A [[unitary operator]] is a fundamental concept in [[mathematics]] and [[quantum mechanics]], describing a transformation that preserves the length and angle between vectors in a [[Hilbert space]], which is a complete vector space equipped with an inner product. Unitary operators are crucial in quantum mechanics because they describe the evolution of quantum systems in a way that conserves probability, ensuring that the total probability of all possible outcomes remains constant over time. ### Mathematical Definition In mathematical terms, an operator $U$ on a [[Hilbert space]] is called [[unitary]] if it satisfies the following conditions: 1. **Preserves Inner Product**: For any vectors $x$ and $y$ in the Hilbert space, $⟨Ux,Uy⟩=⟨x,y⟩$ where $⟨⋅,⋅⟩$ denotes the inner product in the Hilbert space. This condition implies that the operator preserves angles and lengths of vectors, thereby conserving the geometric and probabilistic properties of the space. 2. **Bijective**: The operator $U$ must be bijective, meaning it is both injective (no two different vectors map to the same vector) and surjective (every vector in the space can be reached by applying $U$ to some vector in the space). ### Formal Properties - **Invertibility**: A unitary operator $U$ is always invertible, and its inverse is given by its adjoint (or conjugate transpose), denoted $U^†$. Therefore, $U$ satisfies: $U^†U=UU^†=I$ where $I$ is the identity operator. This equation states that $U^†$ (the inverse of $U$) undoes the action of $U$, and vice versa. - **Norm-Preserving**: Since unitary operators preserve the inner product, they also preserve the norm of any vector. For any vector $x$ in the Hilbert space, $∥Ux∥=∥x∥$ where $∥⋅∥$ denotes the norm derived from the inner product. ### Quantum Mechanics Applications In quantum mechanics, unitary operators are used to describe the time evolution of the state vectors of quantum systems. According to the postulates of quantum mechanics: - **Time Evolution**: The evolution of a closed quantum system is described by a unitary operator. If $∣ψ(t)⟩$ is the state of the system at time $t$, and $∣ψ(0)⟩$ is the initial state, then: $∣ψ(t)⟩=U(t)∣ψ(0)⟩$ where $U(t)$ is a unitary operator dependent on time $t$, typically expressed in terms of the system's Hamiltonian $H$ as: $U(t)=e^{−iHt/ℏ}$ Here, $ℏ$ is the reduced [[Planck constant]], and $H$ is the Hamiltonian operator of the system. - **Quantum Gates**: In quantum computing, quantum gates that manipulate qubits are represented by unitary operators. These gates must be unitary to ensure that the computation remains reversible and that the total probability (sum of probabilities of all possible outcomes) remains $1$. Unitary operators are thus central to both the theoretical framework of quantum mechanics and practical applications in quantum computing, encapsulating key principles of dynamics, state conservation, and probability. # References ```dataview Table title as Title, authors as Authors where contains(subject, "unitary") or contains(subject, "unitarity") or contains(subject, "operator") sort modified desc, authors, title ```