Classical mechanics admits multiple equivalent formulations, from Newton's equations to the variational Lagrange-Hamilton framework and the scalar Hamilton-Jacobi (HJ) theory. In the HJ formulation, classical ensembles evolve through the continuity equation for a real density $ρ= R^{2}$ coupled to Hamilton's principal function $S$ . Here we develop a complementary formulation, the Hamilton-Jacobi-Schrödinger (HJS) theory, by embedding the pair $(R,S)$ into a single complex field. Starting from a completely general complex ansatz $ψ= f(R,S)\, e^{i g(R,S)},$ and imposing two minimal structural requirements, we obtain a unique map $ψ= R\, e^{iS/κ}\, $ together with a linear HJS equation whose $|κ| \to 0$ limit reproduces the HJ formulation exactly. Remarkably, when $\mathrm{Re}(κ)\neq 0$ , essential features of quantum mechanics, superposition, operator algebra, commutators, the Heisenberg uncertainty principle, Born's rule and unitary evolution, follow naturally as structural consistency conditions. HJS thus provides a unified mathematical viewpoint in which classical and quantum dynamics appear as different limits of a single underlying structure.Classical mechanics admits multiple equivalent formulations, from Newton's equations to the variational Lagrange-Hamilton framework and the scalar Hamilton-Jacobi (HJ) theory. In the HJ formulation, classical ensembles evolve through the continuity equation for a real density $ρ= R^{2}$ coupled to Hamilton's principal function $S$. Here we develop a complementary formulation, the Hamilton-Jacobi-Schrödinger (HJS) theory, by embedding the pair $(R,S)$ into a single complex field. Starting from a completely general complex ansatz $ψ= f(R,S)\, e^{i g(R,S)},$ and imposing two minimal structural requirements, we obtain a unique map $ψ= R\, e^{iS/κ}\, $ together with a linear HJS equation whose $|κ| \to 0$ limit reproduces the HJ formulation exactly. Remarkably, when $\mathrm{Re}(κ)\neq 0$, essential features of quantum mechanics, superposition, operator algebra, commutators, the Heisenberg uncertainty principle, Born's rule and unitary evolution, follow naturally as structural consistency conditions. HJS thus provides a unified mathematical viewpoint in which classical and quantum dynamics appear as different limits of a single underlying structure.

quant-pharXiv:2601.22697Feb 2, 20260 citations

Hamilton-Jacobi theory and quantum mechanics. I. The free particle case

Anastasios Kyprianidis

Physics Letters ASep 1, 19881 citation

On the quantum Hamilton-Jacobi formalism

Antonio S. de Castro, ALVARO DE SOUZA DUTRA

Foundations of PhysicsJun 1, 19918 citations

Path Concepts in Hamilton-Jacobi Theory

Rolf W. Landauer

It is pointed out that the classical analog to a wave function is not a single orbit but a set of paths as defined by the solution of the Hamilton-Jacobi equation. A review is given of Hamilton-Jacobi theory in language appropriate to these path concepts. It is shown that there is a classical analog to a wave packet. Quantum-mechanically, constants of motion are not arbitrary but must be characteristic values of commuting operators. It is shown by examples that the possible classical states are similarly restricted. The conditions are discussed under which a classical set of paths is derivable from the solution of a Hamilton-Jacobi equation.It is pointed out that the classical analog to a wave function is not a single orbit but a set of paths as defined by the solution of the Hamilton-Jacobi equation. A review is given of Hamilton-Jacobi theory in language appropriate to these path concepts. It is shown that there is a classical analog to a wave packet. Quantum-mechanically, constants of motion are not arbitrary but must be characteristic values of commuting operators. It is shown by examples that the possible classical states are similarly restricted. The conditions are discussed under which a classical set of paths is derivable from the solution of a Hamilton-Jacobi equation.

American Journal of PhysicsSep 1, 19527 citations

Solutions of the Hamilton-Jacobi Equation and Their Connection to Quantum Mechanics

A. Pareek

Jan 1, 19880 citations

ρ= R^{2}

coupled to Hamilton's principal function

S

. Here we develop a complementary formulation, the Hamilton-Jacobi-Schrödinger (HJS) theory, by embedding the pair

(R,S)

into a single complex field. Starting from a completely general complex ansatz

ψ= f(R,S)\, e^{i g(R,S)},

and imposing two minimal structural requirements, we obtain a unique map $ψ= R\, e^{iS/κ}\, $ together with a linear HJS equation whose

|κ| \to 0

limit reproduces the HJ formulation exactly. Remarkably, when

\mathrm{Re}(κ)\neq 0

, essential features of quantum mechanics, superposition, operator algebra, commutators, the Heisenberg uncertainty principle, Born's rule and unitary evolution, follow naturally as structural consistency conditions. HJS thus provides a unified mathematical viewpoint in which classical and quantum dynamics appear as different limits of a single underlying structure.