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Chapter3_9

# Chapter3_9 - |S ↑ Ψ a> = S ↑ op | Ψ a> is an...

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PHY4604 R. D. Field Department of Physics Chapter3_9.doc University of Florida Operators Relations (2) Raising and Lowering Operators: Consider an hermitian operator H op with satisfying an eigenvalue equation H op | Ψ a > = a| Ψ a >. Suppose that there is an operator S op such that [H op ,S op ] = sS op or H op S op = S op H op + sS op Then H op S op | Ψ a > = H op | S Ψ a > = (S op H op +sS op )| Ψ a > = S op H op | Ψ a >+sS op | Ψ a > = (a+s) S op | Ψ a > = (a+s)|S Ψ a >. Hence, the state |S Ψ a > = S op | Ψ a > is an eigenstate of H op with eigenvalue a+s . The new eigenvalue is shifted by an amount +s ( i.e. raising operator). Also, we see that [H op ,S op ] = -sS op or H op S op = S op H op - sS op Hence, the state
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Unformatted text preview: |S ↑ Ψ a > = S ↑ op | Ψ a > is an eigenstate of H op with eigenvalue a-s . The new eigenvalue is shifted by an amount -s ( i.e. lowering operator). Exponentiation: The operator op A e is defined by the power series expansion as follows: ∑ ∞ = ≡ ! 1 n n op n A A e op It is easy (but tedious) to show that L + + + + = − ]]] , [ , [ , [ ]] , [ , [ ] , [ ! 3 1 ! 2 1 op op op op op op op op op op A op A B A A A B A A B A B e B e op op Hence, op A A op op e e 1 = − . Also, op A op A B e B e op op = − provided ] , [ = op op B A ....
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