In quantum mechanics, the bra vector represents the complex conjugate transpose of the ket vector. Mathematically, if |ψ⟩ is a ket vector representing a quantum state, then its corresponding bra vector is ⟨ψ|, which is obtained by taking the complex conjugate of the elements of |ψ⟩ and transposing them.
Physically, the bra vector represents the complex conjugate of the wave function associated with the quantum state. The wave function describes the probability amplitudes for different possible outcomes of a measurement. By taking the complex conjugate of the wave function and transposing it, the bra vector allows us to calculate the inner product with a ket vector, yielding the probability amplitude for a particular measurement outcome.
In essence, the bra vector provides a way to calculate the probability of obtaining a specific measurement result when operating on a quantum state with a corresponding ket vector. It plays a fundamental role in calculating expectation values, probabilities, and performing calculations in quantum mechanics.