Hermiticity and Hermitian Operators in Quantum Mechanics, Study notes of Physics

Download Hermiticity and Hermitian Operators in Quantum Mechanics and more Study notes Physics in PDF only on Docsity! Lecture 23 Outline - Hermiticity • vectors - Hilbert space [Section 3.1] • transformations - Observables [Section 3.2] Vector Space Example e Problem 3.2 on the board... Hermitian Operators • Express expectation values in inner product notation: • For Q(x, p) with x → x̂ and x → p̂. 〈Q〉 = ∫ Ψ∗Q̂Ψ dx = 〈Ψ|Q̂Ψ〉 • Such operators are linear transformations eg. consider Q̂[af(x) + bg(x)] = aQ̂f(x) + bQ̂g(x) • The outcome of measurements are real so 〈Q〉 = 〈Q〉∗, 〈Ψ|Q̂Ψ〉 = ∫ Ψ∗Q̂Ψ dx = ∫ (Q̂Ψ)∗Ψ dx = 〈Q̂Ψ|Ψ〉 Observables are represented by hermitian operators Hermitian Operators e Example: with Griffith’s equation 3.19 for p Hermitian Operators e Problem 3.4