ON THE OPERATOR EXP - Theoretical and Computational Research in the 21st Century - page 151

8.1 INTRODUCTION

In general, if P and Q are arbitrary operators, then it is not possible to give
a nontrivial splitting for exp (2λ( P + Q )), λ being a parameter. Here, we
study the realization of a factorization of the type:

exp ( 2 λ ( PQ + ) ) = exp( f ( λ ) Q ) exp ( g P ( λ ) ) exp ( h Q ( λ ) ),

(1)

where f, g , and h are functions that determine whether P and Q verify the
following relations between commutators:

[ ] [ ]

(2)

⎡ PQ P , , ⎤ = -2 P , ⎡ PQ Q , , ⎤ = 2 Q

⎣ ⎦ ⎣ ⎦

which are present in the analysis of the time evolution operator for the
harmonic oscillator in one dimension.
In Section 8.2, without restrictions on P and Q , it is proved that f =
h , f (−λ) = − f (λ) and g (−λ) = − g (λ), that is, into Eq. (1) f and g are odd
functions. In Section 8.3, we accept the conditions (Eq. 2) to obtain the
useful expressions:

( )

[ ] [ ]

2

exp( gP Q ) , = g P + g P,Q exp( gP ),

( ) [ ] [ ]

2

exp( fQ ) , P = f Q − f P,Q exp( fQ ),

(3)

[ ]

⎡ exp( fQ ) , P,Q ⎤ =− 2 fQ exp( fQ ).

⎣ ⎦

Section 8.4 is dedicated to the construction of f and g ; therefore,

f () tan, λ = λ g () sin(2 , λ = λ

(4)

Then for Eq. (2), it is valid the splitting:

(5)

exp( 2 λ ( PQ + ) ) = exp( Q tan λ ) exp( P sin(2 λ )) exp( Q tan λ ),

And also it is shown its application to time evolution operator
exp (

U = − i t H /  for the one-dimensional harmonic oscillator. )