Connection to the Schrodinger Equation

Every article in this chapter has been walking toward this page. Hamilton-Jacobi theory recast a particle's motion as a field problem: one function $S(q,t)$ spread over configuration space, its level surfaces sweeping forward like wavefronts, its gradient steering trajectories like rays. The Optical-Mechanical Analogy said what that structure resembles; WKB Approximation and the Eikonal Connection built the expansion machinery. Here we close the loop and state the relationship exactly: the Hamilton-Jacobi equation is the $\hbar\to0$ limit of the Schrödinger equation. Classical mechanics stands to wave mechanics precisely as ray optics stands to wave optics — a shadow theory, exact in the limit of vanishing wavelength. The connection runs productively in both directions. Forward, it is history: in 1926 Schrödinger, armed with de Broglie's $\lambda=h/p$ and Hamilton's analogy, hunted for the wave equation whose eikonal limit is the Hamilton-Jacobi equation — and wrote down the equation that now carries his name. Backward, it is mathematics: substitute $\psi=A\,e^{iS/\hbar}$ into the Schrödinger equation and it splits, exactly, into a continuity equation for $|\psi|^2$ and a Hamilton-Jacobi equation carrying one extra term — the quantum potential $Q=-\frac{\hbar^2}{2m}\frac{\nabla^2A}{A}$ . Every wave phenomenon — interference, tunneling, zero-point motion — lives inside $Q$ ; the classical world is what remains where $Q$ is negligible. This page is the classical-mechanics side of the bridge. Its quantum-field-theory side — how the action reappears in Feynman's path integral and beyond — is the companion capstone The Action Principle as Bridge to Quantum Field Theory.

Key takeaways

Substituting the polar form $\psi=A\,e^{iS/\hbar}$ into the time-dependent Schrödinger equation splits it exactly into two real equations: a quantum Hamilton-Jacobi equation and a continuity equation.
The real part reads $\partial_tS+\frac{(\nabla S)^2}{2m}+V+Q=0$ with quantum potential $Q=-\frac{\hbar^2}{2m}\frac{\nabla^2A}{A}$ ; as $\hbar\to0$ (more precisely $Q\to0$ ), it becomes the classical Hamilton-Jacobi equation and $S$ becomes Hamilton's principal function.
The imaginary part is $\partial_t\rho+\nabla\cdot(\rho\nabla S/m)=0$ : probability flows along the classical velocity field $\mathbf{v}=\nabla S/m$ .
Schrödinger found his equation by running this logic in reverse — demanding a wave equation whose short-wavelength (eikonal) limit is Hamilton-Jacobi theory, with de Broglie's $\lambda=h/p$ setting the wavelength.
The limit is singular, not smooth: $\hbar$ multiplies the highest derivative, so classicality fails at nodes, turning points, and caustics no matter how small $\hbar$ is.
Ehrenfest's theorem is the operational correspondence: quantum expectation values obey Newton-like equations exactly, because the mean quantum force $\langle\nabla Q\rangle$ vanishes identically.
Key idea: writing $\psi=A\,e^{iS/\hbar}$ turns the Schrödinger equation into the classical Hamilton-Jacobi equation for the phase $S$ , corrected by a single term — the quantum potential $Q$ of order $\hbar^2$ ; the classical world is what remains where $Q$ is negligible.

The core insight

Writing $\psi = A\,e^{iS/\hbar}$ turns the Schrödinger equation into the classical Hamilton-Jacobi equation for the phase $S$ , corrected by a term of order $\hbar^2$ (the quantum potential); the classical world is what remains as $\hbar\to0$ .

Where this lives today

The Madelung-Bohm machinery earns its keep computationally. "Quantum trajectory" methods in chemical physics integrate the Madelung equations — classical trajectories plus $Q$ — to simulate proton transfer and nonadiabatic dynamics at a fraction of the cost of full wave-packet propagation; in strong-field and attosecond physics, Bohmian trajectories are used to visualize and time tunneling ionization as electrons exit the laser-tilted barrier. The semiclassical propagator built from $e^{iS_{\mathrm{cl}}/\hbar}$ and the Van Vleck determinant — the time-dependent face of this page's limit — powers spectral calculations in molecules and the periodic-orbit theory of quantum chaos. The deepest modern shift concerns why the world looks classical at all. The $\hbar\to0$ story explains why quantum corrections to a given packet's motion are small; it does not explain why macroscopic objects never show up in superpositions of distant places (nothing in equations 3-4 forbids a two-lobed $\rho$ with enormous $Q$ between the lobes). The missing ingredient is decoherence: environmental scattering measures position continuously, destroying phase coherence between macroscopically distinct branches on timescales as short as $10^{-20}$ s for a dust grain in air. Classicality, on the current understanding, is the joint work of the eikonal limit (this page) and einselection by the environment — with experiments on levitated nanoparticles and matter-wave interference of $10^5$ -amu molecules now probing the crossover directly. The path-integral perspective on the same limit, and its continuation into quantum field theory, is the companion capstone The Action Principle as Bridge to Quantum Field Theory.

