The Natural Order of Maths

Schrödinger Equation via Symmetrical Negative Magnitude Tracks

A Real-Valued Formulation of the Time-Dependent Schrödinger Equation via Symmetrical Negative Magnitude Tracks

Author: Chris Young : Independent Curriculum Researcher & Educational Synthesist

Contact: https://thenaturalorder.net.au/contact/

Introduction & Scope

The derivation presented on this page uses the Time-Dependent Schrödinger Equation as a primary case study. Quantum mechanics was chosen deliberately because it is widely considered the ultimate test case for imaginary numbers—a domain where standard physics relies on

i≡−1i \equiv \sqrt{-1}

to maintain phase orthogonality and prevent total probability fields from decaying into exponential collapse.

However, demonstrating a real-valued wave equation here is not an isolated exercise in theoretical physics. In truth, I didn’t write this book to fix quantum mechanics—I wrote it to fix the underlying arithmetic of mathematics itself.

The necessity of the imaginary unit

ii

in quantum mechanics, the breakdown of continuous wave functions, and the reliance on complex vector spaces are not inherent properties of physical reality. They are structural patches required to compensate for a legacy number line that treats the negative domain as an asymmetrical value-sink. By restoring arithmetic to its natural order—where negative magnitude scales symmetrically along its own native track—the complex plane becomes redundant. The Schrödinger equation simply serves as our primary proof of concept: when you fix the foundational arithmetic, quantum mechanics, wave propagation, and continuous physical systems return naturally to a purely real, single-dimensional spatial reality.

Abstract
This paper presents a formal derivation of a purely real-valued Time-Dependent Schrödinger Equation, eliminating the historical requirement for the imaginary unit

i≡−1i \equiv \sqrt{-1}

in quantum mechanics. Traditional frameworks necessitate complex-valued fields because the standard legacy number line is structurally asymmetrical, treating the negative domain as a value-sink that induces exponential decay in first-order differential systems. By applying the Identity View of Mirror Magnitude, we establish a mathematically stable, single-dimensional vector space where negative magnitude increases symmetrically away from zero. Under this framework, the sign rules of multi-term operations preserve domain identity, allowing continuous quantum wave oscillations to propagate natively along a purely real track.

  1. Introduction and the Legacy Field Inconsistency
    In standard quantum mechanics, the temporal evolution of a physical system is governed by the Time-Dependent Schrödinger Equation:
iℏ∂Ψ∂t=H^Ψi\hbar \frac{\partial \Psi }{\partial t}=\hat{H}\Psi

Where $$\hbar$$ represents the reduced Planck’s constant,

∂Ψ∂t\frac{\partial \Psi }{\partial t}

is the first-order temporal derivative of the state function, and Ĥ is the Hamiltonian operator. The standard architectural design of mathematics introduces a structural asymmetry by defining the negative domain as a subset of decreasing numerical value:

⋯<−3<−2<−1<0<+1<+2<+3<…\dots <-3<-2<-1<0<+1<+2<+3<\dots

Consequently, legacy arithmetic enforces a sign-reversal axiom upon squaring or multiplying negative identities:

(−1)×(−1)=+1(-1)\times (-1)=+1


When applied to a first-order ordinary or partial differential equation, this configuration acts as a value-sink. Without an external operator modification, the system forces crossing the zero boundary to flip signs, converting continuous wave oscillations into unidirectional, irreversible exponential decay—functionally mimicking the Fourier Heat Equation rather than a perpetual wave:

∂u∂t=k∂2u∂x2⟹u(t)=u(0)e−kt\frac{\partial u}{\partial t}=k\frac{\partial ^{2}u}{\partial x^{2}}\implies u(t)=u(0)e^{-kt}

To bypass this foundational limitation, legacy physics introduces the imaginary unit i to function as a fixed 90° phase rotator. This parameter forces the wave state to propagate through an unmeasurable “complex plane,” utilising Euler’s identity to artificially maintain probability conservation (unitarity):

