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# Notes8 - Lesson 8 I Operators In Quantum Mechanics A...

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Unformatted text preview: Lesson 8 I. Operators In Quantum Mechanics A. Hermitian Adjoint - + A ˆ The Hermitian adjoint of an operator A ˆ is the operator + A ˆ which obeys the equation Ψ Ψ = Ψ Ψ + A ˆ A ˆ EXAMPLE: Show that the Hermitian adjoint of a number is its complex conjugate. EXAMPLE: Show that ( 29 + + + = A ˆ B ˆ B ˆ A ˆ (Problem 4.11 (b) in Liboff) B. Hermitian Operator 1. An operator which is its own Hermitian adjoint is called a Hermitian Operator. A ˆ A ˆ = + 2. Why are Hermitian operators important? The operator representing any physical quantity must have real eigenvalues since these are the values we obtain for our measurements. Hermitian operators have real eigenvalues!! Proof: Let n Φ be eigenfunctions of the Hermitian operator A ˆ n n n n A ˆ A ˆ Φ Φ = Φ Φ + Def. of Hermitian Adjoint n n n n A ˆ A ˆ Φ Φ = Φ Φ Using Def. of Hermitian Operator n n n n n n a a Φ Φ = Φ Φ Dirac Algebra n n a a = By comparison of LHS & RHS of Eq. Q.E.D. 3. Since all operators in quantum mechanics will be Hermitian, you can apply the operator either on the bra or the ket. Thus, we can write the expectation value as Ψ A ˆ Ψ or Ψ A ˆ Ψ or Ψ Ψ A ˆ 4. The product of two Hermitian operators A ˆ and B ˆ will be Hermitian only if A ˆ and B ˆ commute!!...
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Notes8 - Lesson 8 I Operators In Quantum Mechanics A...

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