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equations:hamilton-jacobi_equation
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Hamilton-Jacobi Equation
Intuitive
Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.
Concrete
For a very nice explanation of the origin of the Hamilton-Jacobi equation and its meaning see chapter 2 in Sleeping Beauties in Theoretical Physics by Thanu Padmanabhan
The Hamilton-Jacobi equation is essentially a dispersion relation for a complex wave. This is easy to see in the context of non-relativistic quantum mechanics. If a quantum amplitude is expressed in the form ψ = Rexp(iS/h¯), then the Hamilton-Jacobi equation relates p = ∂S/∂q to E = −∂S/∂t by the condition p2(q) = 2m(E −V). This is a relation between the wave vector k = p/h¯ and the frequency ω = E/h¯ of the “matter wave” associated with the particle.chapter 2 in Sleeping Beauties in Theoretical Physics by Thanu Padmanabhan
Take note that while the Hamilton equation for time-evolution is closely related to the Heisenberg equation in quantum mechanics, the Hamilton-Jacobi equation is closely connected to the Schrödinger equation. See, for example, Section 4.8.1 in Tong's lecture notes.
Abstract
The motto in this section is: the higher the level of abstraction, the better.
Why is it interesting?
equations/hamilton-jacobi_equation.1523177810.txt.gz · Last modified: 2018/04/08 08:56 (external edit)
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