The physical picture

Set two theories side by side, each with a wave version and a ray version. On the optics side: Maxwell's wave theory above, geometrical ray optics below, connected by the limit $\lambda\to0$ . On the mechanics side: below sits classical Hamilton-Jacobi theory — a theory of wavefronts $S=\mathrm{const}$ and rays $\mathbf{p}=\nabla S$ , suspiciously shaped like a ray optics. Hamilton drew the horizontal arrow between the two ray theories in 1834. The empty box above classical mechanics sat unlabeled for ninety years: ray theory of what wave theory? Schrödinger's 1926 move was to fill the box by construction — demand the wave equation whose $\hbar\to0$ shadow is Hamilton-Jacobi, and let de Broglie's $\lambda=h/p$ fix the scale.

Figure

The commutative square Schrodinger completed — wave optics over ray optics, wave mechanics over Hamilton-Jacobi mechanics, with the short-wavelength limit running down each column

What does the completed square say a classical particle is? A wave packet whose internal wavelength is far too short to notice. The phase $S/\hbar$ winds through millions of cycles across any region you can resolve, so interference is invisibly fine-grained; the packet's centroid follows the rays of $S$ ; and the packet's shape rides the continuity equation. All specifically quantum behavior is bundled into one object with a name and a formula — the quantum potential $Q=-\frac{\hbar^2}{2m}\frac{\nabla^2A}{A}$ , which measures the curvature of the amplitude. Where the wave is smooth, $Q\approx0$ and Hamilton-Jacobi rules; where the amplitude bends sharply — nodes, slit edges, turning points, barrier walls — $Q$ surges, and there live interference fringes, diffraction, tunneling. One warning shapes the whole subject: the limit is singular. In the Schrödinger equation $\hbar$ multiplies the highest spatial derivative and the time derivative; setting $\hbar=0$ outright deletes the dynamics rather than simplifying it, the same way setting viscosity to zero deletes the boundary layer rather than the drag. Classicality is therefore not a term-by-term Taylor limit but a short-wavelength asymptotic limit — accurate in open smooth regions, always failing somewhere. The art of semiclassics is knowing where.

The exact split

Everything follows from one substitution. Take the time-dependent Schrödinger equation — here simply given, as the wave theory whose classical limit we are auditing:

i\hbar\,\frac{\partial\psi}{\partial t}=-\frac{\hbar^2}{2m}\nabla^2\psi+V(\mathbf{r},t)\,\psi \qquad (1)

where: $\psi(\mathbf{r},t)$ — the complex wavefunction; $\hbar=1.055\times10^{-34}\,\mathrm{J\,s}$ . Note where $\hbar$ sits: on the time derivative and on the Laplacian — the seat of the singular limit. Write the wavefunction in polar (Madelung) form, trading one complex field for two real ones:

\psi=A(\mathbf{r},t)\,\exp\!\Big(\frac{i}{\hbar}S(\mathbf{r},t)\Big),\qquad \rho=A^2=|\psi|^2 \qquad (2)

where: $A\ge0$ — the amplitude; $S$ — the phase, with dimensions of action, whose classical destiny is Hamilton's principal function. Unlike the WKB ansatz of WKB Approximation and the Eikonal Connection (complex $S$ , then truncated in powers of $\hbar$ ), the split (2) is an exact change of variables — nothing will be dropped in the next two equations. Substituting (2) into (1) and separating real and imaginary parts (Derivation A) yields the pair of Madelung equations. The real part is Hamilton-Jacobi with one intruder:

