We study both classical and quantum relation between two Hamiltonian systems which are mutually connected by time-dependent canonical transformation. One is ordinary conservative system and the other is time-dependent Hamiltonian system. The quantum unitary operator relevant to classical canonical transformation between the two systems are obtained through rigorous evaluation. With the aid of the unitary operator, we have derived quantum states of the time-dependent Hamiltonian system through transforming the quantum states of the conservative system. The invariant operators of the two systems are presented and the relation between them are addressed. We showed that there exist numerous Hamiltonians, which gives the same classical equation of motion. Though it is impossible to distinguish the systems described by these Hamiltonians within the realm of classical mechanics, they can be distinguishable quantum mechanically.We study both classical and quantum relation between two Hamiltonian systems which are mutually connected by time-dependent canonical transformation. One is ordinary conservative system and the other is time-dependent Hamiltonian system. The quantum unitary operator relevant to classical canonical transformation between the two systems are obtained through rigorous evaluation. With the aid of the unitary operator, we have derived quantum states of the time-dependent Hamiltonian system through transforming the quantum states of the conservative system. The invariant operators of the two systems are presented and the relation between them are addressed. We showed that there exist numerous Hamiltonians, which gives the same classical equation of motion. Though it is impossible to distinguish the systems described by these Hamiltonians within the realm of classical mechanics, they can be distinguishable quantum mechanically.

Communications in Theoretical PhysicsSep 1, 20091 citation

Invariance of the Canonical Quantization Prescription under Classical Canonical Transformations

Gin-yih Tsaur, Jyhpyng Wang

Dirac's postulate of canonical quantization, [(The Symbol is abbreviated)]=-iħδ(subscript ij) for conjugate canonical variables, has been the most concise and general prescription on how to quantize a classical system. Since classical systems described by variables connected with canonical transformations are equivalent, [(The Symbol is abbreviated)]=-iħδ(subscript ij) must remain invariant under classical canonical transformations This invariance has not been proved except for the limited class of cascaded infinitesimal transformations. In this paper it is shown that if ((The Symbol is abbreviated)) are related to ((The Symbol is abbreviated)) by a classical canonical transformation, then [(The Symbol is abbreviated)]=-iħδ(subscript ij) In other words, the canonical quantization prescription is invariant for variables connected with classical canonical transformations.Dirac's postulate of canonical quantization, [(The Symbol is abbreviated)]=-iħδ(subscript ij) for conjugate canonical variables, has been the most concise and general prescription on how to quantize a classical system. Since classical systems described by variables connected with canonical transformations are equivalent, [(The Symbol is abbreviated)]=-iħδ(subscript ij) must remain invariant under classical canonical transformations This invariance has not been proved except for the limited class of cascaded infinitesimal transformations. In this paper it is shown that if ((The Symbol is abbreviated)) are related to ((The Symbol is abbreviated)) by a classical canonical transformation, then [(The Symbol is abbreviated)]=-iħδ(subscript ij) In other words, the canonical quantization prescription is invariant for variables connected with classical canonical transformations.

Chinese Journal of PhysicsAug 1, 20070 citations

Canonical Transformations for Quantum Systems

WORLD SCIENTIFIC eBooksAug 17, 20150 citations

Canonical Transformations for Quantum Systems

WORLD SCIENTIFIC eBooksMar 1, 20100 citations

Relations of canonical and unitary transformations for a general time-dependent quadratic Hamiltonian system

Kyu-Hwang Yeon, Dukhyeon Kim, Chung‐In Um +2 more

We consider general time-dependent quadratic Hamiltonian systems which are connected by canonical transformations and give the same classical equations of motion. In those systems, we demonstrate that canonical transformations in classical mechanics correspond to unitary transformations in quantum mechanics. The wave functions and the propagators are evaluated using the invariant operator method. However, the values of some functions of the canonical variables q and p are not equal to the values of the same functions of the other canonical variables Q and P, but the values of the functions of q and the kinetic momentum ${\mathrm{p}}_{\mathrm{k}}$ are equal to those of the other Q and ${\mathrm{P}}_{\mathrm{k}}$ in classical mechanics. We prove that these also hold in the quantum treatment. The uncertainty relations of momentum and position are evaluated for the two Hamiltonians.We consider general time-dependent quadratic Hamiltonian systems which are connected by canonical transformations and give the same classical equations of motion. In those systems, we demonstrate that canonical transformations in classical mechanics correspond to unitary transformations in quantum mechanics. The wave functions and the propagators are evaluated using the invariant operator method. However, the values of some functions of the canonical variables q and p are not equal to the values of the same functions of the other canonical variables Q and P, but the values of the functions of q and the kinetic momentum ${\mathrm{p}}_{\mathrm{k}}$ are equal to those of the other Q and ${\mathrm{P}}_{\mathrm{k}}$ in classical mechanics. We prove that these also hold in the quantum treatment. The uncertainty relations of momentum and position are evaluated for the two Hamiltonians.

Physical Review AJun 1, 199752 citations

{\mathrm{p}}_{\mathrm{k}}

are equal to those of the other Q and

{\mathrm{P}}_{\mathrm{k}}

in classical mechanics. We prove that these also hold in the quantum treatment. The uncertainty relations of momentum and position are evaluated for the two Hamiltonians.