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Quantum River
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Particle-Wave duality and Hamilton-Jacobi equation
According to Particle-Wave duality, an observer can't describe a natural object just from its particle-nature or wave-nature, because a particle is always accompanied by a wave and vice versa.
This reminded me some interesting aspects of the classical Hamilton-Jacobi Equation. The Hamilton-Jacobi Equation is also the only formalism of mechanics in which the motion of a particle can be represented as a wave. In this sense, the HJE fulfilled a long-held goal of theoretical physics (dating at least to Johann Bernoulli in the 17th century) of finding an analogy between the propagation of light and the motion of a particle. [1]
Yes, to unite the particle equation and wave equation in one formalism is the original goal of Hamilton when he devised the HJE first. It is apparent from the name/theme from his two original papers [1].
I don't know whether Hamilton really did achieve his original goal even in the case of classical physics. I am interested in the Quantum Mechanics case. Is the Hamilton-Jacobi formalism the natural framework to describe the Particle-Wave duality?
In the one-order linear (or quasi-linear) partial differential equation, the wave of the equation could be buildup by the infinite orbits of particle motion. Could this hold in the two-order partial differential equation? I am curious why Hamilton thought he had achieved his goal, because the Hamiltonian of HJE is obvious not linear (there is the kinetic energy).
[1]:http://en.wikipedia.org/wiki/Hamilton-Jacobi_equation
According to Particle-Wave duality, an observer can't describe a natural object just from its particle-nature or wave-nature, because a particle is always accompanied by a wave and vice versa.
This reminded me some interesting aspects of the classical Hamilton-Jacobi Equation. The Hamilton-Jacobi Equation is also the only formalism of mechanics in which the motion of a particle can be represented as a wave. In this sense, the HJE fulfilled a long-held goal of theoretical physics (dating at least to Johann Bernoulli in the 17th century) of finding an analogy between the propagation of light and the motion of a particle. [1]
Yes, to unite the particle equation and wave equation in one formalism is the original goal of Hamilton when he devised the HJE first. It is apparent from the name/theme from his two original papers [1].
I don't know whether Hamilton really did achieve his original goal even in the case of classical physics. I am interested in the Quantum Mechanics case. Is the Hamilton-Jacobi formalism the natural framework to describe the Particle-Wave duality?
In the one-order linear (or quasi-linear) partial differential equation, the wave of the equation could be buildup by the infinite orbits of particle motion. Could this hold in the two-order partial differential equation? I am curious why Hamilton thought he had achieved his goal, because the Hamiltonian of HJE is obvious not linear (there is the kinetic energy).
[1]:http://en.wikipedia.org/wiki/Hamilton-Jacobi_equation
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