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Revisión del 21:56 19 jul 2007
I. QUANTUM
An invariant for the quantum mechanical, time dependent harmonic oscillator is
2 1 1 ω(t) ˙ 2 I= ω(t)ˆ + q p+ ˆ q ˆ . (1) 2 ω(t) 2ω(t)
where ω(t) = 1/ρ2 with ρ obeying the Ermakov equation
ρ + Ω2 (t)ρ = 1/ρ3 . ¨ (2)
We can define new position and momentum as
Q = T †qT = ω(t)q, (3)
and
1 ω(t) ˙ P = T † pT = p+ q. (4) ω(t) 2ω(t)
with
−i ln (ω(t)) (qp + pq) ω(t) 2 ˙ T (t) = exp exp i q. (5) 4 4 ω(t)
The above operators obey the commutation relation [Q, P] = i, and their evolution equation is
˙ ˙ Q = iω(t)[I, Q] P = iω(t)[I, P], (6)
In terms of Q and P the invariant may be written as
1 P 2 + Q2 . I= (7) 2
II. CLASSICAL
Let us now look at the classical variables, and make the transformation [see (3)]
QC = ω(t)qC , (8)
from where we obtain
ω ˙ ˙ QC = ω(t) q˙ + qC , (9) C 2ω ˙
We now borrow from quantum mechanical evolution equations (6) QC = ωPC and write the classical invariant as
1 Q2 + PC . 2 I= (10) C 2
A. semi-proofs
The classical position obeys the equation
q¨ + Ω2 (t)qC = 0, (11) C dt
and has a relation with the auxiliary Ermakov function of the form qC = ρ cos( ) or
ρ2 QC = cos( ωdt), (12)
therefore [1]
˙ QC PC ≡ = − sin( ωdt), (13) ω
and indeed sin2 ( ωdt) + cos2 ( ωdt) = 1.
d2 QC 1d d
[1] It is like there is a new derivative: → such that + QC = 0
dτ 2 ω dt dτ
[[Categoría:{investigación}]]