Quantum-classical

I. QUANTUM

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

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

                                                               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