Hamilton-Jacobi form
In the non-relativistic quantum mechanics (NRQM), Schoedinger Equation dictates the dynamical evolution of the system, $ i \hbar \partial_t \psi = -\frac{\hbar^2}{2m}\nabla^2\psi + V\psi. $ Schroedinger equation is a linear second-order partial differential equation (PDE). Differential operator $ \hat{\mathbf{p}}=\frac{\hbar}{i}\nabla$ is called momentum, and $\hat H = \frac{\hat{\mathbf{p}}^2}{2m} + V$ Hamiltonian. Let's do a change of variable $\psi = \exp \frac{i}{\hbar}S$. We obtain a non-linear second-order PDE, \[
\partial_t S + \frac{1}{2m}(\nabla S)^2 + V = \frac{i\hbar}{2m}\nabla^2_qS
\] Let's call the complex function $\mathbf{p} = \nabla_q S$ momentum and $H = \frac{\mathbf p^2}{2m} + V$ the Hamiltonian. The above equation can be written as $ \partial_t S + H(q, \nabla_q S, t) = \frac{i\hbar}{2m}\nabla^2 S$. This is the Hamilton-Jacobi form of the wave equation. Function $\mathbf p$ is different from but consistent with the differential operator $\hat{\mathbf{p}}$. It is easy to see, $\hat{\mathbf{p}}\psi = \mathbf{p}\cdot \psi$.
If the Hamiltonian $H$ is time-independent, we separate the time by $S = W - Et$. The resulted equation, $i\hbar \nabla \cdot \mathbf p - \mathbf{p}^2 + 2m(E-V)$, corresponds to the time-independent Schroedinger equation.
\partial_t \hat S + H(\hat q, \partial_{\hat q} \hat S, t) = 0.
\] Here $\hat S = S(\hat q, \hat Q, t)$ is a quantum mechanical operator that depends on the generalized coordinate operator before and after canonical transformation, $\hat{q}$ and $\hat Q$. In order to get the wave equation, define $S = \langle q |\hat S |Q \rangle$. Obviously, $\langle q |\partial_t \hat S |Q \rangle = \partial_t S$, $\langle q |V(\hat q) |Q \rangle = V(q)$. However, one has to be careful with the quadratic term $\langle q |\hat p^2 |Q \rangle$, as $\hat q$ may not commute with $\hat Q$, which means $\hat p^2$ need to be ordered. In general $\hat p = \sum_i a_i(\hat q, t) b_i(\hat Q,t)$. Therefore,
\[ \hat p^2 =
\sum_i \hat p a_i(\hat q, t) b_i(\hat Q,t) = \sum_i a_i(\hat q, t) \hat p b_i(\hat Q,t) + \sum_i [\hat p, a_i(\hat q,t)] b_i (\hat Q,t). \] The first term is well-ordered. For the second term, recall that $[\hat p, A] = -i\partial_q \hbar A$. So $\hat p^2 = :\hat p^2: - i\hbar \partial_q \hat p$. Sandwiched with $\langle q |$ and $|Q\rangle$, we derive the quantum HJ equation.
S = S_0 + \frac{\hbar}{i} S_1 + \left(\frac{\hbar}{i}\right)^2 S_2 + \cdots; \\
W = W_0 + \frac{\hbar}{i} W_1 + \left(\frac{\hbar}{i} \right)^2 W_2 + \cdots.
\] For a semi-classical system, $ \hbar | S_n | \ll |S_{n-1}|$. Therefore, the quantum theory can be solved from improving the classical action order by order. This method is known as the Eikonal approximation (cf. WKB approximation). The classical action obeys the classical HJ equation, $\partial_t S_{cl} + H(q, \partial_q S_{cl}, t) = 0 $. To the first order of $\hbar$, \[\partial_t S_1 + H_1(q, \partial_q S_{cl}, \partial_q S_1, t) = 0,\] here $H_1 = H(q, p_{cl}+p_1,t) - H(q, p_{cl},t) \simeq \partial_p H \partial_q S_1$.
