Abstract
We study classical and quantum dynamics of the symmetric harmonic oscillator and the symmetric bouncer defined in 2-D. For these systems we get each of them for two different constants of motion, two Lagrangians and two Hamiltonians which describe the same classical dynamics. However, the quantization of these systems (using Schrodinger equation), using their two equivalents Hamiltonian, describes different quantum dynamics for each of them. This represents an ambiguity on the Hamiltonian formulation of the Quantum Mechanics.
Highlights
Modern Physics is based on Lagrangian or Hamiltonian [1] [2] [3] as mathematical objects to formulate and study the correspondent behavior of the natural systems
We study classical and quantum dynamics of the symmetric harmonic oscillator and the symmetric bouncer defined in 2-D
Quantum Mechanics has its foundations based on the Hamilton operator and the Schrödinger’s equation, which describes the linear evolution of the wave function
Summary
Modern Physics is based on Lagrangian or Hamiltonian [1] [2] [3] as mathematical objects to formulate and study the correspondent behavior of the natural systems. It is known that for most of the systems (conservative or including electromagnetic interaction) there is no problem to get a well unique Hamiltonian formulation [4] for the system. When one is dealing with dissipative systems, it has several problems [5] [6], and one of the main problems, which we are concerned, is that there can be two different Hamiltonian having the same classical behavior but different quantum behavior [7] [8]. We make the deduction of a pair of Hamiltonian for both systems, and this pair of Hamiltonians for each system describes the same classical dynamics. We show that the associated quantum dynamics of each pair of Hamiltonian is different for each system
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