In quantum mechanics, a unitary operator is a mathematical operator that represents a transformation on a quantum system. It plays a fundamental role in quantum theory and is used to describe the evolution of quantum states over time.
A unitary operator is defined as an operator that preserves the norm of a quantum state vector and preserves the inner product between two quantum states. Mathematically, an operator U is unitary if it satisfies the condition:
U^†U = UU^† = I,
where U^† represents the adjoint or Hermitian conjugate of U, and I is the identity operator.
The property of preserving the norm ensures that the probability of observing a certain outcome when measuring a quantum state remains unchanged under the action of a unitary operator. The property of preserving the inner product ensures that the relative amplitudes and phases between quantum states are preserved.
Unitary operators are used to describe a wide range of quantum phenomena, including time evolution of quantum systems, quantum gates in quantum computing, transformations between different bases, and symmetries in quantum mechanics. They provide a mathematical framework for understanding the dynamics and transformations of quantum states.
It is important to note that unitary operators are reversible, meaning that there exists an inverse operator that can undo the transformation. This reversibility is a key feature of quantum mechanics, distinguishing it from classical physics where some transformations are irreversible.