In quantum mechanics, a bra-ket, also known as a Dirac notation or bra-ket notation, is a symbolic representation used to describe quantum states and their mathematical operations. It was introduced by physicist Paul Dirac.
The notation consists of two parts: a bra and a ket, denoted as ⟨ψ| and |φ⟩, respectively. The bra represents the complex conjugate of a vector or a state, while the ket represents the vector or state itself.
A bra-ket can be written as ⟨ψ|φ⟩, where |ψ⟩ and |φ⟩ represent quantum states. The expression ⟨ψ|φ⟩ represents the inner product or the overlap between the states |ψ⟩ and |φ⟩. Mathematically, the inner product is defined as the complex conjugate of the first state, ⟨ψ|, multiplied by the second state, |φ⟩.
The bra-ket notation is used to express various operations and concepts in quantum mechanics, such as:
State vectors: Quantum states are represented as kets, for example, |ψ⟩ or |φ⟩. These kets can describe the position, momentum, spin, or any other observable property of a quantum system.
Dual space: The bra vectors, such as ⟨ψ|, represent the complex conjugate transpose of a ket. They belong to the dual space of the ket space and are used to represent linear functionals or operators.
Inner product: The bra-ket notation is used to compute the inner product between two states, as mentioned earlier. The inner product provides a measure of the similarity or overlap between the states.
Operators: Quantum mechanical operators, such as observables or transformations, are represented using the bra-ket notation. For example, the position operator would be represented as ^x, and the corresponding eigenstate as |x⟩.
Overall, the bra-ket notation simplifies the mathematical representation of quantum states, operations, and calculations in quantum mechanics, providing a concise and elegant way to express complex concepts.