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lecture21 - Chem 120A Spring 2007 READING The Harmonic...

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Chem 120A The Harmonic Oscillator (II) and molecular vibrations 03/09/07 Spring 2007 Lecture 21 READING: Ratner and Schatz, Ch. 5 1 Dirac notation, parity, expectation values Here we take a look at the harmonic oscillator in Dirac notation and look at parity. In Dirac notation the eigenstates and eigenvalues are derived from the dimensionless Hamiltonian: ˆ p 2 2 m + m ω 2 2 ˆ x 2 [ ˆ p 2 + ˆ q 2 ] ¯ h ω 2 ˆ H (1) ˆ p ( m ω ¯ h ) - 1 / 2 ˆ p ˆ q r m ω ¯ h ˆ x . In Dirac notation, the eigenstates are written as { ± ± n ² } , ˆ H ± ± n ² = ¯ h ω ( n + 1 2 ) ± ± n ² , n = 0 , 1 , 2 , . . . (2) See Ratner and Schatz, Ch. 5 for a derivation of this using the raising and lowering operators b = ( m ω h ) 1 / 2 ( x + i m ω p ) and b = ( m ω h ) 1 / 2 ( x - i m ω p ) . As we will show, ˆ P ψ n ( q ) = ( - 1 ) n ψ n ( q ) , (3) where ˆ P is the parity operator. Remember that ˆ P = ˆ P because the parity operator is Hermitian, and ˆ P 2 = 1. Now, ³ q ± ± n ² = ψ n ( q ) and ³ q ± ± ˆ P ± ± n ² = ψ n ( - q ) since ³ q ± ± ˆ P = ³ - q ± ± . But , ψ n ( q ) = ( - 1 ) n ψ n ( q ) ˆ P ± ± n ² = ( - 1 ) n ± ± n ² . So the eigenstates have either even ( n is even) or odd ( n is odd) parity. Now we must see how ˆ P commutes Chem 120A, Spring 2007, Lecture 21 1
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with ˆ H . In order to do this we must examine how ˆ P acts on ˆ p and ˆ q . We will consider ˆ q first: ˆ q ± ± q ² = q ± ± q ² ˆ P ˆ q ± ± q ² = q ˆ P ± ± q ² where q is a number Now since ˆ P 2 = ˆ P , ˆ P ˆ q ˆ P 2 = ˆ P ˆ q ˆ P ˆ P ± ± q ² ˆ P ˆ q ˆ P ˆ P = q ˆ P ± ± q ² ˆ P ˆ q ˆ P ± ± - q ² = q ± ± - q ² But , ˆ q ± ± - q ² = - q ± ± - q ² ˆ P ˆ q ˆ P = - ˆ q . (4) Now, we consider how the parity operator effects the momentum operator: From [ ˆ x , ˆ p ] = i ¯ h , [ ˆ q , ˆ p ] = i We now multiply both sides of the commutator by the parity operator: ˆ P [ ˆ q , ˆ p ] ˆ P = i ˆ P 2 ˆ P ( ˆ q ˆ p - ˆ p ˆ q ) ˆ P = i . Inserting ˆ P 2 between ˆ q and ˆ p in both terms of the commutator: ˆ P ˆ q ˆ P ˆ P ˆ p ˆ P - ˆ P ˆ p ˆ P ˆ P ˆ q ˆ P = i ⇒ - ˆ q ˆ P ˆ p ˆ P - ˆ P ˆ p ˆ P ( - ˆ q ) = i Or , ˆ q ˆ P ˆ p ˆ P - ˆ P ˆ p ˆ P ˆ q = - i But ˆ q ˆ p - ˆ p ˆ q = i , Therefore , ˆ P ˆ p ˆ P = - ˆ p . (5) Now we are ready to consider the commutation of the parity operator and the Hamiltonian: [ ˆ P , ˆ H ] = ˆ P ³ ˆ p 2 2 + ˆ q 2 2 ´ - ³ ˆ p 2 2 + ˆ q 2 2 ´ ˆ P = ˆ P ³ ˆ p 2 2 + ˆ q 2 2 ´ ˆ P 2 - ˆ P 2 ³ ˆ p 2 2 + ˆ q 2 2 ´ ˆ P Now , ˆ P ˆ p 2 ˆ P = ˆ P ˆ p ˆ P ˆ P ˆ p ˆ P = ˆ p 2 And ˆ P ˆ q 2 ˆ P = ˆ P ˆ q ˆ P ˆ P
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