\frac{\partial S}{\partial t}+\frac{(\nabla S)^2}{2m}+V+Q=0,\qquad Q\equiv-\frac{\hbar^2}{2m}\frac{\nabla^2A}{A} \qquad (3)

where: $Q$ — the quantum potential (Bohm potential): an effective potential energy generated by curvature of the amplitude itself. Everything non-classical is here. $Q$ is independent of the strength of $\psi$ (double $A$ and $Q$ is unchanged) but exquisitely sensitive to its shape — it diverges near nodes, which is exactly where classical reasoning dies. Set $Q=0$ and equation (3) is the Hamilton-Jacobi equation of Hamilton-Jacobi Equation, with $\mathbf{p}=\nabla S$ . The imaginary part contains no $\hbar$ at all once written in terms of $\rho$ :

\frac{\partial\rho}{\partial t}+\nabla\!\cdot\!\Big(\rho\,\frac{\nabla S}{m}\Big)=0 \qquad (4)

where: $\mathbf{v}=\nabla S/m$ — the velocity field of the probability fluid. Probability is conserved and flows along the rays of the phase — a statement that survives the classical limit untouched, becoming the transport of a density of classical trajectories. Together (3)+(4) are fully equivalent to (1): quantum mechanics as a classical-looking fluid with one extra internal pressure term. For a stationary state, $S=W(\mathbf{r})-Et$ and (3) reduces to $\frac{(\nabla W)^2}{2m}+V+Q=E$ — the time-independent Hamilton-Jacobi equation of Time-Independent Hamilton-Jacobi Equation, again $Q$ -corrected. The classical limit in all cases is the regime

|Q|\ll|E-V| \quad\Longleftrightarrow\quad \lambda=\frac{2\pi\hbar}{|\nabla S|}\ \text{short and slowly varying} \qquad (5)

where: the equivalence with the WKB validity condition is no accident — $Q$ small is precisely what truncating the $\hbar$ -expansion requires. The classical limit is a regime of the solution, not merely a value of a constant.

Figure

One complex equation becomes two real ones — the Madelung split routes the phase into Hamilton-Jacobi dynamics and the amplitude into probability transport

Derivation AThe Madelung split of the Schrodinger equation

We are performing the exact decomposition of the time-dependent Schrödinger equation under $\psi=Ae^{iS/\hbar}$ into its real part (quantum Hamilton-Jacobi) and imaginary part (continuity). The strategy is direct substitution and careful bookkeeping of which terms are real and which imaginary; the only subtlety is keeping every term until the final sort, since the split — unlike WKB — makes no approximation. Watch how the $1/\hbar^2$ from $(\nabla S)^2$ cancels the explicit $\hbar^2$ of the kinetic operator, leaving a genuinely classical term.

Step 1. Time side. With both $A$ and $S$ time-dependent:

i\hbar\,\partial_t\psi=\big(i\hbar\,\partial_tA-A\,\partial_tS\big)e^{iS/\hbar}

Step 2. Space side. Two applications of the gradient give

-\frac{\hbar^2}{2m}\nabla^2\psi=\Big[-\frac{\hbar^2}{2m}\nabla^2A+\frac{A(\nabla S)^2}{2m}-\frac{i\hbar}{2m}\big(2\nabla A\cdot\nabla S+A\nabla^2S\big)\Big]e^{iS/\hbar}

Why the middle term is the important one: $(\nabla S)^2/\hbar^2$ from differentiating the phase twice meets the prefactor $\hbar^2$ and survives the classical limit — it is the kinetic energy. The first term keeps its $\hbar^2$ and will become $Q$ ; the last, carrying $i$ , is headed for the continuity equation.

Step 3. Equate, divide by $e^{iS/\hbar}$ , and collect the REAL terms:

-A\,\partial_tS=-\frac{\hbar^2}{2m}\nabla^2A+\frac{A(\nabla S)^2}{2m}+VA

Divide by $A$ :

\partial_tS+\frac{(\nabla S)^2}{2m}+V-\frac{\hbar^2}{2m}\frac{\nabla^2A}{A}=0

— equation (3), the Hamilton-Jacobi equation plus the quantum potential $Q$ .

Step 4. Collect the IMAGINARY terms:

\hbar\,\partial_tA=-\frac{\hbar}{2m}\big(2\nabla A\cdot\nabla S+A\nabla^2S\big)

Cancel $\hbar$ , multiply by $2A$ , and recognize total derivatives:

\partial_t(A^2)+\nabla\!\cdot\!\Big(A^2\,\frac{\nabla S}{m}\Big)=0

— equation (4), conservation of probability along $\mathbf{v}=\nabla S/m$ .

