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equations:hamilton-jacobi_equation [2018/04/15 11:49] ida [Abstract] |
equations:hamilton-jacobi_equation [2018/05/03 13:04] jakobadmin ↷ Links adapted because of a move operation |
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<blockquote>The Hamilton-Jacobi equation is essentially a dispersion relation for | <blockquote>The Hamilton-Jacobi equation is essentially a dispersion relation for | ||
a complex wave. This is easy to see in the context of non-relativistic | a complex wave. This is easy to see in the context of non-relativistic | ||
- | [[theories:quantum_mechanics|quantum mechanics]]. If a quantum amplitude is expressed in the | + | [[theories:quantum_mechanics:canonical_quantum_mechanics|quantum mechanics]]. If a quantum amplitude is expressed in the |
form ψ = Rexp(iS/h¯), then the Hamilton-Jacobi equation relates | 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). | p = ∂S/∂q to E = −∂S/∂t by the condition p2(q) = 2m(E −V). | ||
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where $p_1$ is the momentum of the particle going from $q_0$ to $q_1$, at time $t_1$, and | where $p_1$ is the momentum of the particle going from $q_0$ to $q_1$, at time $t_1$, and | ||
- | \begin{align*} | + | $$ \frac{\partial W}{\partial q_0^i}= -(p_0)_i ,\qquad\text{(-momentum at time $t_0$)} $$ |
- | \frac{\partial W}{\partial q_0^i}&= -(p_0)_i,\qquad\text{(-momentum at time $t_0$)}\\ | + | $$ \frac{\partial W}{\partial t_1}= -H_1,\qquad\text{(-energy at time $t_1$)} $$ |
- | \frac{\partial W}{\partial t_1}&= -H_1,\qquad\text{(-energy at time $t_1$)}\\ | + | $$ \frac{\partial W}{\partial t_0}= H_0,\qquad\text{(+energy at time $t_0$)} |
- | \frac{\partial W}{\partial t_0}&= H_0,\qquad\text{(+energy at time $t_0$)} | + | $$ |
- | \end{align*} | + | |
Take note that $H_1=H_0$ since energy is conserved. These last four equations are the Hamilton--Jacobi equations. | Take note that $H_1=H_0$ since energy is conserved. These last four equations are the Hamilton--Jacobi equations. |