Hamilton–Jacobi Theory Preview

Hamilton–Jacobi theory is the classical language most directly adapted to quantum phase. It packages a family of classical trajectories into an action function $S$, then recovers momenta from gradients of that function. Semiclassical quantum mechanics uses the dimensionless phase $S/\hbar$ while adding amplitudes, interference, boundary conditions, and caustic corrections that classical Hamilton–Jacobi theory does not supply.

The full classical construction belongs to Hamilton–Jacobi Theory. WKB Approximation owns the one-dimensional approximation method, and Semiclassical Propagator owns the full Van Vleck treatment. This page owns the dynamics bridge: why an action appears as a quantum phase, what approximation is involved, and where the identification fails. Classical Action and Quantum Phase uses that bridge as an operational dictionary for wavefronts, energy and momentum gradients, interference, and gauge-covariant motion.

Hamilton’s Principal Function

Let a classical path $q_\gamma(t)$ connect the endpoints $(q_i,t_i)$ and $(q_f,t_f)$. Hamilton’s principal function for that trajectory branch is the on-shell action

$$S_\gamma(q_f,t_f;q_i,t_i) = \int_{t_i}^{t_f} dt\, L(q_\gamma,\dot q_\gamma,t).$$

The subscript matters. More than one classical trajectory may connect the same endpoints, so the principal function can have several branches.

Varying an on-shell action leaves only endpoint terms:

$$\begin{aligned} \delta S_\gamma &= p_f\mathbin{\cdot}\delta q_f - H_f\,\delta t_f \\ &\quad- p_i\mathbin{\cdot}\delta q_i + H_i\,\delta t_i. \end{aligned}$$

Consequently,

$$p_f = \nabla_{q_f}S_\gamma, \qquad p_i = - \nabla_{q_i}S_\gamma,$$

and

$$\frac{\partial S_\gamma}{\partial t_f} = -H_f, \qquad \frac{\partial S_\gamma}{\partial t_i} = H_i.$$

These identities make $S_\gamma$ a generating function for the classical endpoint map. Its first derivatives encode endpoint momenta; its second derivatives encode how neighboring trajectories respond to changed endpoints.

The Hamilton–Jacobi Equation

Hold the initial endpoint fixed and abbreviate a smooth branch by $S(q,t)$. Since

$$p = \nabla S,$$

and

$$\frac{\partial S}{\partial t} = -H(q,p,t),$$

the principal function obeys

$$\frac{\partial S}{\partial t} + H\left( q,\nabla S,t \right) = 0.$$

For a particle with

$$H(q,p,t) = \frac{p^2}{2m} + V(q,t),$$

this becomes

$$\frac{\partial S}{\partial t} + \frac{(\nabla S)^2}{2m} + V = 0.$$

This is a nonlinear first-order partial differential equation. Its characteristics are Hamiltonian trajectories. A smooth solution describes a family of trajectories locally, but folds of that family can make $S$ multivalued or singular in a chosen coordinate representation.

For a time-independent Hamiltonian, the separated form

$$S(q,t) = W(q;E)-Et$$

gives Hamilton’s characteristic equation

$$H\left( q,\nabla W \right) = E.$$

In one dimension,

$$\frac{1}{2m} \left( \frac{dW}{dx} \right)^2 + V(x) = E,$$

so

$$\frac{dW}{dx} = \pm p(x), \qquad p(x) = \sqrt{2m\left[E-V(x)\right]}.$$

The two signs are distinct local momentum branches. Their action integrals become the two oscillatory WKB phases.

Free-Particle Check

For a free particle in $d$ dimensions and $T=t_f-t_i\gt0$, the unique straight path has

$$S_{\rm free} = \frac{m\lvert q_f-q_i\rvert^2}{2T}.$$

Its final endpoint derivative is

$$\nabla_{q_f}S_{\rm free} = \frac{m(q_f-q_i)}{T} = p_f,$$

while

$$\frac{\partial S_{\rm free}}{\partial t_f} = - \frac{m\lvert q_f-q_i\rvert^2}{2T^2} = - \frac{p_f^2}{2m}.$$

It therefore satisfies the free Hamilton–Jacobi equation. The exact free-particle kernel is

$$\begin{aligned} K_{\rm free} &= \left( \frac{m}{2\pi i\hbar T} \right)^{d/2} \\ &\quad\times \exp\left[ \frac{i}{\hbar} S_{\rm free} \right]. \end{aligned}$$

This example displays the semiclassical architecture without approximation: the principal function supplies the phase, while the prefactor supplies normalization and spreading. Quadratic actions are special because the fluctuation expansion is Gaussian and terminates exactly.