In the time-independent problem, the classical momentum is just $\mathbf p_{cl}^2 = 2m (E - V)$. Up to the first order of $\hbar$, $\mathbf p = \mathbf p_{cl} + \frac{\hbar}{i}\mathbf p_1$. Substitute it to the HJ equation and drop the second order terms, $ \mathbf p_{cl} \cdot \nabla S_1 = -\frac{1}{2}\nabla \cdot \mathbf p_{cl}$. $S_1$ is a hyper surface composed of the integral curves \[
\dot{\mathbf{q}} = \mathbf{p}_{cl}, \quad
\dot{S_1} = -\frac{1}{2}\nabla \cdot \mathbf{p}_{cl}
\] The wavefunction is, \[
\psi = \exp\left( \frac{i}{\hbar} \int \mathrm{d} \mathbf{q} \cdot \mathbf{p}_{cl} + S_1 \right)
\] For example, in the 1D case, $S_1 = -\frac{1}{2} \log p_{cl} + c$ and the full solution is, \[
\psi = \frac{C_+}{\sqrt{p_{cl}}} \exp\left( \frac{i}{\hbar}\int p_{cl} \mathrm dq \right) +
\frac{C_-}{\sqrt{p_{cl}}} \exp\left(- \frac{i}{\hbar}\int p_{cl} \mathrm dq \right).
\]
In the classically forbidden region $(E<V)$, $p_{cl} = \pm i \sqrt{2m|E-V|}$ becomes imaginary and the wavefunction developed an exponential-decline factor. This is the phenomenon of quantum tunneling.
Some higher-dimensional system can be reduced to one-dimensional by the separation of variables. The notable example is the central potential: $V = V(r)$. We can separate variables by $S = W_r + W_\theta + W_\phi - Et$ and get, \[
i\hbar W''_\phi - {W'}_\phi^2 = -m_s^2 \hbar^2; \\
i\hbar \frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta W'_\theta \right) - \frac{m_s^2\hbar^2}{\sin^2\theta} - {W'}_\theta^2 = -l(l+1) \hbar^2; \\
i\hbar \frac{1}{2m r^2} \frac{\partial}{\partial r}\left( r^2 W'_r \right) - \frac{1}{2m r^2}l(l+1)\hbar^2 = V(r) - E + {W'_r}^2 \] with the boundary condition $ W \to W_{cl} $ if $\hbar \to 0, W_\phi(\phi + 2\pi) = W_\phi(\phi)$.
The first equation has the solution $W_\phi(\phi) = \frac{\hbar}{i}\log \cos m_s\phi$ with $m_s = 0, \pm 1, \pm 2, \cdots$.
The second equation is a Riccati equation, that can be rewritten to a linear second order differential equation. The resultant equation is just the Bessel equation. Meanwhile, if we do a change of variable $W'_\theta = p_\theta = i\hbar ( u_\theta + \frac{1}{2}\cot\theta)$, it becomes, \[
u'_\theta = u_\theta^2 + \frac{1-4m_s^2}{4\sin^2\theta} + (l+ \frac{1}{2})^2
\] The third equation is also a Riccati equation. With a change of variable $W'_r = p_r = \frac{i\hbar}{2m}( u_r + \frac{1}{r})$, the equation becomes \[
u'_r = u_r^2 + \frac{4 m^2}{\hbar^2} (E - V(r)) - \frac{2 m l(l+1)}{r^2}
\]
J_q = \oint_C \mathrm{d}q \; p_q
\] where $p_q = \frac{\partial}{\partial q}W$. The value of $J_q$ only depends on the analytic structure of the solution $p_q$ of the Hamilton-Jacobi equation. The poles are the "good" singular points that gives finite result for $J_q$. The "bad" singularities include the branch points and the essential singular points are called the critical points. $J_q$ gains contributions from both poles and critical points.
Non-linear first order differential equations $\frac{\mathrm{d}w}{\mathrm{d}z} = F(z, w)$ where $z \in \mathbb{C}$, and $F(z, w)$ is locally analytic can have "internal singularities" or the movable singular points. The location of the singular points depends on integration constant. A leading example is the equation \[
\frac{\mathrm{d}w}{\mathrm{d}z} = w^2 \quad \Rightarrow \quad w(z) = \frac{1}{c - z}. \] Riccati equations do not have movable critical singularities. In fact, in 1884 W. Fuchs showed that Riccati equations are the only class of first order differential equations without movable critical singular points [2].
