Chapter3_11

# Chapter3_11 - − Φ = Φ − = Φ − and where the Φ ±...

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PHY4604 R. D. Field Department of Physics Chapter3_11.doc University of Florida The Parity Operator Parity Operator: The parity operator is the operator that takes x -x as follows: ) ( ) ( x x P op = ψ ψ . Eigenvalues of the Parity Operator: In general, ) ( ) ( x x P op λψ ψ = and ) ( ) ( ) ( 2 2 x x x P op ψ ψ λ ψ = = . It is easy to show that the parity operator is hermitian, P =P , and P 2 = 1 and hence λ is real which implies λ = ±1 ( i.e. the eigenvalues are ±1). Expansion Theorem: The eigenfunctions of any hermitian operator form a complete set of states, in terms of which any wave function can be expanded. This is easy to see for the parity operator since any function can be written as follows: ) ( ) ( )) ( ) ( ( )) ( ) ( ( ) ( 2 1 2 1 x x x x x x x + Φ + Φ = Φ Φ + Φ + Φ = Φ , where )) ( ) ( ( ) ( ) 1 ( ) ( 2 1 2 1 x x x P x op Φ + Φ = Φ + = Φ + )) ( ) ( ( ) ( ) 1 ( ) ( 2 1 2 1 x x
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Unformatted text preview: − Φ = Φ − = Φ − and where the Φ ± are eigenfunctions of the parity operator with eigenvalues ±1 as follows: ) ( ) ( x x P op ± ± Φ ± = Φ . Parity Conservation: Since the P operator has no explicit time dependence we have ] , [ op op H P dt P d i = > < h Thus, if the Hamiltonian H is symmetric under x →-x then [P op ,H op ] = 0 parity is a constant of the motion ( i.e. it is conserved ). Projection Operators: The operators ) 1 ( 2 1 op op P P ± = ± project out the states with definite parity. In general, projection operators have the properties: ± ± = op op P P 2 ) ( and = = + − − + op op op op P P P P ....
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