PS5sol - Solutions to Problem Set 5 Physics 341 by Callum...

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Unformatted text preview: Solutions to Problem Set 5 Physics 341 by: Callum Quigley 1 Completeness Relation for Coherent States For this problem, I’ll use the normalized coherent states from the last homework, namely: | λ i = e-| λ | 2 / 2 e λa † | i = e-| λ | 2 / 2 ∑ n ( λ n / √ n !) | n i . Writing λ = x + iy = re iθ we can compute Z d x d y | λ ih λ | = Z d x d y X m,n e-| λ | 2 λ m ( λ * ) n √ m ! n ! | m ih n | = X m,n Z r d r d θ e- r 2 r m + n e i ( m- n ) θ √ m ! n ! | m ih n | = 2 π X n 1 n ! Z d re- r 2 r 2 n +1 | n ih n | (1.1) = π X n | n ih n | = π 1 where I’ve used the facts that R d θe i ( m- n ) θ = 2 πδ n,m and R d rr 2 n +1 e- r 2 = 1 2 Γ( n + 1) = 1 2 n !. And so, 1 = Z d x d yf ( x,y ) | λ ih λ | ⇐⇒ f ( x,y ) = 1 π (1.2) Note: If I’d used the unnormalized coherent states | λ i = e λa † | i as they’re given in Shankar then we’d have f ( λ,λ * ) = e-| λ | 2 /π 2 Properties of Coherent States Note: We’ll now switch to Shankar’s conventions for coherent states | z i = e za † | i , which are normalized as h z | z i = e | z | 2 . Using the facts that a | z i = z | z i and a = βX + ( i/ 2 β ~ ) P where β 2 = mω/ 2 ~ , we have zψ z ( x ) = z h x | z i = h x | a | z i = βxψ z ( x ) + 1 2 β ∂ x ψ z ( x ) (2.1) which we can integrate to find ψ z ( x ) = ψ z (0) e- β 2 x 2 +2 βzx . (2.2) 1 To determine ψ z (0) we’ll consider the norm e | z | 2 = h z | z i = Z d x | ψ z ( x ) | 2 = | ψ z (0) | 2 Z d xe- 2 β 2 x 2 +2 βx ( z + z * ) = 1 2 β | ψ z (0) | 2 e 1 2 ( z + z * ) 2 Z ∞-∞ d y e- y 2 / 2 = √ 2 π 2 β | ψ z (0) | 2 e | z | 2 e ( z 2 + z * 2 ) / 2 (2.3) ⇒ ψ z (0) = s 2 β √ 2 π e- z 2 / 2 , where in the second line I switch integration variables to y = 2 βx- z- z * . Putting this all together, ψ z ( x ) = mω π ~ 1 4 e- z 2 / 2 e- ( mω/ 2 ~ ) x 2 + √ 2 mω/ ~ zx . (2.4) If we focus on the x dependence of the wavefunction ψ z ( x ) ∝ e- ( βx- z ) 2 ∝ e- β 2 ( x- Re( z ) /β ) 2 e 2 i Im( z ) βx (2.5) then, comparing to (9.3.7) in Shankar, we can read off the expectation values h X i z = Re( z ) /β h P i z = 2 β ~ Im( z )...
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PS5sol - Solutions to Problem Set 5 Physics 341 by Callum...

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