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MCS,PHD
Argosy University/ Phoniex University/
Nov-2005 - Oct-2011
Professor
Phoniex University
Oct-2001 - Nov-2016
Problem 3.35  Coherent states of the harmonic oscillator.  Among  the stationary states of  the  harmonic  oscillator  (In)  = v,,,(x), Equation  2.67) only  n  = 0 bits  the uncertainty  limit  ( ax ap  = h / 2);  in  general,  ax ap  =  ( 2n + l)h / 2,  as  you  found
in Problem 2.12. But certain linear combinations (known as coherent states) also minimize the uncertainty product. They are (as it turns out) eigenfunctions of the lowering operator :32
a_ la )  = a la ) (the eigenvalue  a  can  be  any  complex  number).
(a)  Calculate  (x),  (x2 ) ,  (p),  (p2 )  in  the  state  la).  Hint:  Use  the  technique  in Example  2.5, and remember  that a+ is the hermitian  conjugate  of a_ . Do not
assume a is  real.
(b)  Find ax  and ap ; show that ax ap = h / 2.
(c)Â Â Â Â Â Like any other wave function, a coherent state can be expanded in terms of energy eigenstates:
00
l a) Â =Â Â Â Â Â c11 In).
11=0
Show that the expansion coefficients  are
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(d)      Determine  co by  normalizing  la ) . Answer:  exp(-la l 2 / 2). ( e)  Now  put  in  the  time  dependence:
In) """""7 Â e-i E,,t / h l n ) ,
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and show  that  la (t ) )  remains  an eigenstate  of a_ , but  the eigenvalue  evolves in  time:
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So a coherent state stays coherent, and continues to minimize the uncertainty product.
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( f) Is the ground state (In = 0)) itself a coherent state? If so, what is the eigen value?
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