HWI - PHY 4221 Quantum Mechanics Fall Term Of 2005 Problem...

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PHY 4221 Quantum Mechanics Fall Term Of 2005 Problem Set No.1 (Suggested Solutions) September 27, 2005 Question 1: You could expand () ( ) exp exp A BA ξ as following: ( ) 00 1 exp exp !! n nn mm nm AB A A B A ξξ ∞∞ == −= ∑∑ We will pick out the terms containing k and thus the above equation could be rewritten as: ( ) 1 exp exp 1 1 ! ! ! ! n n k kn n k k k n n k k n k k A A A nk n ABA n k k n k ABA A k n k −− =⋅ =− ⎩⎭ = {} ! k n nnn k k CA B A A −⋅ (1) The following is to prove that { } [] 0 ,,, ,, k n k k n total number of A is k B A A B A A A A = ⎡⎤ ⋅ ⋅ = ⋅⋅⋅ ⎣⎦ ±²²²²³²²²²´ (2) To achieve this, we firstly notice that when k =1 and 0, this relation obviously holds. Then we assume that when k = t , the relation still holds, and try to see what happen when k = t +1. Now we have { } 0 t n t t n total number of A is t B A A B A A A A = = and when k = t +1, the right hand side becomes: [ ] [ ] [ ] 1 total number of A is t total number of A is t total number of A is t B AA BAA AAAA BAA + = ±²²²²³²²²² ´± ² ² ² ²³²²²²´ Correspondingly, the left hand side becomes: { } { } tt t t B A A A A B A A (3)
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We could do some calculation about equation (3): () { } { } {} 00 11 1 1 10 1 1 1 1 1 tt nn nnn t t t n t t n n CA B A A A A B A A C A B A A C A B A A BAA B A B A A B A B A B A −− == −+ + + + −⋅ = ⋅⋅ + ⎧⎫ =− + ⎨⎬ ⎩⎭ ∑∑ 1 1 1 1 1 1 1 1 1 t t pp t ppp p t t p t t pt p p t t t p AB A A B B A C A B A A B A A B C A BA C A BA A B CC A B AA B A A ++ + + + + + + + + + + = +− = + = + + ⎡⎤ =+
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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.

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HWI - PHY 4221 Quantum Mechanics Fall Term Of 2005 Problem...

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