Phase of the Wavefunction

Consider the scalar-potential Schrödinger equation

$$i\hbar\frac{\partial\psi}{\partial t} = \left[ - \frac{\hbar^2}{2m}\nabla^2 + V \right]\psi.$$

On a region where $\psi$ does not vanish, write

$$\psi(q,t) = A(q,t) \exp\left[ \frac{i}{\hbar}S(q,t) \right],$$

with real $A\geq0$ and real phase function $S$. Substitution and separation into real and imaginary parts give two exact equations.

The real part is

$$\frac{\partial S}{\partial t} + \frac{(\nabla S)^2}{2m} + V + Q = 0,$$

where

$$Q = - \frac{\hbar^2}{2m} \frac{\nabla^2A}{A}.$$

The term $Q$ is often called the quantum-potential or quantum-pressure correction. The name does not require any particular interpretation of quantum mechanics; here it is simply the exact amplitude-dependent term produced by the Schrödinger equation.

The imaginary part is the transport equation

$$\frac{\partial A^2}{\partial t} + \nabla\mathbin{\cdot} \left( A^2\frac{\nabla S}{m} \right) = 0.$$

Since $\rho=A^2$, this is the probability continuity equation with current

$$j = \rho\frac{\nabla S}{m}.$$

The wavefunction phase therefore resembles a classical principal function, but the resemblance is not an identity. The amplitude feeds back into the exact phase equation through $Q$. At nodes, $A=0$ and this local decomposition becomes singular. For a superposition of several semiclassical branches, one should not expect the phase of the total sum to equal any one branch action.

Controlled Semiclassical Reduction

Suppose $A$ varies on a length scale $L_A$ and the local phase gradient has magnitude $p=\lvert\nabla S\rvert$. Then

$$\lvert Q\rvert \sim \frac{\hbar^2}{2mL_A^2},$$

whereas the kinetic term is

$$\frac{p^2}{2m}.$$

Their ratio scales as

$$\frac{\lvert Q\rvert}{p^2/(2m)} \sim \left( \frac{\hbar}{pL_A} \right)^2.$$

Thus a local condition for the classical Hamilton–Jacobi equation is

$$\frac{\hbar}{pL_A} \ll 1,$$

together with control of time variation and any other scales in the problem. This is the statement that the amplitude changes slowly compared with the local de Broglie phase.

Dropping $Q$ at leading order gives

$$\frac{\partial S_0}{\partial t} + \frac{(\nabla S_0)^2}{2m} + V = 0.$$

The transport equation then evolves the leading amplitude along the classical flow generated by $S_0$. Phase and amplitude play different roles:

• Hamilton–Jacobi theory determines the leading eikonal phase;
• transport or fluctuation equations determine the amplitude;
• boundary and matching conditions determine which branches occur;
• coherent addition of branches determines interference.

The condition fails near turning points, nodes, caustics, sharp boundaries, and regions where the amplitude changes on the scale of a wavelength. It can also fail after long evolution when classical flow creates fine structure, even if it held initially.

WKB Preview

For a stationary one-dimensional problem, write

$$\psi(x) = A(x) \exp\left[ \frac{i}{\hbar}W(x) \right].$$

At leading order,

$$\frac{[W'(x)]^2}{2m} + V(x) = E,$$

which is the time-independent Hamilton–Jacobi equation. Hence

$$W_\pm(x) = \pm \int^x dx'\,p(x').$$

The next-order transport equation is

$$\frac{d}{dx} \left[ A^2(x)p(x) \right] = 0,$$

so in a classically allowed region,

$$A(x) \propto \frac{1}{\sqrt{\lvert p(x)\rvert}}.$$

The two local WKB branches therefore have the form

$$\begin{aligned} \psi_{\rm WKB}(x) &\approx \frac{C_+}{\sqrt{p(x)}} \exp\left[ \frac{i}{\hbar} \int^x p(x')\,dx' \right] \\ &\quad+ \frac{C_-}{\sqrt{p(x)}} \exp\left[ - \frac{i}{\hbar} \int^x p(x')\,dx' \right]. \end{aligned}$$

Hamilton–Jacobi theory explains the phase. It does not by itself provide the $1/\sqrt{p}$ amplitude, turning-point connection formulas, quantization conditions, or tunneling continuation. Those belong to WKB Approximation and its neighboring method pages.