Riccati equation $w' = w^2 + f(z)$ admits a solution $w(z) = -\frac{v'(z)}{v(z)}$ where $v(z)$ is a solution of the second order linear equation $v'' + f(z) v = 0$. Then, the original solution $w(z) = - \frac{v'(z)}{v(z)} = - \frac{g(z) + (z-z_1) g'(z)}{(z-z_1)g(z)} = -\frac{1}{z-z_1} - \frac{g'(z)}{g(z)} = -\sum_i \frac{1}{z-z_i} - \varphi(z)$ ($\varphi(z)$ only has critical singularities ) can only have first order non-movable singularities. Then the angular action is quantized with the famous Bohr-Sommerfeld condition \[
J_q = 2 \pi i n (-1) i \hbar = 2 \pi n \hbar + C, \quad n = 0,1,2,... \] where $n$ is the number of single poles lying on the real axis of the solution $w(z)$ hence $p_q$. $C$ is constant. With any luck, the solution has no critical singularities and then $C = 0$. A nice feature of this analysis is the manifest of the correspondence principle. In the classic mechanics, $p_q = \sqrt{2m (E-V)}$ that has a branch cut. Whereas in quantum theory in the classical limit $n \to \infty$, the poles on the real axis behaves like the branch cut. Along this line, the ground state is the solute with the minimal pole.
- - -
[1]:http://eqworld.ipmnet.ru/en/solutions/ode/ode0123.pdf
[2]:http://www.encyclopediaofmath.org/index.php/Painlev%C3%A9-type_equations#References
[3]:http://www.risc.jku.at/publications/download/risc_2773/prep2.pdf
[4]:Robert A. Leacock and Michael J. Padgett, Hamilton-Jacobi Theory and the Quantum Action Variable, Phys. Rev. Lett. 50, 3–6 (1983) URL: http://prl.aps.org/abstract/PRL/v50/i1/p3_1
[5]:Marco Roncadelli and L. S. Schulma, Quantum Hamilton-Jacobi Theory, PRL, 99, 170406 (2007), arxiv URL: http://arxiv.org/pdf/0712.0307v1.pdf
[6]: https://michaelberryphysics.files.wordpress.com/2013/07/berry023.pdf
\partial_t S + \frac{1}{2m}(\nabla S)^2 + V = \frac{i\hbar}{2m}\nabla^2_qS
\] Let's call the complex function $\mathbf{p} = \nabla_q S$ momentum and $H = \frac{\mathbf p^2}{2m} + V$ the Hamiltonian. The above equation can be written as $ \partial_t S + H(q, \nabla_q S, t) = \frac{i\hbar}{2m}\nabla^2 S$. This is the Hamilton-Jacobi form of the wave equation. Function $\mathbf p$ is different from but consistent with the differential operator $\hat{\mathbf{p}}$. It is easy to see, $\hat{\mathbf{p}}\psi = \mathbf{p}\cdot \psi$.
If the Hamiltonian $H$ is time-independent, we separate the time by $S = W - Et$. The resulted equation, $i\hbar \nabla \cdot \mathbf p - \mathbf{p}^2 + 2m(E-V)$, corresponds to the time-independent Schroedinger equation.
Second quantization
The quantum HJ equation differs the classical one by an extra term $\frac{i\hbar}{2m}\nabla \cdot \mathbf{p}$. It has been argued by Roncadelli and Schulman (2007) that this term arises from second-quantization of the classical HJ equation, \[\partial_t \hat S + H(\hat q, \partial_{\hat q} \hat S, t) = 0.
\] Here $\hat S = S(\hat q, \hat Q, t)$ is a quantum mechanical operator that depends on the generalized coordinate operator before and after canonical transformation, $\hat{q}$ and $\hat Q$. In order to get the wave equation, define $S = \langle q |\hat S |Q \rangle$. Obviously, $\langle q |\partial_t \hat S |Q \rangle = \partial_t S$, $\langle q |V(\hat q) |Q \rangle = V(q)$. However, one has to be careful with the quadratic term $\langle q |\hat p^2 |Q \rangle$, as $\hat q$ may not commute with $\hat Q$, which means $\hat p^2$ need to be ordered. In general $\hat p = \sum_i a_i(\hat q, t) b_i(\hat Q,t)$. Therefore,
\[ \hat p^2 =
\sum_i \hat p a_i(\hat q, t) b_i(\hat Q,t) = \sum_i a_i(\hat q, t) \hat p b_i(\hat Q,t) + \sum_i [\hat p, a_i(\hat q,t)] b_i (\hat Q,t). \] The first term is well-ordered. For the second term, recall that $[\hat p, A] = -i\partial_q \hbar A$. So $\hat p^2 = :\hat p^2: - i\hbar \partial_q \hat p$. Sandwiched with $\langle q |$ and $|Q\rangle$, we derive the quantum HJ equation.
