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AnswerToTheTakeHomeQuiz2

AnswerToTheTakeHomeQuiz2 - PHY4221 Quantum Mechanics I(Fall...

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PHY4221 Quantum Mechanics I (Fall 2004) Take Home Quiz 2 (Answer) For simplicity, we have set 1 m ϖ = = = 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 δ δ δ + + + = - = - - - = - + - - - (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 0 0 4 2 1 1 1 2 1 2 1 4 0 0 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 δ δ δ δ δ + + + + + - + - Φ Φ = - Φ + - - + Φ = - Φ + - - + Φ + + Φ Φ = - + - Φ Φ - Φ Φ - + Φ Φ = - + + + - - + - Φ Φ (3) We have proved before that the eigen functions of harmonic oscillators have fixed parity (if you do not know how to obtain this, please see my appendix of this document): ( 29 ( 29 ( 29 ( 29 ( 29 2 1 2 1/ 4 2 !
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