Course:PHYS350/Hamilton-Jacobi Equation

Let's take yet another viewpoint on classical mechanics which will make the connection to quantum mechanics pretty obvious. It is known as Hamilton-Jacobi theory. Let's start with the definition of the action for a particular path $q_i(t)$,

$S=\underset{0}{\overset{T}{\int }}L(q_i,{\dot{q}}_i,t)dt$

that we evaluate for all paths with fixed end points

$q_i(0)=q_i^{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}},q_i(T)=q_i^{\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}$

where the true path is an extremum of the action: $\delta S=0$.

If we take a new perspective and define a function of the initial and final positions and time that gives the action for the classical path

$W(q_i^{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}},q_i^{\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}},T)=S[q_i^{\mathrm{c}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}}(t)].$

Let's keep the initial positions fixed but vary the end point, we have

$\delta S=\underset{0}{\overset{T}{\int }}dt[\frac{\partial L}{\partial q_i}-\frac{d}{dt}\left(\frac{\partial L}{\partial {\dot{q}}_i}\right)]\delta q_i(t)+{\frac{\partial L}{\partial {\dot{q}}_i}\delta q_i(t)|}_0^T.$

For the classical path, the first term vanishes. This leaves

$\frac{\partial W}{\partial q_i^{\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}}={\frac{\partial L}{\partial {\dot{q}}_i}|}_{t=T}=p_i^{\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}.$

Let's trying running the evolution for a little bit longer, so we have

$\frac{dW}{dT}=\frac{\partial W}{\partial T}+\sum _i\frac{\partial W}{\partial q_i^{\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}}{\dot{q}}_i^{\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}=\frac{\partial W}{\partial T}+\sum _i{p}_i^{\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}{\dot{q}}_i^{\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}.$

We know that $dS/dT=L$ so we have

$\frac{dW}{dT}=L(q_i^{\mathrm{c}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}}(T),{\dot{q}}_i^{\mathrm{c}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}}(T),T)=L(q_i^{\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}},{\dot{q}}_i^{\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}).$

If we combine this results we have

$\frac{\partial W}{\partial T}=-(\sum _i{p}_i^{\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}{\dot{q}}_i^{\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}-L(q_i^{\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}},{\dot{q}}_i^{\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}))=-H(q_i^{\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}},p_i^{\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}).$

There is nothing particularly special about the final time $T$ so we can replace $T\to t$ to give the following equations

$\frac{\partial W}{\partial q_i}=p_i\mathrm{a}\mathrm{n}\mathrm{d}\frac{\partial W}{\partial t}=-H(q_i,p_i,t)$

or by combining the two we have

$\frac{\partial W}{\partial t}=-H(q_i,\frac{\partial W}{\partial q_i},t)$

which is known as the Hamilton-Jacobi Equation.

The Hamilton-Jacobi Equation and Hamilton's Equations

Let's suppose that we have a solution to the Hamilton-Jacobi equation, $W(q_i,t)$. We can use Hamilton's equations to determine the velocity of the particles at any position and time,

${\dot{q}}_i={\frac{\partial H}{\partial p_i}|}_{p_i=\partial W/\partial q_i}.$

We should check that the second of Hamilton's equations is also satisfied. We have

${\dot{p}}_i=\frac{d}{dt}\left(\frac{\partial W}{\partial q_i}\right)=\sum _j\frac{{\partial }^{2}W}{\partial q_i\partial q_j}{\dot{q}}_j+\frac{{\partial }^{2}W}{\partial t\partial q_i}.$

Let's differentiate the Hamilton-Jacobi equation to look at the last term

${\dot{p}}_i=\frac{d}{dt}\left(\frac{\partial W}{\partial q_i}\right)=\sum _j\frac{{\partial }^{2}W}{\partial q_i\partial q_j}{\dot{q}}_j+(-\frac{\partial H}{\partial q_i}-\sum _j\frac{\partial H}{\partial p_j}\frac{{\partial }^{2}W}{\partial q_i\partial q_j})$

      $=\frac{d}{dt}\left(\frac{\partial W}{\partial q_i}\right)=\sum _j\frac{{\partial }^{2}W}{\partial q_i\partial q_j}{\dot{q}}_j+(-\frac{\partial H}{\partial q_i}-\sum _j{\dot{q}}_j\frac{{\partial }^{2}W}{\partial q_i\partial q_j})=-\frac{\partial H}{\partial q_i}.$

Using the Hamilton-Jacobi Equation

Let's suppose that the Hamiltonian is constant in time. We can solve for the time dependence of the function $W$ as

$W(q_i,t)=W^0(q_i)-Et$

giving the Hamilton-Jacobi equation

$H(q_i,\frac{\partial W_0}{\partial q_i})=E.$

Let's consider one-dimensional motion, so we have

$H=\frac{p_x^2}{2m}+V(x)$

and

$\frac{1}{2m}{\left(\frac{\partial W^0}{\partial x}\right)}^2+V(x)=E.$

Rearranging gives

$\frac{\partial W}{\partial x}=\pm \sqrt{2m[E-V(x)]}$

so

$W=\int \sqrt{2m[E-V(x)]}dx$

and

$P_x=\frac{1}{\pi }[W\left(x_{\mathrm{m}\mathrm{a}\mathrm{x}}\right)-W\left(x_{\mathrm{m}\mathrm{i}\mathrm{n}}\right)]$

Schrodinger Equation

In quantum mechanics we have

$i\hslash \frac{\partial \psi }{\partial t}=\hat{H}\psi =-\frac{h^2}{2m}\frac{{\partial }^2\psi }{\partial q^2}+V(q)\psi .$

Let's write the wavefunction $\psi$ as an amplitude and a phase

$\psi (q,t)=R(q,t)e^{iW(q,t)/\hslash }$

and plug into the Schrodinger equation

$i\hslash [\frac{\partial R}{\partial t}+\frac{iR}{\hslash }\frac{\partial W}{\partial t}]=-\frac{{\hslash }^2}{2m}[\frac{{\partial }^{2}R}{\partial q^2}+\frac{2i}{\hslash }\frac{\partial R}{\partial q}\frac{\partial W}{\partial q}-\frac{R}{{\hslash }^2}{\left(\frac{\partial W}{\partial q}\right)}^2+\frac{iR}{\hslash }\frac{{\partial }^{2}W}{\partial q^2}]+VR.$

Let's take the classical limit that

$\hslash \left|\frac{{\partial }^{2}W}{\partial q^2}\right|\ll {\left(\frac{\partial W}{\partial q}\right)}^2$

and collecting the terms leading in $\hslash$ we get

$\frac{\partial W}{\partial t}+\frac{1}{2m}{\left(\frac{\partial W}{\partial q}\right)}^2+V(q)={𝒪}(\hslash )$

or

$\frac{\partial W}{\partial t}=-H(q_i,\frac{\partial W}{\partial q_i},t)+{𝒪}(\hslash ).$

We see that the solution to the Hamilton-Jacobi equation is the phase of the wavefunction in the classical limit. We can also interpret what it means for the momentum to become imaginary -- the corresponding wave function decreases exponentially in space (the WKB approximation).