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 $q_{i}(t)$,

$S=\int _{0}^{T}L(q_{i},{\dot {q}}_{i},t)dt$

that we evaluate for all paths with fixed end points

$q_{i}(0)=q_{i}^{\rm {initial}},q_{i}(T)=q_{i}^{\rm {final}}$

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

$W(q_{i}^{\rm {initial}},q_{i}^{\rm {final}},T)=S\left[q_{i}^{\rm {classical}}(t)\right].$

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

$\delta S=\int _{0}^{T}dt\left[{\frac {\partial L}{\partial q_{i}}}-{\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {q}}_{i}}}\right)\right]\delta q_{i}(t)+\left.{\frac {\partial L}{\partial {\dot {q}}_{i}}}\delta q_{i}(t)\right|_{0}^{T}.$

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

${\frac {\partial W}{\partial q_{i}^{\rm {final}}}}=\left.{\frac {\partial L}{\partial {\dot {q}}_{i}}}\right|_{t=T}=p_{i}^{\rm {final}}.$

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

${\frac {dW}{dT}}={\frac {\partial W}{\partial T}}+\sum _{i}{\frac {\partial W}{\partial q_{i}^{\rm {final}}}}{\dot {q}}_{i}^{\rm {final}}={\frac {\partial W}{\partial T}}+\sum _{i}p_{i}^{\rm {final}}{\dot {q}}_{i}^{\rm {final}}.$

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

${\frac {dW}{dT}}=L(q_{i}^{\rm {classical}}(T),{\dot {q}}_{i}^{\rm {classical}}(T),T)=L\left(q_{i}^{\rm {final}},{\dot {q}}_{i}^{\rm {final}}\right).$

If we combine this results we have

${\frac {\partial W}{\partial T}}=-\left(\sum _{i}p_{i}^{\rm {final}}{\dot {q}}_{i}^{\rm {final}}-L\left(q_{i}^{\rm {final}},{\dot {q}}_{i}^{\rm {final}}\right)\right)=-H\left(q_{i}^{\rm {final}},p_{i}^{\rm {final}}\right).$

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

${\frac {\partial W}{\partial q_{i}}}=p_{i}~~{\rm {and}}~~{\frac {\partial W}{\partial t}}=-H(q_{i},p_{i},t)$

or by combining the two we have

${\frac {\partial W}{\partial t}}=-H\left(q_{i},{\frac {\partial W}{\partial q_{i}}},t\right)$

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,

${\dot {q}}_{i}=\left.{\frac {\partial H}{\partial p_{i}}}\right|_{p_{i}=\partial W/\partial q_{i}}.$

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

${\dot {p}}_{i}={\frac {d}{dt}}\left({\frac {\partial W}{\partial q_{i}}}\right)=\sum _{j}{\frac {\partial ^{2}W}{\partial q_{i}\partial q_{j}}}{\dot {q}}_{j}+\left(-{\frac {\partial H}{\partial q_{i}}}-\sum _{j}{\frac {\partial H}{\partial p_{j}}}{\frac {\partial ^{2}W}{\partial q_{i}\partial q_{j}}}\right)$

      $={\frac {d}{dt}}\left({\frac {\partial W}{\partial q_{i}}}\right)=\sum _{j}{\frac {\partial ^{2}W}{\partial q_{i}\partial q_{j}}}{\dot {q}}_{j}+\left(-{\frac {\partial H}{\partial q_{i}}}-\sum _{j}{\dot {q}}_{j}{\frac {\partial ^{2}W}{\partial q_{i}\partial q_{j}}}\right)=-{\frac {\partial H}{\partial q_{i}}}.$

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

$H\left(q_{i},{\frac {\partial W_{0}}{\partial q_{i}}}\right)=E.$

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

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

so

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

and

$P_{x}={\frac {1}{\pi }}\left[W\left(x_{\rm {max}}\right)-W\left(x_{\rm {min}}\right)\right]$

Schrodinger Equation

In quantum mechanics we have

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

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

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

and plug into the Schrodinger equation

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

Let's take the classical limit that

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

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

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

or

${\frac {\partial W}{\partial t}}=-H\left(q_{i},{\frac {\partial W}{\partial q_{i}}},t\right)+{\mathcal {O}}(\hbar ).$

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).