ON THE OPERATOR EXP - Theoretical and Computational Research in the 21st Century - page 155

2

1 ih d imw

2

λ = wt , P = , Q = − x ,

(20)

2 2 mw d x

2

2 h

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

⎛ i mw wt ⎞ ⎛ i  d

2

⎞ ⎛ i mw wt ⎞

⎛⎞ ⎛⎞

( )

2 2

U = exp − x tan exp sin w t exp − x tan ,

(21) ⎜ ⎜⎟ ⎟ ⎜ ⎟ ⎜ ⎜⎟ ⎟

2  2 2 mw d x

2

2  2

⎝⎠ ⎝⎠

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

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

8.3 CONCLUSION

In this work, we have showed that if the operators P and Q satisfy the
conditions

[ ] [ ]

, then it is valid the

⎡ PQ P , , ⎤ =− 2 P and ⎡ PQ Q , , ⎤ = 2 Q

⎣ ⎦ ⎣ ⎦

relation exp( 2 . This

result is important in the study of the time evolution operator for the one-
dimensional harmonic oscillator.

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

KEYWORDS

• Arbitrary operators

• Harmonic oscillator

• Time evolution operator

REFERENCE

1. Merzbacher, E.; Quantum Mechanics. New York: John Wiley & Sons; Chapter 8,
1970, 167 p.