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Lecture19Notes - Chem 120A Spring 2006 READING The Harmonic...

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Chem 120A The Harmonic Oscillator, part II 03/03/06 Spring 2006 Lecture 19 READING: Engel (QCS): Chapter 7 and 8.3 Dirac notation, parity, expectation values, and time evolution Here we take a look at the harmonic oscillator in Dirac notation and look at parity, ’1D chirality,’ and some dynamics. 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 2 ˆ H (1) ˆ p ( m ¯ h ) 1 / 2 ˆ p ˆ q r m ¯ h ˆ x . In Dirac notation, the eigenstates are written as { ± ± n ² } , ˆ H ± ± n ² = ¯ h ( n + 1 2 ) ± ± n ² , n = 0 , 1 , 2 ,... (2) As we will show, ˆ P ψ n ( q )=( 1 ) n 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 = n ( q ) and ³ q ± ± ˆ P ± ± n ² = n ( q ) since ³ q ± ± ˆ P = ³ q ± ± . But , n ( q 1 ) n n ( q ) ˆ P ± ± n ² =( 1 ) n ± ± n ² . The eigenstates have either even or odd parity. Now we must see how ˆ P commutes with ˆ H . In order to do Chem 120A, Spring 2006, Lecture 19 1
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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 +
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Lecture19Notes - Chem 120A Spring 2006 READING The Harmonic...

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