Step 5. Classical limit. If the amplitude varies on classical length scales $L$ while the phase varies on $\lambda\ll L$ , then $Q\sim\hbar^2/(mL^2)$ is smaller than the kinetic term $\sim p^2/m$ by $(\lambda/L)^2$ : equation (3) collapses to the classical Hamilton-Jacobi equation and $S$ is Hamilton's principal function. Verified: matches Goldstein §10.8 and Madelung's 1927 hydrodynamic equations; the continuity equation reproduces the standard probability current $\mathbf{j}=\frac{\hbar}{m}\mathrm{Im}(\psi^*\nabla\psi)=\rho\nabla S/m$ . Dimensions: $[Q]=$ energy, since $\hbar^2/(m\,\mathrm{length}^2)=$ energy.

Go deeper · Derivation B · 1 minSchrodinger's 1926 construction run forward

Derivation BSchrodinger's 1926 construction run forward

We are reconstructing the historical inference: given the time-independent Hamilton-Jacobi equation and de Broglie's postulate, find the wave equation whose eikonal limit is classical mechanics. The strategy mirrors optics — a medium's ray equation plus a wavelength law determine its wave equation — and the honest conclusion is a construction, not a proof: the simplest wave equation consistent with the limit, whose ultimate justification is experiment. Watch how little freedom there is once the wavelength is fixed.

Step 1. The mechanical "medium". For energy $E$ in potential $V$ , the ray theory is $|\nabla W|^2=2m(E-V)$ : an effective index proportional to the local momentum $p(\mathbf{r})=\sqrt{2m(E-V)}$ — the dictionary of Optical-Mechanical Analogy.

Step 2. The wavelength law. De Broglie (1924): a particle of momentum $p$ carries wavelength $\lambda=h/p$ , hence local wavenumber

k(\mathbf{r})=\frac{2\pi}{\lambda}=\frac{p(\mathbf{r})}{\hbar}=\frac{\sqrt{2m(E-V(\mathbf{r}))}}{\hbar}

Step 3. The generic stationary wave equation. Any scalar wave with position-dependent wavenumber obeys, in the regime where rays make sense, the Helmholtz form $\nabla^2\psi+k^2(\mathbf{r})\,\psi=0$ . Insert Step 2's $k$ :

\nabla^2\psi+\frac{2m}{\hbar^2}\big(E-V(\mathbf{r})\big)\psi=0

Step 4. Rearrange:

-\frac{\hbar^2}{2m}\nabla^2\psi+V\psi=E\psi

— the time-independent Schrödinger equation, obtained with no input beyond Hamilton-Jacobi theory and $\lambda=h/p$ . Why this is a construction and not a derivation: Step 3 chose the simplest wave equation with the right rays; infinitely many others share the same eikonal limit (add any term vanishing as $\hbar\to0$ ). Schrödinger's bet was minimality. The bet was settled by the hydrogen spectrum in the same 1926 paper, and by electron diffraction a year later (worked example 2).

Step 5. Consistency check backwards: insert $\psi=Ae^{iW/\hbar}$ into Step 4's equation and expand — order $\hbar^0$ returns $|\nabla W|^2=2m(E-V)$ , confirming the eikonal limit is exactly the classical theory we started from. The square in the hero figure commutes. Verified: the construction follows Schrödinger's second 1926 communication (Ann. Phys. 79, 489), which cites Hamilton explicitly; the eikonal check matches the WKB order- $\hbar^0$ result of the sibling article.

Go deeper · Derivation C · 1 minEhrenfest's theorem — the mean quantum force vanishes

Derivation CEhrenfest's theorem — the mean quantum force vanishes

We are proving the operational correspondence principle: quantum expectation values obey Newton-like equations, because the quantum potential exerts zero average force on any normalizable state. The strategy uses the Madelung fluid: compute the rate of change of mean position and mean momentum by moving derivatives through the continuity equation, then show $\int\rho\,\nabla Q\,\mathrm{d}^3r=0$ identically — quantum corrections cancel in the mean, not merely approximately. We work in 1D for clarity of the final integration by parts.

Step 1. Mean position. With $\langle x\rangle=\int x\rho\,\mathrm{d}x$ and continuity $\partial_t\rho=-\partial_x(\rho v)$ , $v=\partial_xS/m$ :

\frac{\mathrm{d}\langle x\rangle}{\mathrm{d}t}=-\int x\,\partial_x(\rho v)\,\mathrm{d}x=\int\rho v\,\mathrm{d}x=\frac{\langle p\rangle}{m}

after one integration by parts (boundary terms vanish for normalizable $\rho$ ).