Semiclassical Propagator Link

The same structure appears in a propagator. Schematically,

$$\begin{aligned} K_{\rm sc}(q_f,t_f;q_i,t_i) &\sim \sum_\gamma \mathcal A_\gamma \\ &\quad\times \exp\left[ \frac{i}{\hbar}S_\gamma - \frac{i\pi}{2}\nu_\gamma \right]. \end{aligned}$$

Each classical trajectory branch $\gamma$ supplies a principal function $S_\gamma$. The prefactor $\mathcal A_\gamma$ contains stability information, commonly through a Van Vleck determinant, and $\nu_\gamma$ tracks caustic phases. Hamilton–Jacobi theory determines the branch phases and endpoint momenta but not the complete quantum kernel.

The action also composes by stationary matching. For an intermediate time $t_m$,

$$\begin{aligned} S(q_f,t_f;q_i,t_i) &= \underset{q_m}{\operatorname{stat}} \Big[ S(q_f,t_f;q_m,t_m) \\ &\qquad\qquad+ S(q_m,t_m;q_i,t_i) \Big]. \end{aligned}$$

Stationarity with respect to $q_m$ equates the momentum arriving from the first segment with the momentum leaving on the second. The quantum propagator composes by integrating over $q_m$; stationary phase turns that integral into classical momentum matching at leading order.

Stationary Phase owns the asymptotic formula. Semiclassical Propagator Preview develops the trajectory sum, determinant, Maslov phase, and caustic caveats.

What the Preview Does Not Claim

• A classical action phase does not turn a wavefunction into a classical probability distribution.
• One Hamilton–Jacobi branch does not replace a coherent sum over several branches.
• The leading phase does not determine normalization or probability transport.
• Real classical trajectories do not capture ordinary forbidden-region tunneling without analytic continuation or complex trajectories.
• A caustic divergence belongs to the approximation, not to the exact wavefunction.
• Small $\hbar$ is meaningful only relative to the action and variation scales held fixed.
• Writing $j=\rho\nabla S/m$ does not by itself select an interpretation of quantum mechanics.
• With electromagnetic vector potentials, $\nabla S$ is canonical momentum; the kinetic momentum is gauge-covariantly shifted.

Canonical Boundaries

Question	Canonical page
How is Hamilton–Jacobi theory derived and used classically?	Hamilton–Jacobi Theory
Why is classical action stationary?	Action Principles
Why do action differences become quantum phases?	Action and Phase
How is stationary phase calculated?	Stationary Phase
How is one-dimensional WKB used in practice?	WKB Approximation
How are trajectory stability and caustic phases included?	Semiclassical Propagator

Common Mistakes

• Confusing Hamilton’s principal function $S(q,t)$ with the time-independent characteristic function $W(q;E)$.
• Identifying the exact wavefunction phase with a classical action while omitting $Q$.
• Forgetting that $S_\gamma$ is branch-dependent when several classical paths connect the endpoints.
• Treating $\nabla S$ as a globally smooth momentum field across nodes and caustics.
• Keeping the WKB phase while discarding the transport amplitude.
• Interpreting $1/\sqrt{p}$ at a turning point as a physical divergence.
• Assuming the Hamilton–Jacobi equation alone imposes quantum boundary conditions or quantizes energy.
• Replacing a sum of amplitudes by a classical mixture of branch probabilities.

References

• W. R. Hamilton, “On a General Method in Dynamics,” Philosophical Transactions of the Royal Society of London 124, 247–308, 1834.
• H. Goldstein, C. Poole, and J. Safko, Classical Mechanics, 3rd ed., Addison-Wesley, 2002.
• V. I. Arnold, Mathematical Methods of Classical Mechanics, 2nd ed., Springer, 1989.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Butterworth-Heinemann, 1977.
• M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics,” Reports on Progress in Physics 35, 315–397, 1972, doi:10.1088/0034-4885/35/1/306.
• R. G. Littlejohn, “The semiclassical evolution of wave packets,” Physics Reports 138, 193–291, 1986, doi:10.1016/0370-1573(86)90103-1.
• J. H. Van Vleck, “The correspondence principle in the statistical interpretation of quantum mechanics,” Proceedings of the National Academy of Sciences 14, 178–188, 1928, doi:10.1073/pnas.14.2.178.

Exercises

1. Derive the final-endpoint Hamilton–Jacobi equation from the on-shell endpoint variation.

Solution

For a classical path,

$$\delta S = p_f\mathbin{\cdot}\delta q_f - H_f\,\delta t_f$$

when the initial endpoint is fixed. Therefore

$$\nabla_{q_f}S = p_f,$$

and

$$\frac{\partial S}{\partial t_f} = -H_f.$$

Eliminate $p_f$ using $p_f=\nabla_{q_f}S$:

$$\frac{\partial S}{\partial t_f} + H\left( q_f,\nabla_{q_f}S,t_f \right) = 0.$$

This is the Hamilton–Jacobi equation for the chosen smooth trajectory branch.

