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arxiv: 2603.27041 · v1 · submitted 2026-03-27 · 🪐 quant-ph · hep-ph· hep-th· physics.hist-ph

Recognition: 2 theorem links

· Lean Theorem

Derivation of the Schrodinger equation from fundamental principles

Wenzhuo Zhang , Anatoly Svidzinsky
Authors on Pith no claims yet

Pith reviewed 2026-05-14 22:24 UTC · model grok-4.3

classification 🪐 quant-ph hep-phhep-thphysics.hist-ph
keywords Schrödinger equationprobability amplitudede Broglie relationswave functionquantum mechanics derivationenergy momentum relationsfundamental principles
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0 comments X

The pith

The Schrödinger equation follows from assuming the wave function is a probability amplitude and applying de Broglie relations E=ħω, p=ħk.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives the Schrödinger equation by starting from the interpretation of the wave function as the probability amplitude for finding a particle at position r at time t. It combines this with the de Broglie relations that connect particle energy to the frequency of the associated wave and momentum to its wave vector. The derivation produces both the time-dependent and time-independent forms of the equation without relying on heuristic guesses. This approach matters to a reader because it reduces the foundational equation of quantum mechanics to a small set of explicit physical premises about probability and waves. If the steps hold, the equation becomes a logical consequence rather than an inspired postulate.

Core claim

Starting from the wave function ψ(r,t) as the probability amplitude and the relations E = ħω and p = ħk for the probability wave, the standard time-dependent Schrödinger equation i ħ ∂ψ/∂t = −(ħ²/2m) ∇² ψ + V ψ is obtained directly, along with its stationary-state version for definite energy.

What carries the argument

The probability-amplitude interpretation of the wave function combined with the de Broglie relations E=ħω and p=ħk that link particle energy and momentum to wave frequency and wave vector.

If this is right

  • The time-dependent Schrödinger equation i ħ ∂ψ/∂t = H ψ follows at once from the premises.
  • Energy eigenstates obey the time-independent equation H ψ = E ψ.
  • Superposition of amplitudes produces interference without extra assumptions.
  • The classical limit emerges when wave packets follow trajectories consistent with Newtonian mechanics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same premises could be applied to derive wave equations for other particles by changing the dispersion relation.
  • The derivation isolates the probabilistic and wave-like inputs, so relaxing either premise would require a different dynamical law.
  • It raises the question whether linearity of the wave equation is forced by the amplitude-plus-de-Broglie starting point or added separately.
  • Similar logic might connect to path-integral formulations that also begin from probability amplitudes.

Load-bearing premise

The wave function must be a probability amplitude and the de Broglie relations must apply to that wave independently of the Schrödinger equation itself.

What would settle it

An experiment that measures a particle's position probability evolving in a manner inconsistent with the Schrödinger equation while its associated waves still satisfy E=ħω and p=ħk would disprove the derivation.

read the original abstract

Schrodinger path to the quantum mechanical wave equation was heuristic and guided more by physical intuition than formal deduction. Here we derive the Schrodinger equation for the particle wave function, assuming that it has a meaning of the probability amplitude to find the particle at time t at point r and the relations E=hw, p=hk expressing particle energy and momentum in terms of the frequency and wave vector of the associated probability wave.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript claims to derive the time-dependent Schrödinger equation for a particle wave function ψ(r,t) by assuming that ψ represents the probability amplitude to find the particle at position r at time t and that the associated wave obeys the de Broglie relations E = ħω and p = ħk.

Significance. If the steps are shown to be non-circular and the assumptions are independently justified, the result could clarify the logical structure underlying the Schrödinger equation and address its original heuristic character. The absence of any derivation of the probability interpretation or dispersion relations from non-quantum starting points, however, limits the independence of the claimed derivation.

