Metaplectic operator approach to a time-dependent generalized harmonic oscillator

Abstract

The metaplectic operator is a unitary operator corresponding to a time-dependent classical linear canonical transformation. We present the explicit expression for the metaplectic operator. The parameter in the metaplectic operator is expressed in terms of the solution of a classical time-dependent generalized harmonic oscillator. The time evolution operator, Lewis and Riesenfeld invariant, and the wave function of the time-dependent generalized harmonic oscillator are studied using the metaplectic operator, and the results are compared with those from other studies. The metaplectic operator method is a simpler method for studying the time-dependent generalized harmonic oscillator.

Metaplectic operator approach to Gutzwiller’s trace formula

“…In this work, we have derived the Gutzwiller's trace formula by considering the time-evolution operator of the Hamiltonian which describes an N-dimensional time-dependent generalized harmonic oscillator (TDGHO) with general force terms The Hamiltonian is obtained by considering the wave packet quadratically expanded in position and momentum operators around a reference trajectory (a periodic orbit) as done in [8,20]. The timeevolution operator is composed of Weyl-Heisenberg and time-dependent metaplectic operators, where the metaplectic operator is known as the time-evolution operator for TDGHO [14].…”

“…Using the explicit form of the metaplectic operator recently obtained in [14], we have derived the well known properties, the law of concatenation (8) and winding number relation (9), of the metaplectic operator. And we also derived the coordinate representation of the metaplectic operator, which is the same form as that in [8].…”

“…It is noted that the mapping relation between S and M S ˆ( ) is two-to-one because that both +M S ˆ( ) and -M S ˆ( ) give the same symplectic transformation (3). In [14], it is also shown that the operator M S t ˆ( ( )) is the time evolution operator of the Hamiltonian…”

“…In [8], by expanding S around the identity I, it was shown that the operator M S ˆ( ) obeys equation (6). But, in [14] equation (6) was derived by directly using the formula for the derivative of an exponential operator [18].…”

“…The Hamiltonian governing the wave packet is for an Ndimensional time-dependent generalized harmonic oscillator (TDGHO) with general force terms. The explicit form of the time evolution operator for the TDGHO is recently constructed in terms of symplectic matrix, position, and momentum operators [14]. The time evolution operator is just the metaplectic operator.…”