Step 2. Mean momentum. The momentum density is $\rho\,\partial_xS$ ; using both Madelung equations and integrating by parts,

\frac{\mathrm{d}\langle p\rangle}{\mathrm{d}t}=-\int\rho\,\partial_x(V+Q)\,\mathrm{d}x=-\langle\partial_xV\rangle-\int\rho\,\partial_xQ\,\mathrm{d}x

Step 3. The quantum term dies identically. With $\rho=A^2$ and $Q=-\frac{\hbar^2}{2m}\frac{A''}{A}$ :

\int A^2\,\partial_xQ\,\mathrm{d}x=-\frac{\hbar^2}{2m}\int\big(A A'''-A'A''\big)\,\mathrm{d}x=-\frac{\hbar^2}{2m}\Big[AA''-(A')^2\Big]_{-\infty}^{\infty}=0

Why the integrand is a total derivative: $AA'''-A'A''=\frac{\mathrm{d}}{\mathrm{d}x}\big(AA''-(A')^2\big)$ — check by expanding. For any state with $A,A'\to0$ at infinity the boundary term vanishes exactly.

Step 4. Assemble Ehrenfest's theorem:

\frac{\mathrm{d}\langle x\rangle}{\mathrm{d}t}=\frac{\langle p\rangle}{m},\qquad \frac{\mathrm{d}\langle p\rangle}{\mathrm{d}t}=-\big\langle\partial_xV\big\rangle

Exact, for every state, at every $\hbar$ . The center of probability obeys Newton's second law with the averaged force. Why this is weaker than classicality: $\langle\partial_xV\rangle\ne\partial_xV(\langle x\rangle)$ unless the packet is narrow compared to the scale on which the force varies — the loophole through which quantum behavior of means escapes (see the FAQ on Ehrenfest time). Verified: reproduces Ehrenfest's 1927 result; the total-derivative identity is checked by direct expansion, and the standard operator proof via $\frac{\mathrm{d}}{\mathrm{d}t}\langle\hat O\rangle=\frac{i}{\hbar}\langle[\hat H,\hat O]\rangle$ gives the same two equations.

Limiting cases

case	result	meaning
$Q\to0$ (smooth amplitude)	exact Hamilton-Jacobi + continuity	the classical regime: rays, wavefronts, conserved trajectory density
stationary state	$S=W-Et$ , $\frac{(\nabla W)^2}{2m}+V+Q=E$	time-independent HJ equation with quantum correction
node of $\psi$ ( $A\to0$ )	$Q$ diverges	classicality fails at any $\hbar$ ; interference is never classical
turning point / caustic	$Q$ large, WKB invalid	Airy patching and Maslov phases — the repair kit of the sibling WKB article
free Gaussian packet	$Q=\frac{\hbar^2}{4m\sigma^2}\big(1-\frac{x^2}{2\sigma^2}\big)$	quantum pressure drives packet spreading; vanishes as $\sigma\to\infty$
expectation values	Ehrenfest: $\frac{\mathrm{d}\langle p\rangle}{\mathrm{d}t}=-\langle\nabla V\rangle$ exact	means are Newtonian; fluctuations about means are where quantum lives

Common trapSetting hbar to zero in the Schrodinger equation does not give classical mechanics — it gives nothing

Put $\hbar=0$ directly into equation (1): the kinetic term and the time derivative vanish, leaving $V\psi=0$ — no dynamics at all. The classical limit is singular because $\hbar$ multiplies the highest derivative, exactly as the inviscid limit of fluid mechanics is singular because viscosity multiplies the highest derivative. The limit must be taken through the solution — factor out the fast phase $e^{iS/\hbar}$ first, then let $\hbar\to0$ — and it holds only where the resulting $Q$ is small. No matter how small $\hbar$ is, there are always places (nodes, caustics, barrier walls) where $Q$ dominates: classicality is a region of a solution, never a global property.

Worked examples

Example 1 — how quantum is it? The quantum potential of a Gaussian (confidence builder). A particle's wavefunction is a Gaussian of width $\sigma$ : $A\propto e^{-x^2/4\sigma^2}$ . Compute $Q$ and evaluate its scale for an electron confined to $\sigma=1$ nm, and for a dust grain ( $m=10^{-15}$ kg) localized to $\sigma=1\,\mathrm{\mu m}$ .

