PHYS4221(Fall2010)_Solution_5

PHYS4221(Fall2010_S - 1 PHYS4221 Suggested Solution of Homework 5 Question 1(a The Hamiltonian H = p 2 x p 2 y 2 m m ω 2 2 x 2 y 2 λxy can be

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Unformatted text preview: 1 PHYS4221 Suggested Solution of Homework 5 Question 1 (a) The Hamiltonian H = p 2 x + p 2 y 2 m + m ω 2 2 ( x 2 + y 2 ) + λxy can be easily diagonized by a change of variable. We let x ± = x ± y √ 2 , p ± = p x ± p y √ 2 . This change of variables corresponds to a unitary transformation , because the new variables preserve the commutation relations [ x + ,p + ] = [ x- ,p- ] = i ¯ h [ x + ,p- ] = [ x- ,p + ] = 0 [ x- ,x + ] = [ p- ,p + ] = 0 . As an example, using [ x,p x ] = [ y,p y ] = i ¯ h and [ x,p y ] = [ y,p x ] = 0, [ x + ,p + ] = 1 2 ([ x + y,p x + p y ]) = 1 2 ([ x,p x ] + [ y,p y ]) = 1 2 ( i ¯ h + i ¯ h ) = i ¯ h [ x + ,p- ] = 1 2 ([ x + y,p x- p y ]) = 1 2 ([ x,p x ]- [ y,p y ]) = 0 . Furthermore, we can write H as H = p 2 + + p 2- 2 m + m ω 2 2 ( x 2 + + x 2- ) + λ 2 ( x 2 +- x 2- ) = p 2 + 2 m + 1 2 m ω 2 + x 2 + + p 2- 2 m + 1 2 m ω 2- x 2- ≡ H + + H- where H ± = p 2 ± 2 m + 1 2 m ω 2 ± x 2 ± , ω ± = r ω 2 ± λ m . To summarize, in normal-mode coordinates { x + ,x- } , the Hamiltonian describes two decoupled harmonic oscillators with frequencies ω + and ω- respectively. Hence the eigenstates of H are given by {| n + ,n- i} , with eigenvalues H | n + ,n- i = ¯ hω + n + + 1 2 + ¯ hω- n- + 1 2 | n + ,n- i , n ± = 0 , 1 , 2 ,.... (i) In coordinate space ( x + ,x- ), the eigenfunction of H is just the product of the eigenfunctions of H + and H- : Ψ n + ,n- ( x + ,x- ) ≡ h x + ,x- | n + ,n- i = ψ (+) n + ( x + ) * ψ (- ) n- ( x- ) , where ψ ( ± ) n ( x ) = 1 q 2 n n ! √ πx ± H n x x ± exp (- 1 2 x x ± 2 ) , x ± = s ¯ h m ω ± . (c.f. lecture notes no.3, eq.(42)). Rewriting the required eigenfunction in terms of ( x,y ), Ψ n + ,n- ( x,y ) = ψ (+) n + x + y √ 2 ψ (- ) n- x- y √ 2 . 2 (ii) Similarly, the eigenfunction in momentum space is...
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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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PHYS4221(Fall2010_S - 1 PHYS4221 Suggested Solution of Homework 5 Question 1(a The Hamiltonian H = p 2 x p 2 y 2 m m ω 2 2 x 2 y 2 λxy can be

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