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Unformatted text preview: PHY4221 Quantum Mechanics I (Fall 2004) Take Home Quiz 2 (Answer) For simplicity, we have set 1 m = = = h . Thus the momentum and position operator could be written as: ( 29 ( 29 1 , 2 2 i x a a p a a + + = + = (1) So the Hamiltonian could be expressed in terms of the raising and lowering operators: ( 29 ( 29 ( 29 ( 29 ( 29 2 2 2 2 1 2 4 1 2 1 4 p H x a a x a a a a x + + + = =  =  + (2) So we have: ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 2 2 2 2 2 , 2 , 2 , 1 2 1 4 1 2 1 4 1 2 1 1 4 2 1 1 1 2 1 2 1 4 m n m n m n m n m n m n m n m n m n m n m n H a a a a x a a a a x n n n n n n n n n n + + + + + + =  + + =  + + + + =  +   + =  + + + +  (3) We have proved before that the eigen functions of harmonic oscillators have fixed parity (if you...
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This note was uploaded on 12/10/2011 for the course PHYS 4221 taught by Professor Cflo during the Spring '11 term at CUHK.
 Spring '11
 CFLO
 Momentum

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