ON THE OPERATOR EXP - Theoretical and Computational Research in the 21st Century - page 154

8.2.3 CONSTRUCTION OF F AND G

From the derivative of Eq. (10) with respect to λ ,
( ) ( )

,

' ' fQ

⎡

g P

⎤

g P fQ '

⎡

fQ

⎤

fQ g P fQ

2 ( PQT fQT fe + ) = + e Q Qe e , + + g e P Pe e e , +

⎣ ⎦ ⎣ ⎦

( )

(

( ) ( )

)

( )

[ ] [ ] [ ]

( 3) 2 fQ gP g f Q f PQ

'

++

' ' 2

− , T + fg

' 2

⎡ e

fQ

, P ⎤ + Pe

fQ

+ fg e

'

⎡

fQ

, PQ , ⎤ + PQe ,

fQ

e e

g P fQ

⎣ ⎦ ⎣ ⎦

( )

( ) ( ) ( ) [

]

(3) g

'

+ fg P

' 2

+ 2 f

'

+ gf

' 2

− 2 f fg fg f

'

+

' 2 2

Q + fg gf

'

−

'

− fg f

' 2

PQ , T

Then, from the compatibility of the coefficients of P , Q , and [ P , Q ] in both
members, we obtain differential equations for f and g :

, (17.a)

' ' 2

gf + g = 2

( )

' ' ' 2

, (17.b)

fg f g − + fg = 0

( )

( )

2 ' ' 2 21

f gf + g + f − f g = 2 (17.c)

By Substituting Eq. (17.a) in Eqs. (17.b) and (17.c) implies

2 f

, (18.a)

g =

'

f

( )

2 '

f + f 1 − f g = 1

(18.b)

' 2

f =+ 1 f

And substituting Eq. (18.a) in Eq. (18.b) leads to the equation

whose integration gives f ( λ ) = tan( λ + c ) , but c = 0 because f is an odd

f () tan λ = λ

f () tan λ = λ

function; thus, . There-

fore, it is a complete proof of Eqs. (4) and (5), in accordance with Eq. (10).
The time evolution operator for the harmonic oscillator in one-dimen-
sion is given by the following equation:

and due to Eq. (18.a)

2 2

 d 1

22

U = exp( − i t H /  ) , H = - + m w x ,

(19)

2

2 mdx 2

Then, U has the structure exp (2λ ( P + Q )) with