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This follows from the definition of the Poisson bracket and is the basic definition of equations of motion in Hamiltonian mechanics. Any textbook covering Hamiltonian mechanics should tell you this at most 10 pages after starting the discussion on the subject.oristo42 said:
The harmonic oscillator Hamiltonian equation is a mathematical expression that describes the motion of a particle in an oscillating system. It is given by H = p2/2m + 1/2mω2x2, where p is the momentum of the particle, m is its mass, and ω is the frequency of the oscillation.
The harmonic oscillator Hamiltonian equation is a fundamental equation in quantum mechanics that is used to describe the behavior of many physical systems, such as atoms, molecules, and solid materials. It allows us to understand and predict the energy states and properties of these systems.
The harmonic oscillator Hamiltonian equation is derived using principles from classical mechanics and quantum mechanics. It can be obtained by applying the Schrödinger equation to a particle in an oscillating potential energy field.
The harmonic oscillator Hamiltonian equation assumes that the potential energy of the system is quadratic, the system is in a stable equilibrium, and there is no external force acting on the system. It also assumes that the system is in a vacuum and that there is no energy dissipation.
The harmonic oscillator Hamiltonian equation is used in many areas of physics, including quantum mechanics, statistical mechanics, and solid-state physics. It is used to calculate the energy levels and wave functions of a particle in an oscillating system, and to study the behavior of physical systems such as atoms and molecules.