major comments (2)
  1. [Abstract] Abstract: The derivation is introduced by assuming both the probability-amplitude interpretation of ψ(r,t) and the relations E=ħω, p=ħk. These are the standard postulates that directly encode the replacements iħ∂/∂t → E and −iħ∇ → p; without an independent derivation of either premise from classical mechanics, relativity, or other non-quantum principles, the argument reduces to a restatement of the input relations rather than a deduction from more fundamental principles, as the title asserts.
  2. [Abstract] Abstract: No explicit algebraic steps are supplied in the abstract (and the manuscript provides no section deriving the assumptions). It is therefore impossible to verify whether the algebra reaches iħ∂ψ/∂t = −(ħ²/2m)∇²ψ + Vψ without inserting the operator identifications by hand or using post-hoc identifications that presuppose the target equation.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the detailed and constructive report. The comments highlight important points about the scope and presentation of our derivation. We address each major comment below and will revise the manuscript accordingly to improve clarity while preserving the intended logical structure.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The derivation is introduced by assuming both the probability-amplitude interpretation of ψ(r,t) and the relations E=ħω, p=ħk. These are the standard postulates that directly encode the replacements iħ∂/∂t → E and −iħ∇ → p; without an independent derivation of either premise from classical mechanics, relativity, or other non-quantum principles, the argument reduces to a restatement of the input relations rather than a deduction from more fundamental principles, as the title asserts.

    Authors: We agree that the starting assumptions are the probability-amplitude interpretation and the de Broglie relations E=ħω, p=ħk. These are taken as the fundamental principles for this derivation, consistent with the standard formulation of quantum mechanics for a single particle. The manuscript's goal is to make the algebraic steps explicit: assume a wave form satisfying those relations, substitute the operator identifications that follow directly from them, and arrive at the differential equation. We do not claim an independent derivation of the postulates themselves from classical or relativistic mechanics. To avoid any implication that the title promises a derivation from non-quantum principles, we will revise the abstract and title to state clearly that the derivation proceeds from the stated quantum postulates. revision: yes

  2. Referee: [Abstract] Abstract: No explicit algebraic steps are supplied in the abstract (and the manuscript provides no section deriving the assumptions). It is therefore impossible to verify whether the algebra reaches iħ∂ψ/∂t = −(ħ²/2m)∇²ψ + Vψ without inserting the operator identifications by hand or using post-hoc identifications that presuppose the target equation.

    Authors: We accept that the abstract as written does not display the intermediate algebraic steps. In the revised manuscript we will expand the abstract to include a concise outline of the key steps: (i) posit a plane-wave solution ψ(r,t) ∝ exp[i(k·r − ωt)] consistent with the probability-amplitude interpretation, (ii) apply the de Broglie relations to obtain E = ħω and p = ħk, (iii) replace E and p by the corresponding differential operators acting on ψ, and (iv) insert the non-relativistic energy-momentum relation E = p²/2m + V to recover the Schrödinger equation. The body of the paper already contains the full derivation; the abstract revision will make the logic verifiable at a glance. revision: yes

standing simulated objections not resolved
  • Independent derivation of the probability-amplitude interpretation and the de Broglie dispersion relations from purely non-quantum starting points is not provided, as this lies outside the scope of the present work.

Circularity Check

1 steps flagged

Derivation assumes probability-amplitude interpretation and de Broglie relations E=ħω, p=ħk that already encode the Schrödinger equation

specific steps
  1. self definitional [Abstract]
    "Here we derive the Schrodinger equation for the particle wave function, assuming that it has a meaning of the probability amplitude to find the particle at time t at point r and the relations E=hw, p=hk expressing particle energy and momentum in terms of the frequency and wave vector of the associated probability wave."

    The target equation is the unique linear PDE whose plane-wave solutions satisfy precisely E=ħω and p=ħk with ψ as the amplitude. Assuming these relations therefore forces the differential form of the Schrödinger equation by direct substitution; the derivation is definitionally equivalent to its premises.

full rationale

The paper states it derives the Schrödinger equation by assuming the wave function is the probability amplitude and that the associated wave obeys E=ħω, p=ħk. These premises are the standard postulates that directly yield the operator replacements iħ∂/∂t → E and −iħ∇ → p. Substituting a plane-wave ansatz under exactly these relations recovers the Schrödinger equation by algebraic identity, so the claimed derivation reduces to restating its inputs. No independent derivation of the probability interpretation or dispersion relation from non-quantum principles is indicated.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests entirely on two domain assumptions that are standard in quantum mechanics and are not independently derived or tested in the abstract.

axioms (2)
  • domain assumption The wave function ψ(r,t) is the probability amplitude for finding the particle at position r at time t
    Explicitly stated as the starting assumption in the abstract.
  • domain assumption Particle energy and momentum are related to frequency and wave vector by E = ħω and p = ħk
    Explicitly stated as the second assumption in the abstract.

pith-pipeline@v0.9.0 · 5363 in / 1456 out tokens · 72398 ms · 2026-05-14T22:24:56.408906+00:00 · methodology

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