#### 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

\partial_t S + \frac{1}{2m}(\nabla S)^2 + V = \frac{i\hbar}{2m}\nabla^2_qS

\] Let's call the

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

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

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

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,

[6]: https://michaelberryphysics.files.wordpress.com/2013/07/berry023.pdf

*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.

#### 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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