# Quantum-classical

I. QUANTUM

An invariant for the quantum mechanical, time dependent harmonic oscillator is

${\displaystyle I={\frac {1}{2}}\omega ^{2}}$

                                                               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