( a)   Generalize  Problem  7.2, using  the  trial  wave  function 14

A

1/l (x ) = (x 2 + b2 )n '

 

 

 

 

 

for arbitrary  11 . Partial  answer:  The best  value of b is given by

Ii = }!    [ 11 ( 411  - 1)(4n - 3) J l / l

111Ct)                 2(211 + 1)

(b)    Find the least upper bound on the first excited state of the harmonic oscillator using a trial function of the form

Bx

1/1 (x ) = (x2 + bl ) " .

 

Pa rtial answer:  The best  value  of b is given by

 

 

J  - Â

I ('

b2 =  !!     [ n ( 4n - 5 )(4 11 - 3)

 

nu.v               2(2n

 

+ 1)

 

 

(c)    Notice  that  the  bounds  approach  the  exact  energies  as n        oo. Why is that? Hint: Plot the trial wave functions for n = 2, 11 = 3, and n = 4, and

compare them with the true wave functions (Equations 2.59 and 2.62). To  do it analytically, start with the  identity

 

e"-

 

= Hm

 

(1+-Z )" .

 

11-H)O            11