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Category > Applied Sciences Posted 06 May 2017 My Price 7.00

Coherent states of the harmonic oscillator.

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)      Like any other wave function, a coherent state can be expanded in terms of energy eigenstates:

L00

l a)  =      c11 In).

11=0

Show  that  the  expansion  coefficients  are

 

(d)       Determine  co by  normalizing  la ) . Answer:  exp(-la l 2 / 2). ( e)   Now  put  in  the  time  dependence:

In) """""7  e-i E,,t / h l n ) ,

 

and show  that  la (t ) )  remains  an eigenstate  of a_ , but  the eigenvalue  evolves in  time:

 

 

So a coherent state stays coherent, and continues to minimize the uncertainty product.

 

 

( f) Is the ground state (In = 0)) itself a coherent state? If so, what is the eigen­ value?

 

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Status NEW Posted 06 May 2017 11:05 AM My Price 7.00

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