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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

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.

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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