ABSTRACT

And if in Eq. (1) only we realize the change λ λ→ − in accordance with Eq. (7):

Again, as P and Q are arbitrary, we obtain the total compatibility between Eqs. (8) and (9) if f = h; therefore, the factorization of Eq. (1) is equivalent to

( ) º exp( 2 ( ) ) exp( ) exp( ) exp ( ),T P Q f Q g P f Qλ λ + = (10)

8.2.2 P AND Q VERIFY

with the function g as a parameter; it is clear that

From the double derivative of Eq. (11) with respect to g, we obtain

' ( ) 2 ( , )M g g P N P Q= + , (15)

8.2.3 CONSTRUCTION OF F AND G

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

The time evolution operator for the harmonic oscillator in one-dimension is given by the following equation:

1exp( i t / ) , - , 2 2

dU H H m w x m d x

= − = +  (19)

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

verifying Eq. (2). Therefore, Eq. (5) permits to factorize this evolution operator:

which can be useful in the calculation of the propagator (Green function) for the harmonic oscillator.