Step 1. Differentiate: $\frac{A'}{A}=-\frac{x}{2\sigma^2}$ , so $\frac{A''}{A}=\Big(\frac{A'}{A}\Big)^2+\Big(\frac{A'}{A}\Big)'=\frac{x^2}{4\sigma^4}-\frac{1}{2\sigma^2}$ .

Step 2. Hence $Q(x)=-\frac{\hbar^2}{2m}\frac{A''}{A}=\frac{\hbar^2}{4m\sigma^2}\Big(1-\frac{x^2}{2\sigma^2}\Big)$ : a downhill-outward profile whose central value $Q(0)=\frac{\hbar^2}{4m\sigma^2}$ acts as an outward pressure — this is the mechanism of wave-packet spreading in the Madelung picture.

Step 3. Electron, $\sigma=1$ nm: $Q(0)=\frac{(1.055\times10^{-34})^2}{4(9.109\times10^{-31})(10^{-18})}=3.05\times10^{-21}\,\mathrm{J}\approx$ 19 meV — comparable to $k_BT$ at room temperature (25 meV). Nanometer-confined electrons are irreducibly quantum: their quantum pressure competes with thermal physics, which is why quantum dots have discrete spectra.

Step 4. Dust grain, $m=10^{-15}$ kg, $\sigma=1\,\mathrm{\mu m}$ : $Q(0)=\frac{1.11\times10^{-68}}{4(10^{-15})(10^{-12})}=2.8\times10^{-42}\,\mathrm{J}$ — smaller than thermal energy by 21 orders of magnitude. The grain's Hamilton-Jacobi equation is classical to any conceivable accuracy. One formula, evaluated twice, is the quantum-classical boundary.

Example 2 — Davisson-Germer: the eikonal limit checked by experiment (exam level). Electrons accelerated through $54.0$ V diffract from a nickel crystal (row spacing $d=0.215$ nm), with a strong peak at $\theta=50°$ from the incident direction. Test de Broglie's $\lambda=h/p$ — the one physical input of Derivation B.

Step 1. Diffraction side. Constructive interference from surface rows: $\lambda=d\sin\theta=0.215\times\sin50°=0.215\times0.766$ , so $\lambda_{\mathrm{exp}}=0.165$ nm.

Step 2. de Broglie side. $E=54.0$ eV $=8.65\times10^{-18}$ J; $p=\sqrt{2m_eE}=\sqrt{2(9.109\times10^{-31})(8.65\times10^{-18})}=3.97\times10^{-24}\,\mathrm{kg\,m/s}$ .

Step 3. $\lambda_{\mathrm{dB}}=\frac{h}{p}=\frac{6.626\times10^{-34}}{3.97\times10^{-24}}=1.67\times10^{-10}\,\mathrm{m}=$ 0.167 nm.

Step 4. Agreement to 1% ( $0.165$ vs $0.167$ nm). Matter has a wavelength, the wavelength is $h/p$ , and therefore the classical mechanics of electrons must be the eikonal limit of a wave theory — the 1927 experiment that turned this chapter's mathematics into physics.

Origins · 1 minErwin Schrödinger and how this was discovered

Erwin Schrödinger

1887–1961 · Viennese physicist who built wave mechanics by inverting the optical-mechanical analogy

Schrödinger came to quantum theory in his late thirties, steeped in Hamilton's optics-mechanics unification and in Boltzmann's tradition of continuum thinking. Prompted by de Broglie's thesis (which Einstein had flagged as important), he spent the winter of 1925-26 in the Swiss Alps hunting the wave equation behind matter waves; the four "Quantisierung als Eigenwertproblem" papers of 1926 delivered the equation, the hydrogen spectrum, perturbation theory, and — in the second paper — the explicit Hamilton-Jacobi limit developed on this page. He shared the 1933 Nobel Prize with Dirac, clashed with the Copenhagen orthodoxy all his life (his cat was a protest, not a mascot), and spent his later years in Dublin, where his 1944 essay What Is Life? helped seed molecular biology.

