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Chem120A+Notes+8-29-08

Chem120A+Notes+8-29-08 - Chem120A Notes Schrdinger Equation...

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Chem120A Notes 8/29/08 Schrödinger Equation → ) ( ) ( ˆ x E x H Ψ = Ψ (note: “^” signifies an operator) → This is an eigenvalue equation H ˆ is the Hamiltonian operator Ψ (x) is an eigenfunction → E is the eigenvalue → Not all functions are eigenfunctions, only those which give you a number → Operator operating on a function gives you that function multiplied by a number (Operator ∙ Function = Number ∙ Function → Eigenvalue Equation) Examples of Operators D ˆ → Differentiation operator with respect to x f (x) → differentiable equation ) ( ) ( ) ( ˆ x f x f dx d x f D = x x e x e x D 3 2 ) 3 ( ˆ 2 + = + Sum of two operators A ˆ and B ˆ ) ( ˆ ) ( ˆ ) ( ) ˆ ˆ ( x f B x f A x f B A + + Product of two operators A ˆ and B ˆ [ ] ) ( ˆ ˆ ) ( ˆ ˆ x f B A x f B A A ˆ operates on the result of B ˆ operating on f (x) Operators obey associative law of multiplication i.e. C B A C B A ˆ ) ˆ ˆ ( ) ˆ ˆ ( ˆ = However, order of operations is important → For numbers, ba ab = → For operators, A B B A ˆ ˆ ˆ ˆ ?
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