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.