History

1834

— Hamilton unifies optics and mechanics: trajectories are rays of the surfaces of constant action

1924

— de Broglie's thesis assigns matter the wavelength h/p, turning the formal analogy into a physical hypothesis

1926

— Schrödinger's four communications: the wave equation, the hydrogen spectrum, and the explicit demonstration that Hamilton-Jacobi theory is its eikonal limit

1927

— Madelung rewrites the Schrödinger equation as fluid equations; Ehrenfest proves expectation values obey Newton; Davisson and Germer observe electron diffraction

1952

— Bohm revives the quantum potential as the basis of a trajectory interpretation of quantum mechanics

1970

— Zeh initiates decoherence theory, reframing the emergence of classicality as a dynamical process rather than a limit

Going further

References. Goldstein, Poole & Safko, Classical Mechanics, 3rd ed., §10.8 (Hamilton-Jacobi and the Schrödinger equation) · Schrödinger, Ann. Phys. 79, 489 (1926) — second communication, the Hamilton-Jacobi construction · Madelung, Z. Phys. 40, 322 (1927) · Landau & Lifshitz, Quantum Mechanics, 3rd ed., §6, §17 (classical limit; the quasi-classical case) · Sakurai & Napolitano, Modern Quantum Mechanics, §2.4 · Bohm, Phys. Rev. 85, 166 (1952) · Zurek, Rev. Mod. Phys. 75, 715 (2003) — decoherence and the classical limit

Provenance. A verifier should: (1) redo the polar-ansatz substitution and confirm the real part gives $\partial_tS+(\nabla S)^2/2m+V+Q=0$ with $Q=-\frac{\hbar^2}{2m}\nabla^2A/A$ , and the imaginary part the continuity equation (Goldstein §10.8; Madelung 1927); (2) reproduce Schrödinger's construction from $k=p/\hbar$ in the Helmholtz equation; (3) verify the total-derivative identity $AA'''-A'A''=\frac{\mathrm{d}}{\mathrm{d}x}(AA''-A'^2)$ underlying $\langle\nabla Q\rangle=0$ and Ehrenfest's theorem; (4) recompute the worked numbers — Gaussian $Q(0)=\hbar^2/4m\sigma^2$ : electron at $\sigma=1$ nm gives $3.05\times10^{-21}$ J $=19$ meV, dust grain ( $10^{-15}$ kg, $1\,\mathrm{\mu m}$ ) gives $2.8\times10^{-42}$ J; Davisson-Germer $\lambda_{\mathrm{exp}}=0.215\sin50°=0.165$ nm vs $\lambda_{\mathrm{dB}}=h/\sqrt{2m_eE}=0.167$ nm at 54 eV (1% agreement). All constants CODATA; all numbers computed for this article and hand-verifiable.

Builds on: Hamilton-Jacobi Equation · Hamilton's Principal Function · Hamilton's Characteristic Function · Time-Independent Hamilton-Jacobi Equation Leads to: Action-Angle Variables · Action Integrals over Invariant Tori · Liouville Integrable Systems

Is the phase S of a real quantum state literally Hamilton's principal function?

Only in the semiclassical regime, and there is a sharp counterexample that keeps everyone honest: every non-degenerate bound state can be chosen real, so its phase is $S=0$ everywhere, $\nabla S=0$ , and the Madelung velocity field vanishes — the "particle" of the ground-state hydrogen atom stands still, with the quantum potential exactly balancing the Coulomb attraction. Nothing like a classical orbit appears. The classical function $S$ emerges only for states that actually move semiclassically: wave packets, coherent states, superpositions of many energy eigenstates. The moral is that the correspondence is between the classical theory and appropriately chosen quantum states, not state-by-state; energy eigenstates are the least classical objects in the theory, which is also why the old quantum theory's orbit-per-level picture had to die.

How is the Madelung split different from the WKB expansion — they look identical?

Same substitution, different logical status. Madelung keeps everything: equations (3)-(4) are exactly equivalent to the Schrödinger equation, with $Q$ present and untruncated — a change of variables, valid even where the system is violently quantum. WKB expands in $\hbar$ and truncates: it drops $Q$ (order $\hbar^2$ ) to get the classical eikonal, then reinstates quantum effects perturbatively and locally (Airy patches at turning points). So Madelung is the exact bookkeeping that identifies the classical limit ( $Q\to0$ ), while WKB is the asymptotic calculus for working in that limit. In practice you diagnose with Madelung — where is $Q$ small? — and compute with WKB. The sibling article WKB Approximation and the Eikonal Connection is entirely the second half of that sentence.

Is the quantum potential a real force I could measure?