2. Verify both endpoint momentum signs for the free-particle principal function.

Solution

With

$$S_{\rm free} = \frac{m\lvert q_f-q_i\rvert^2}{2T},$$

differentiate with respect to the final endpoint:

$$\nabla_{q_f}S_{\rm free} = \frac{m(q_f-q_i)}{T} = p.$$

Differentiating with respect to the initial endpoint gives

$$\nabla_{q_i}S_{\rm free} = - \frac{m(q_f-q_i)}{T} = -p.$$

Hence

$$p_f = \nabla_{q_f}S_{\rm free}, \qquad p_i = - \nabla_{q_i}S_{\rm free}.$$

The initial and final mechanical momenta are equal for free motion, while the generating-function derivatives carry opposite endpoint signs.

3. Starting from $\psi=Ae^{iS/\hbar}$, derive the exact phase and transport equations.

Solution

The time derivative is

$$\frac{\partial\psi}{\partial t} = e^{iS/\hbar} \left( \frac{\partial A}{\partial t} + \frac{iA}{\hbar} \frac{\partial S}{\partial t} \right).$$

The Laplacian is

$$\begin{aligned} \nabla^2\psi &= e^{iS/\hbar} \Bigg[ \nabla^2A + \frac{2i}{\hbar} \nabla A\mathbin{\cdot}\nabla S \\ &\qquad+ \frac{iA}{\hbar}\nabla^2S - \frac{A}{\hbar^2} (\nabla S)^2 \Bigg]. \end{aligned}$$

Insert these expressions into the Schrödinger equation and divide by $e^{iS/\hbar}$. The real part gives

$$\frac{\partial S}{\partial t} + \frac{(\nabla S)^2}{2m} + V - \frac{\hbar^2}{2m} \frac{\nabla^2A}{A} = 0.$$

The imaginary part gives

$$\frac{\partial A}{\partial t} + \frac{1}{m} \nabla A\mathbin{\cdot}\nabla S + \frac{A}{2m}\nabla^2S = 0.$$

Multiplying the latter by $2A$ and using product rules yields

$$\frac{\partial A^2}{\partial t} + \nabla\mathbin{\cdot} \left( A^2\frac{\nabla S}{m} \right) = 0.$$

4. Estimate when the quantum correction is small if $A$ varies over $L_A$ and the local momentum is $p$.

Solution

Dimensional estimation gives

$$\frac{\nabla^2A}{A} \sim \frac{1}{L_A^2},$$

so

$$\lvert Q\rvert \sim \frac{\hbar^2}{2mL_A^2}.$$

Relative to the kinetic scale,

$$\frac{\lvert Q\rvert}{p^2/(2m)} \sim \frac{\hbar^2}{p^2L_A^2} = \left( \frac{\hbar}{pL_A} \right)^2.$$

The correction is locally small when $pL_A\gg\hbar$. This criterion fails when $p$ approaches zero or when the amplitude varies rapidly.

5. Derive the leading one-dimensional WKB amplitude from probability transport.

Solution

For a stationary branch, $\partial_tA^2=0$ and $W'(x)=p(x)$. The transport equation becomes

$$\frac{d}{dx} \left[ A^2(x)\frac{p(x)}{m} \right] = 0.$$

Thus $A^2p$ is constant along the branch. Away from a turning point,

$$A(x) \propto \frac{1}{\sqrt{\lvert p(x)\rvert}}.$$

The divergence as $p\to0$ signals failure of the local WKB representation and the need for a turning-point uniform approximation.

6. Show that stationary composition of two principal functions enforces momentum matching.

Solution

Let

$$\Phi(q_m) = S_2(q_f,t_f;q_m,t_m) + S_1(q_m,t_m;q_i,t_i).$$

The initial-endpoint rule for $S_2$ gives

$$\nabla_{q_m}S_2 = -p_m^{(2)},$$

while the final-endpoint rule for $S_1$ gives

$$\nabla_{q_m}S_1 = p_m^{(1)}.$$

Therefore

$$\nabla_{q_m}\Phi = p_m^{(1)}-p_m^{(2)}.$$

The stationarity condition $\nabla_{q_m}\Phi=0$ requires

$$p_m^{(1)} = p_m^{(2)}.$$

The two segments join into one differentiable classical trajectory. In the quantum composition integral, this same condition is selected by stationary phase.