equations:hamilton-jacobi_equation
Differences
This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision Next revision | Previous revision | ||
|
equations:hamilton-jacobi_equation [2018/04/15 11:48] ida [Abstract] |
equations:hamilton-jacobi_equation [2018/05/05 12:29] (current) 63.143.42.253 ↷ Links adapted because of a move operation |
||
|---|---|---|---|
| Line 15: | Line 15: | ||
| <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]]. 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). | ||
| Line 43: | Line 43: | ||
| and | and | ||
| \[ | \[ | ||
| - | S(q) = \int_{t_0}^{t_1} L\Bigl(q(t),\qdot(t)\Bigr)\,dt . | + | S(q) = \int_{t_0}^{t_1} L\Bigl(q(t),\dot{q}(t)\Bigr)\,dt . |
| \] | \] | ||
| Here $W$ is the least action for some path from $(q_0,t_0)$ to $(q_1,t_1)$. We have | Here $W$ is the least action for some path from $(q_0,t_0)$ to $(q_1,t_1)$. We have | ||
| \[ | \[ | ||
| - | \frac{\pa}{\pa q_1^i}W(q_0,q_1) = (p_1)_i, | + | \frac{\partial}{\partial q_1^i}W(q_0,q_1) = (p_1)_i, |
| \] | \] | ||
| - | \[ | + | |
| - | \xy | + | |
| - | (48,-3)*+{(q_0,t_0)},(50,0)*\dir{*};(90,10), | + | |
| - | **\crv{(60,0)&(70,10)&(85,10)&(90,10)}, | + | |
| - | (90,7)*+{p_1}, (85,10),(85,10) {\ar@{*\dir{*}->} (90,10)} | + | |
| - | \POS(79.5,7) \ar @{-} (79.5,13) | + | |
| - | \POS(81,7) \ar @{-} (81,13) \POS(82.5,7) \ar @{-} (82.5,13) | + | |
| - | \POS(84,7) \ar @{-} (84,13) \POS(85.5,7), | + | |
| - | (82.5,5)*+{(q_1,t_1)} | + | |
| - | \endxy\] | + | |
| 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{\pa W}{\pa 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{\pa W}{\pa t_1}&= -H_1,\qquad\text{(-energy at time $t_1$)}\\ | + | $$ \frac{\partial W}{\partial t_0}= H_0,\qquad\text{(+energy at time $t_0$)} |
| - | \frac{\pa W}{\pa t_0}&= H_0,\qquad\text{(+energy at time $t_0$)} | + | $$ |
| - | \end{align*} | + | |
| - | ($H_1=H_0$ as 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. | ||
| | | ||
equations/hamilton-jacobi_equation.1523785687.txt.gz · Last modified: 2018/04/15 09:48 (external edit)
Page Tools
Except where otherwise noted, content on this wiki is licensed under the following license: CC Attribution-Share Alike 4.0 International
