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:

00

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?