Classical limit
The quantum Hamilton-Jacobi equation reduces to the classical Hamilton-Jacobi equation as $\hbar \to 0$. This observation suggests an expansion of the quantum action $S$ around the classical action $S_{cl} \equiv S_0$ with respect to $\hbar$: \[S = S_0 + \frac{\hbar}{i} S_1 + \left(\frac{\hbar}{i}\right)^2 S_2 + \cdots; \\
W = W_0 + \frac{\hbar}{i} W_1 + \left(\frac{\hbar}{i} \right)^2 W_2 + \cdots.
\] For a semi-classical system, $ \hbar | S_n | \ll |S_{n-1}|$. Therefore, the quantum theory can be solved from improving the classical action order by order. This method is known as the Eikonal approximation (cf. WKB approximation). The classical action obeys the classical HJ equation, $\partial_t S_{cl} + H(q, \partial_q S_{cl}, t) = 0 $. To the first order of $\hbar$, \[\partial_t S_1 + H_1(q, \partial_q S_{cl}, \partial_q S_1, t) = 0,\] here $H_1 = H(q, p_{cl}+p_1,t) - H(q, p_{cl},t) \simeq \partial_p H \partial_q S_1$.
 |
| Fig. The solution surface of $S_1$ is the collection of the integral curve $\dot{\mathbf{q}} = \mathbf{p}_{cl}$, $\dot{S}_1 = -\frac{1}{2}\nabla \cdot \mathbf{p}_{cl}$. |
In the time-independent problem, the classical momentum is just $\mathbf p_{cl}^2 = 2m (E - V)$. Up to the first order of $\hbar$, $\mathbf p = \mathbf p_{cl} + \frac{\hbar}{i}\mathbf p_1$. Substitute it to the HJ equation and drop the second order terms, $ \mathbf p_{cl} \cdot \nabla S_1 = -\frac{1}{2}\nabla \cdot \mathbf p_{cl}$. $S_1$ is a hyper surface composed of the integral curves \[
\dot{\mathbf{q}} = \mathbf{p}_{cl}, \quad
\dot{S_1} = -\frac{1}{2}\nabla \cdot \mathbf{p}_{cl}
\] The wavefunction is, \[
\psi = \exp\left( \frac{i}{\hbar} \int \mathrm{d} \mathbf{q} \cdot \mathbf{p}_{cl} + S_1 \right)
\] For example, in the 1D case, $S_1 = -\frac{1}{2} \log p_{cl} + c$ and the full solution is, \[
\psi = \frac{C_+}{\sqrt{p_{cl}}} \exp\left( \frac{i}{\hbar}\int p_{cl} \mathrm dq \right) +
\frac{C_-}{\sqrt{p_{cl}}} \exp\left(- \frac{i}{\hbar}\int p_{cl} \mathrm dq \right).
\]
In the classically forbidden region $(E<V)$, $p_{cl} = \pm i \sqrt{2m|E-V|}$ becomes imaginary and the wavefunction developed an exponential-decline factor. This is the phenomenon of quantum tunneling.
Separation of variables
In one-dimension, the time-independent HJ equations, $i\hbar p' = p^2 - 2m(E-V)$, is a Riccati equation. If given a particular solution $p_0$, the general solution is \[
p = p_0 + \frac{ \exp\left( \frac{2}{i\hbar} \int^x \mathrm dy p_0(y) \right) }{C + \frac{i}{\hbar} \int^x \mathrm dy \exp\left( \frac{2}{i\hbar} \int^y \mathrm dz p_0(z) \right) }.
\]
p = p_0 + \frac{ \exp\left( \frac{2}{i\hbar} \int^x \mathrm dy p_0(y) \right) }{C + \frac{i}{\hbar} \int^x \mathrm dy \exp\left( \frac{2}{i\hbar} \int^y \mathrm dz p_0(z) \right) }.
\]
i\hbar W''_\phi - {W'}_\phi^2 = -m_s^2 \hbar^2; \\
i\hbar \frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta W'_\theta \right) - \frac{m_s^2\hbar^2}{\sin^2\theta} - {W'}_\theta^2 = -l(l+1) \hbar^2; \\
i\hbar \frac{1}{2m r^2} \frac{\partial}{\partial r}\left( r^2 W'_r \right) - \frac{1}{2m r^2}l(l+1)\hbar^2 = V(r) - E + {W'_r}^2 \] with the boundary condition $ W \to W_{cl} $ if $\hbar \to 0, W_\phi(\phi + 2\pi) = W_\phi(\phi)$.
