fixed by [Wigner's theorem](https://www.edgepedia.org/wigners-theorem), formulated and proved by [Eugene Wigner](https://www.edgepedia.org/eugene-wigner) in 1931. The theorem states that any invertible map of the space of pure states (unit rays) of a quantum system onto itself that preserves transition probabilities is induced either by a linear unitary transformation or by an antilinear antiunitary transformation on the underlying Hilbert space; invariance of all transition probabilities demands that probability amplitudes be either preserved or complex conjugated uniformly<sup>[6](https://www.sciencedirect.com/science/article/pii/S0375960108014709)</sup>. The dominant role of unitary group representations in quantum physics traces back to this result<sup>[6](https://www.sciencedirect.com/science/article/pii/S0375960108014709)</sup>.

## Unitary operations in quantum computing

Within quantum information theory, every unitary matrix indexed by a system's classical states in both its rows and its columns qualifies as a legitimate quantum operation on that system. For composite systems, the rows and columns correspond to the [Cartesian product](https://www.edgepedia.org/cartesian-product) of the individual systems' classical state sets, and unitary operations performed independently on several systems combine by tensor product, \( U_{n-1} \otimes \cdots \otimes U_0 \)<sup>[12](https://quantum.cloud.ibm.com/learning/en/courses/basics-of-quantum-information/multiple-systems/quantum-information)</sup>. Operations that are both deterministic and unitary are called reversible operations; the conjugate transpose of such a unitary matrix represents the inverse operation, as expected from \( U^{\dagger} = U^{-1} \)<sup>[12](https://quantum.cloud.ibm.com/learning/en/courses/basics-of-quantum-information/multiple-systems/quantum-information)</sup>.

## References

1. [Definition 9.2, Unitary operators, Mathematics for Quantum Physics, TU Delft](https://interactivetextbooks.tudelft.nl/mqp-v2/operators.html)
2. [Mathematical interlude: Hilbert spaces and linear operators, Lecture notes for PHYS223, Lancaster University](https://www.lancaster.ac.uk/staff/schomeru/lecturenotes/Quantum%20Mechanics/S8.html)
3. [Quantum Physics II, Lecture Notes 6, MIT OpenCourseWare (8.05, Fall 2013)](https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/79a8091ef4f18d8e3ff76a097e8db33c_MIT8_05F13_Chap_06.pdf)
4. [Time Evolution in Quantum Mechanics, Chapter 6, MIT OpenCourseWare (22.02, Spring 2012)](https://ocw.mit.edu/courses/22-02-introduction-to-applied-nuclear-physics-spring-2012/b5106a499ae03e36b5a2e002355668f9_MIT22_02S12_lec_ch6.pdf)
5. [Lectures on Analysis, Lecture 23: Theorems on Unbounded Self-Adjoint Operators, Penn State](https://bpb-us-e1.wpmucdn.com/sites.psu.edu/dist/1/4020/files/2017/12/analysis-slides-278829v.pdf)
6. [Two elementary proofs of the Wigner theorem on symmetry in quantum mechanics, Physics Letters A](https://www.sciencedirect.com/science/article/pii/S0375960108014709)
7. [Unitary operators, PHY 721 lecture notes, University of Nevada, Las Vegas](https://www.physics.unlv.edu/~bernard/phy721_99/tex_notes/node9.html)
8. [Advanced Quantum Mechanics II PHYS 40202, Section 3.4, University of Manchester](https://oer.physics.manchester.ac.uk/AQM2/Notes/Notes-3.4.html)
9. [Foundations of Quantum Mechanics, lecture notes, Universität Heidelberg](https://www.thphys.uni-heidelberg.de/~wetterich/Vorlesungen/Foundations%20of%20quantum%20mechanics%202021/Foundations_of_Quantum_Mechanics.pdf)
10. [Companion Notes on Daniel Freed's lectures, geometric quantization](https://math-terek.github.io/texts/GQT-companion.pdf)
11. [Matrix Exponential and One-Parameter Flows, physics2u](https://www.physics2u.com/vault/matrix-exponential-generator-flow.html)
12. [Quantum information, IBM Quantum Learning](https://quantum.cloud.ibm.com/learning/en/courses/basics-of-quantum-information/multiple-systems/quantum-information)
13. [The spectral theorem, Math foundations](https://math.hodapp.com/lectures/higher/functional-analysis/spectral-theorem.html)

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*Topic: Encyclopedia › Physical world and mathematics › Physics › Quantum physics › Quantum mechanics › Quantum formalism and states › Operators, observables, and angular momentum › Unitary evolution operators and time evolution*

*Initially written Sep 17, 2026 · Reviewed: Sep 19, 2026; Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

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