A 2017 physics-education paper shows that a second-order wave equation for the classical action of a free particle follows from the free-particle invariances of the Hamiltonian, momentum and velocity together with the Legendre relations linking the action to these quantities. Writing the action in the complex plane-wave form S=−iSˉφS=-\mathrm{i}\bar S\varphi, where Sˉ\bar S is a real amplitude that is constant in space and time and φ\varphi is a phase factor, produces wave-like Hamiltonian and momentum fields whose amplitudes obey H=SˉωH=\bar S\omega and P=Sˉk\boldsymbol{P}=\bar S\boldsymbol{k}; discretising a scalar potential into small, locally constant segments then yields an equation of motion of the same form as the time-dependent Schrödinger equation. The construction leaves three gaps open, which this problem isolates as its open core: (a) the derivation is valid only where the potential is smooth, and the matching conditions connecting the wave-like solutions across a potential discontinuity are not derived; (b) the action magnitude Sˉ\bar S is not determined within the classical theory but must be externally fixed to the Planck constant ℏ\hbar, and neither a derivation nor a rigorous non-determinability argument for Sˉ\bar S exists within the framework; (c) the comparison with quantum mechanics is carried out entirely in the wave-mechanics (Schrödinger) picture, and no extension to other formulations of quantum mechanics, such as the Heisenberg picture or the path-integral formulation, has been given. None of the three sub-questions has been addressed in the later literature, and a separate published dispute over the framework's overall validity remains unresolved. The problem is open.

Background

The Schrödinger equation, iℏ∂tψ=H^ψi\hbar\partial_t\psi=\hat H\psi, is normally postulated or motivated heuristically in quantum mechanics courses; deriving it from classical mechanics is a long-standing pedagogical and foundational theme. One standard route runs through the Hamilton–Jacobi equation, ∂tS+H(x,∇S,t)=0\partial_t S+H(\boldsymbol{x},\nabla S,t)=0, where SS is Hamilton's principal function (the classical action): writing ψ=A eiS/ℏ\psi=A\,e^{iS/\hbar} and substituting into the Schrödinger equation recovers Hamilton–Jacobi in the limit where the amplitude AA varies slowly, which is the core of the semiclassical (Wentzel–Kramers–Brillouin, WKB) connection between the two theories. A 2017 article in a physics-education journal (the source of this problem) takes a different route: instead of recovering classical structure from quantum mechanics, it asks what differential equation the classical action itself satisfies for a free particle. Combining the free-particle invariances dH/dt=0\mathrm{d}H/\mathrm{d}t=0, dP/dt=0\mathrm{d}\boldsymbol{P}/\mathrm{d}t=0 and dr˙/dt=0\mathrm{d}\dot{\boldsymbol{r}}/\mathrm{d}t=0 with the Legendre relations H=−∂S/∂tH=-\partial S/\partial t and P=∂S/∂r\boldsymbol{P}=\partial S/\partial\boldsymbol{r} that link the action to the Hamiltonian HH and momentum P\boldsymbol{P}, the paper derives a second-order wave equation for the action. Its non-trivial solutions are complex plane waves, S=−iSˉφS=-\mathrm{i}\bar S\varphi with φ(r,t)=ei(k⋅r−ωt)\varphi(\boldsymbol{r},t)=e^{i(\boldsymbol{k}\cdot\boldsymbol{r}-\omega t)}, where k\boldsymbol{k} is the wave vector and ω\omega the angular frequency. Propagating this form through the Legendre relations gives wave-like Hamiltonian and momentum fields with amplitudes H=SˉωH=\bar S\omega and P=Sˉk\boldsymbol{P}=\bar S\boldsymbol{k}, which carry the same form as the Planck–Einstein and de Broglie relations E=ℏωE=\hbar\omega and p=ℏk\boldsymbol{p}=\hbar\boldsymbol{k}, with Sˉ\bar S playing the role of the Planck constant. To include a scalar potential V(r)V(\boldsymbol{r}), the potential is discretised into small segments in each of which it is approximately constant; imposing on each segment the operator-like equalities iSˉ ∂tφn=Hnφn\mathrm{i}\bar S\,\partial_t\varphi_n=H_n\varphi_n and −iSˉ ∇φn=Pnφn-\mathrm{i}\bar S\,\nabla\varphi_n=\boldsymbol{P}_n\varphi_n together with the classical energy relation Hn=Pn2/2m+VnH_n=\boldsymbol{P}_n^2/2m+V_n yields a segment equation of the same form as the time-dependent Schrödinger equation, iSˉ ∂tφ=−(Sˉ2/2m)∇2φ+V(r)φ\mathrm{i}\bar S\,\partial_t\varphi=-(\bar S^2/2m)\nabla^2\varphi+V(\boldsymbol{r})\varphi, and in regions where the potential is smooth enough the segment index can be dropped. The author's program continued in follow-up papers (2018 on electromagnetic potentials, a 2021 preprint on a wave-particle duality model, 2024 on acoustic and elastic wave models, 2025 on heat-transport models), all resting on the same construction. Three gaps that follow from this construction — as described in the paper's abstract and restated in the author's follow-up papers — remain open despite the follow-up work, and constitute the present problem: first, the segment-wise derivation assumes a smooth potential, and additional continuity conditions would be required where the potential is discontinuous (the analogue of standard matching conditions for the Schrödinger equation at potential steps, familiar from one-dimensional quantum scattering); second, the magnitude Sˉ\bar S is left undetermined by the classical theory and is set to ℏ\hbar by hand, raising the question of whether it can be derived within the framework or proven classically inaccessible; third, the comparisons drawn between the classical relations and quantum mechanics are confined to the Schrödinger picture (wave mechanics), leaving the Heisenberg picture (where operators, not states, carry the dynamics) and the path-integral formulation (where amplitudes are summed over histories) untouched. The framework's overall soundness is contested: a 2025 comment in the same journal reports 'obvious and severe flaws' in the 2017 and 2018 papers, and a 2026 reply defends the construction, arguing that the commentator considered only the trivial solution of the dynamical variables and missed the non-trivial wave-like solution; neither the comment nor the reply engages the three sub-questions above.