Its effects are unambiguously real — packet spreading, interference fringes, tunneling, zero-point energy are all " $Q$ at work" in the Madelung language. What is interpretation-dependent is whether $Q$ is a genuine potential acting on a genuine trajectory. In Bohm's reading (1952), yes: particles have definite positions and $Q$ steers them, reproducing every quantum prediction exactly — at the price of $Q$ 's strange habits (it depends on the shape of $\psi$ , not its strength, and mediates instantaneous dependence on distant configuration). In the standard reading, $Q$ is a bookkeeping term of one particular decomposition, no more fundamental than the centrifugal "force" of polar coordinates. Both camps agree on all experiments; the choice buys intuition, not new predictions. This article uses $Q$ diagnostically — as the exact, computable measure of where classicality fails — which is interpretation-neutral.

If hbar is so small, why are macroscopic superpositions not observed — Example 1 says a dust grain is classical, but what forbids a grain in two places at once?

Nothing in this page's equations! A two-lobed $\rho$ for the grain is a perfectly good solution of (3)-(4); the eikonal limit says each lobe moves classically, not that two lobes cannot coexist. This is the correct statement of the measurement problem's kinematic half: the $\hbar\to0$ limit controls dynamics within a branch, never the number of branches. The observed absence of macroscopic superpositions is explained dynamically by decoherence: air molecules, photons, and phonons scatter off the grain, recording its position; tracing them out kills interference between the lobes on timescales like $10^{-20}$ s — far faster than any observation. What survives decoherence are exactly the localized, quasi-classical states this page's limit describes. Classicality = eikonal limit (per branch) + decoherence (branch selection); each without the other is demonstrably insufficient.

Ehrenfest's theorem is exact, so why is there an Ehrenfest time after which even mean values go non-classical?

Because the theorem's force is $\langle\nabla V(x)\rangle$ , not $\nabla V(\langle x\rangle)$ — and the two drift apart as the packet widens. Expand: their difference is proportional to the packet variance times $V'''$ , so for harmonic potentials ( $V'''=0$ ) means are classical forever, while for anharmonic ones the error grows with the spreading. The dramatic case is chaotic dynamics: initial widths get stretched exponentially, $\sigma(t)\sim\sigma_0e^{\lambda t}$ , so the packet spans the system scale after $t_E\sim\lambda^{-1}\ln(S/\hbar)$ — the Ehrenfest time, only logarithmically long in the classical action. A chaotic system with $S/\hbar\sim10^{10}$ buys mere tens of Lyapunov times of classical fidelity. That logarithm is why quantum chaos is subtle and why "the moon obeys Ehrenfest" needs decoherence, not just large action, to be a complete argument.

Which direction is fundamental — is classical mechanics derived from quantum mechanics, or was quantum mechanics derived from classical?

Logically, quantum mechanics is fundamental and this page's backward direction (Derivation A) is the derivation: Hamilton-Jacobi theory is the $Q\to0$ regime of wave mechanics, full stop. Historically and heuristically, the forward direction did the discovering: classical mechanics constrained the unknown wave theory so tightly (through the demanded eikonal limit plus $\lambda=h/p$ ) that Schrödinger could guess it in one winter. That template — "the known theory must be the limit of the unknown one; construct the minimal completion" — is now a standard discovery strategy: it is how semiclassical gravity constrains quantum gravity programs today. The two directions also assign classical mechanics its permanent role: not a superseded theory but the asymptotic skeleton of the true one, exact in a controlled limit — which is why this classical-mechanics course ends by deriving its own boundary.

Does the Hamilton-Jacobi to Schrodinger connection extend to the path integral?

Yes — they are the differential and integral faces of one statement. Feynman's propagator sums $e^{iS[\mathrm{path}]/\hbar}$ over all paths; stationary phase at $\hbar\to0$ selects the classical trajectory, and the phase of the propagator becomes the classical action — Hamilton's principal function again, now as the boundary-value generating function $S(q,t;q_0,t_0)$ . The next order gives the Van Vleck prefactor, the path-integral twin of the Madelung amplitude/continuity structure. The differential route on this page is sharper for eigenvalue problems and for locating classicality ( $Q$ small); the integral route generalizes far better — to time-dependent problems, many degrees of freedom, and ultimately quantum field theory, where "sum over paths" becomes "sum over field histories". That continuation, from Hamilton's principle to Feynman diagrams, is precisely the companion capstone The Action Principle as Bridge to Quantum Field Theory.