The first equation has the solution $W_\phi(\phi) = \frac{\hbar}{i}\log \cos m_s\phi$ with $m_s = 0, \pm 1, \pm 2, \cdots$.
The second equation is a Riccati equation, that can be rewritten to a linear second order differential equation. The resultant equation is just the Bessel equation. Meanwhile, if we do a change of variable $W'_\theta = p_\theta = i\hbar ( u_\theta + \frac{1}{2}\cot\theta)$, it becomes, \[
u'_\theta = u_\theta^2 + \frac{1-4m_s^2}{4\sin^2\theta} + (l+ \frac{1}{2})^2
\] The third equation is also a Riccati equation. With a change of variable $W'_r = p_r = \frac{i\hbar}{2m}( u_r + \frac{1}{r})$, the equation becomes \[
u'_r = u_r^2 + \frac{4 m^2}{\hbar^2} (E - V(r)) - \frac{2 m l(l+1)}{r^2}
\]
Bohr-Sommerfeld quantization
Recall the angular action $J_q$ is defined as \[J_q = \oint_C \mathrm{d}q \; p_q
\] where $p_q = \frac{\partial}{\partial q}W$. The value of $J_q$ only depends on the analytic structure of the solution $p_q$ of the Hamilton-Jacobi equation. The poles are the "good" singular points that gives finite result for $J_q$. The "bad" singularities include the branch points and the essential singular points are called the critical points. $J_q$ gains contributions from both poles and critical points.
Non-linear first order differential equations $\frac{\mathrm{d}w}{\mathrm{d}z} = F(z, w)$ where $z \in \mathbb{C}$, and $F(z, w)$ is locally analytic can have "internal singularities" or the movable singular points. The location of the singular points depends on integration constant. A leading example is the equation \[
\frac{\mathrm{d}w}{\mathrm{d}z} = w^2 \quad \Rightarrow \quad w(z) = \frac{1}{c - z}. \] Riccati equations do not have movable critical singularities. In fact, in 1884 W. Fuchs showed that Riccati equations are the only class of first order differential equations without movable critical singular points [2].
Riccati equation $w' = w^2 + f(z)$ admits a solution $w(z) = -\frac{v'(z)}{v(z)}$ where $v(z)$ is a solution of the second order linear equation $v'' + f(z) v = 0$. Then, the original solution $w(z) = - \frac{v'(z)}{v(z)} = - \frac{g(z) + (z-z_1) g'(z)}{(z-z_1)g(z)} = -\frac{1}{z-z_1} - \frac{g'(z)}{g(z)} = -\sum_i \frac{1}{z-z_i} - \varphi(z)$ ($\varphi(z)$ only has critical singularities ) can only have first order non-movable singularities. Then the angular action is quantized with the famous Bohr-Sommerfeld condition \[
J_q = 2 \pi i n (-1) i \hbar = 2 \pi n \hbar + C, \quad n = 0,1,2,... \] where $n$ is the number of single poles lying on the real axis of the solution $w(z)$ hence $p_q$. $C$ is constant. With any luck, the solution has no critical singularities and then $C = 0$. A nice feature of this analysis is the manifest of the correspondence principle. In the classic mechanics, $p_q = \sqrt{2m (E-V)}$ that has a branch cut. Whereas in quantum theory in the classical limit $n \to \infty$, the poles on the real axis behaves like the branch cut. Along this line, the ground state is the solute with the minimal pole.
- - -
[1]:http://eqworld.ipmnet.ru/en/solutions/ode/ode0123.pdf
[2]:http://www.encyclopediaofmath.org/index.php/Painlev%C3%A9-type_equations#References
[3]:http://www.risc.jku.at/publications/download/risc_2773/prep2.pdf
[4]:Robert A. Leacock and Michael J. Padgett, Hamilton-Jacobi Theory and the Quantum Action Variable, Phys. Rev. Lett. 50, 3–6 (1983) URL: http://prl.aps.org/abstract/PRL/v50/i1/p3_1
[5]:Marco Roncadelli and L. S. Schulma, Quantum Hamilton-Jacobi Theory, PRL, 99, 170406 (2007), arxiv URL: http://arxiv.org/pdf/0712.0307v1.pdf
[6]: https://michaelberryphysics.files.wordpress.com/2013/07/berry023.pdf
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