HW4-sol - PHYSICS 4455 QUANTUM MECHANICS Problem Set 4...

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PHYSICS 4455 — QUANTUM MECHANICS Problem Set 4 — Solutions. 1. Becoming more familiar with operators. Liboff Problem 3.3 Nonlinear operators are: P , Ĝ - why? P [ a ϕ 1 ( x )+ b ϕ 2 ( x )] = [ a ϕ 1 ( x )+ b ϕ 2 ( x )] 3 3 [ a ϕ 1 ( x )+ b ϕ 2 ( x )] 2 4 which is not equal to : aP ϕ 1 ( x )+ bP ϕ 2 ( x )= a 1 3 ( x )− 3 ϕ 1 2 ( x )− 4 }+ b 2 3 ( x )− 3 ϕ 2 2 ( x )− 4 } Ĝ [ a ϕ 1 ( x )+ b ϕ 2 ( x )] = 8 which is not equal to : a Ĝ ϕ 1 ( x )+ b Ĝ ϕ 2 ( x )= a × 8 + b × 8 All others are linear. I’m a bit lazy here: to get full credit for your homework, you should go through the operators one by one, and check the linearity condition. I’ll skip that - if any of these others give you trouble, please ask about them during office hours. Liboff Problem 3.16 We have  ϕ n ( x )= a n ϕ n ( x ) and a function f ( x ) with expansion f ( x )= l = 0 b l x l We want to show that ϕ n ( x ) is an eigenfunction of f (  )≡ l = 0 b l  l . To do so, note that  3 ϕ n ( x )=  2 a n ϕ n ( x )= a n  2 ϕ n ( x )= a n Âa n ϕ n ( x )= ... = a n 3 ϕ n ( x ) etc., for any integer power of  .So , f (  n ( x )= l = 0 b l  l ϕ n ( x )= l = 0 b l a n l ϕ n ( x )= f ( a n n ( x ) . 2. and in particular, the displacement operator: Liboff Problem 3.4 The displacement operator is defined as D f ( x )≡ f ( x +ζ) This is of course a silly definition - what is ζ ? For the purposes of this definition, it is simply a (given) constant. It would be much less obscure to define D ζ as the displacement operator by an amount ζ
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This note was uploaded on 11/09/2009 for the course PHY 4604 taught by Professor Mucciolo during the Spring '09 term at University of Central Florida.

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HW4-sol - PHYSICS 4455 QUANTUM MECHANICS Problem Set 4...

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