Problem Statement

Within the classical framework of the 2017 source paper — in which a wave equation for the action of a free particle is derived from the free-particle invariances of the Hamiltonian, momentum and velocity together with the Legendre relations H=−∂S/∂tH=-\partial S/\partial t and P=∂S/∂r\boldsymbol{P}=\partial S/\partial\boldsymbol{r} (with the smoothness condition ∂t∇S−∇∂tS=0\partial_t\nabla S-\nabla\partial_t S=0); the action is taken in the complex plane-wave form S=−iSˉφS=-\mathrm{i}\bar S\varphi with constant amplitude Sˉ\bar S, giving wave-like Hamiltonian and momentum fields with H=SˉωH=\bar S\omega and P=Sˉk\boldsymbol{P}=\bar S\boldsymbol{k}; and a segment-wise discretisation of a scalar potential, valid where the potential is smooth, yields the Schrödinger-form equation iSˉ ∂tφ=−(Sˉ2/2m)∇2φ+V(r)φ\mathrm{i}\bar S\,\partial_t\varphi=-(\bar S^2/2m)\nabla^2\varphi+V(\boldsymbol{r})\varphi — solve the three questions the framework leaves open: (a) derive, within the same classical setting, the matching or continuity conditions that connect the wave-like solutions across a surface where the potential is discontinuous (the derivation in the source is valid only in regions where the potential is smooth); (b) determine the magnitude Sˉ\bar S within the classical theory, or prove rigorously that it cannot be determined classically within this framework — where "this framework" means the assumption set enumerated above (the free-particle invariances, the Legendre relations, the smoothness condition ∂t∇S−∇∂tS=0\partial_t\nabla S-\nabla\partial_t S=0, the constant-amplitude plane-wave ansatz, and the segment-wise discretisation with its per-segment operator identifications) — thereby justifying or refuting the external identification Sˉ=ℏ\bar S=\hbar, which in the source and all follow-up work is imposed by substitution rather than derived; and (c) extend the comparison between these classical relations and quantum mechanics beyond the Schrödinger picture to at least one non-Schrödinger formulation, for instance the Heisenberg picture or the path-integral formulation. For each of (a)–(c), an answer must be internal to the source framework's assumptions (or demonstrate precisely where the relevant assumption fails) and must not be a restatement of standard quantum mechanics; all three sub-questions must be addressed.

References

  1. Chen Yang, 'Wave-like variables of a classical particle and their connections to quantum mechanics', European Journal of Physics 38 (2017) 015401. DOI: 10.1088/0143-0807/38/1/015401 (source paper of this problem). URL: https://doi.org/10.1088/0143-0807/38/1/015401
  2. Chen Yang, 'Analogy between electromagnetic potentials and wave-like dynamic variables with connections to quantum theory', European Journal of Physics 39 (2018) 035406. DOI: 10.1088/1361-6404/aa948e (author's follow-up extending the construction to electromagnetic potentials). URL: https://doi.org/10.1088/1361-6404/aa948e
  3. Chen Yang and S. Olutunde Oyadiji, 'Modelling Wave-Particle Duality of Classical Particles and Waves', Research Square preprint (2021). DOI: 10.21203/rs.3.rs-475920/v1. URL: https://doi.org/10.21203/rs.3.rs-475920/v1
  4. Chen Yang, 'Quantization of linear acoustic and elastic wave models in characterizations of isomorphism', Research Square preprint (December 2023; preprint of the 2024 Scientific Reports article). DOI: 10.21203/rs.3.rs-3697150/v2. URL: https://doi.org/10.21203/rs.3.rs-3697150/v2
  5. Chen Yang, 'Quantization of linear acoustic and elastic wave models in characterizations of isomorphism', Scientific Reports 14 (2024) 8759. DOI: 10.1038/s41598-024-57092-0. URL: https://doi.org/10.1038/s41598-024-57092-0
  6. Chen Yang, 'Quantization of nonequilibrium heat transport models based on isomorphism and gauge symmetry', Scientific Reports 15 (2025) 14820. DOI: 10.1038/s41598-025-93640-y. URL: https://doi.org/10.1038/s41598-025-93640-y
  7. C. Baumgarten, 'Comments on "Wave-like variables of a classical particle and their connections to quantum mechanics" (Eur. J. Phys. 38, 015401, 2017)', European Journal of Physics 46 (2025) 038001. DOI: 10.1088/1361-6404/adbf73 (critical comment). URL: https://doi.org/10.1088/1361-6404/adbf73
  8. Chen Yang, 'Reply to Comments on "Wave-like variables of a classical particle and their connections to quantum mechanics" (Eur. J. Phys. 2017, 38, 015401)', European Journal of Physics 47 (2026) 028001. DOI: 10.1088/1361-6404/ae126b (author's reply). URL: https://doi.org/10.1088/1361-6404/ae126b
  9. C. Baumgarten, 'A Two-Page "Derivation" of Schroedinger's Equation', arXiv:1907.09917 (2019; adjacent work by the same commentator, arguing within a different derivation that an action-unit constant is a unit-conversion factor that cannot be derived theoretically). URL: https://arxiv.org/abs/1907.09917