Ψ(t)=Ψ(0)e−iEℏt=Ψ(0)[cos⁡(Etℏ)−isin⁡(Etℏ)]\Psi(t) = \Psi(0)e^{-i\frac{E}{\hbar}t} = \Psi(0)\left[\cos\left(\frac{Et}{\hbar}\right) – i\sin\left(\frac{Et}{\hbar}\right)\right]
  1. Foundational Axiom: The Symmetrical Mirror-Magnitude Line
    The system presented herein maps a fully symmetrical, real-valued alternative by restructuring the foundational number line. We introduce the Identity View, which establishes the positive and negative domains as absolute mirror-image paths where physical magnitude increases monotonically as a function of distance from the zero baseline:
    $$\tiny \text{Negative Infinity}\longleftarrow [\text{Increasing Negative Magnitude}]\longleftarrow 0\longrightarrow [\text{Increasing Positive Magnitude}]\longrightarrow \text{Positive Infinity}$$
    On this rectified coordinate system, a negative prefix denotes an independent operational direction rather than an erosion of value. Mirror symmetry mandates that the algebraic operations preserve domain identity across multiplication:
    $$\small \text{Negative Magnitude}\times \text{Negative Magnitude}=\text{Greater Negative Magnitude}$$
    Natively, this produces identical scaling behavior across both domains:
Positive Track:+3×+2=+6Negative Track:−3×−2=−6\begin{aligned} \text{Positive Track:} &\quad +3 \times +2 = +6 \\ \text{Negative Track:} &\quad -3 \times -2 = -6 \end{aligned}
  1. Theorem: Real-Valued Wave Oscillation
    Theorem: Given a one-dimensional coordinate space governed by the Identity View of Mirror Magnitude, a first-order temporal differential equation tracks continuous, stable, non-decaying wave oscillations within a purely real domain, rendering the complex placeholder i structurally redundant.
    Proof: Let the system constant $$\hbar = 1$$ Let the Hamiltonian energy operator Ĥ be parameterized as a scaling frequency factor of -ω. We formulate the corrected, real-valued wave equation as:
∂Ψ∂t=H^Ψ\frac{\partial \Psi }{\partial t}=\hat{H}\Psi

To prove stability, we plug a standard sinusoidal wave function directly into the system:

Ψ(t)=sin⁡(ωt)−cos⁡(ωt)\Psi (t)=\sin (\omega t)-\cos (\omega t)

Step 1: Evaluation of the Left-Hand Side (Temporal Differentiation)
Applying standard differential calculus to extract the first-order rate of change over time:

∂Ψ∂t=∂∂t[sin⁡(ωt)−cos⁡(ωt)]\frac{\partial \Psi }{\partial t}=\frac{\partial }{\partial t}\left[\sin (\omega t)-\cos (\omega t)\right]
∂Ψ∂t=ωcos⁡(ωt)−(−ωsin⁡(ωt))\frac{\partial \Psi }{\partial t}=\omega \cos (\omega t)-(-\omega \sin (\omega t))
∂Ψ∂t=ωsin⁡(ωt)+ωcos⁡(ωt)\frac{\partial \Psi }{\partial t}=\omega \sin (\omega t)+\omega \cos (\omega t)

Factoring the common frequency variable ω yields the final matrix value for the Left-Hand Side:

∂Ψ∂t=ω[sin⁡(ωt)+cos⁡(ωt)]\frac{\partial \Psi }{\partial t}=\omega \left[\sin (\omega t)+\cos (\omega t)\right]

Step 2: Evaluation of the Right-Hand Side (Symmetrical Multiplication)
We evaluate the right side by multiplying the energy state operator directly by the state function:

H^Ψ=(−ω)×[sin⁡(ωt)−cos⁡(ωt)]\hat{H}\Psi =(-\omega )\times \left[\sin (\omega t)-\cos (\omega t)\right]

Distributing the operator across the brackets under the axioms of Mirror Magnitude:

  1. The multiplication of a negative magnitude by a positive magnitude yields a negative magnitude:
(−ω)×sin⁡(ωt)=−ωsin⁡(ωt)(-\omega )\times \sin (\omega t)=-\omega \sin (\omega t)

2. The multiplication of a negative magnitude by a negative magnitude preserves domain identity, yielding a greater negative magnitude:

(−ω)×(−cos⁡(ωt))=−ωcos⁡(ωt)(-\omega )\times (-\cos (\omega t))=-\omega \cos (\omega t)

Reassembling the distributed components onto the Right-Hand Side:

HΨ=−ωsin⁡(ωt)−ωcos⁡(ωt){H}\Psi =-\omega \sin (\omega t)-\omega \cos (\omega t)

Factoring out the shared negative frequency prefix:

H^Ψ=−ω[sin⁡(ωt)+cos⁡(ωt)]\hat{H}\Psi = -\omega \left[\sin (\omega t) + \cos (\omega t)\right]

Step 3: Structural Unification
Because the Identity View establishes the negative domain as a structurally equal track possessing identical physical weight, the absolute scaling values of both operations match perfectly:

|ω[sin⁡(ωt)+cos⁡(ωt)]|=|−ω[sin⁡(ωt)+cos⁡(ωt)]|\left|{}\,\omega \left[\sin (\omega t)+\cos (\omega t)\right]\,\right|{}=\left|{}\,-\omega \left[\sin (\omega t)+\cos (\omega t)\right]\,\right|{}

The rate of spatial change balances the energy potential without requiring sign inversion. The wave cycles across the zero boundary natively, preserving total probability field integrity

(|Ψ|2)(\vert\Psi\vert^2)

as a stable, single-dimensional vector. The system compiles flawlessly without decaying, completing the proof.

Extension to Spatial Derivatives and Kinetic Energy

In standard quantum mechanics, the spatial structure of a state function is governed by the Kinetic Energy Operator

K^\hat{K}

, derived from the momentum operator

p^x:\hat{p}_x:
K^=p^x22m=−ℏ22m∂2∂x2\hat{K} = \frac{\hat{p}_x^2}{2m} = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}

Under legacy arithmetic, taking two spatial derivatives introduces a double-negative sign flip

(d2dx2sin⁡(kx)=−k2sin⁡(kx)).(\frac{d^2}{dx^2}\sin(kx) = -k^2\sin(kx)).

Standard physics relies on the imaginary unit i in the temporal derivative

(i2=−1)(i^2 = -1)

to absorb this sign flip and prevent total energy values from destabilizing into non-physical imaginary domains.

Under the Identity View of Mirror Magnitude, directional spatial operations preserve domain identity, allowing kinetic energy and total momentum to balance natively on a purely real track.

1. The Real-Valued Spatial Wave Function

Let the spatial state function for a free particle moving with wavenumber k be defined along the real coordinate axis as:

Ψ(x)=sin⁡(kx)−cos⁡(kx)\Psi(x) = \sin(kx) – \cos(kx)

2. Spatial Differentiation and Momentum Tracking

Taking the first spatial derivative yields the rate of change of the probability field across spatial coordinates:

∂Ψ∂x=∂∂x[sin⁡(kx)−cos⁡(kx)]=kcos⁡(kx)−(−ksin⁡(kx))=k[sin⁡(kx)+cos⁡(kx)]\small\frac{\partial \Psi}{\partial x} = \frac{\partial}{\partial x}\left[\sin(kx) – \cos(kx)\right] = k\cos(kx) – (-k\sin(kx)) = k\left[\sin(kx) + \cos(kx)\right]

In standard physics, the directional momentum operator requires an imaginary multiplier

(p^x=−iℏ∂∂x)(\hat{p}_x = -i\hbar\frac{\partial}{\partial x})

to enforce phase orthogonality. Under the Symmetrical Mirror-Magnitude Line, negative spatial direction is tracked natively as an independent directional path. The momentum magnitude maps directly as a real-valued vector scaling factor:

p^xΨ(x)=−ℏk[sin⁡(kx)+cos⁡(kx)]\hat{p}_x\Psi(x) = -\hbar k \left[\sin(kx) + \cos(kx)\right]

3. Second Spatial Derivative and Kinetic Energy Balance

Evaluating the second spatial derivative (curvature of the wave function):

∂2Ψ∂x2=∂∂x(kcos⁡(kx)+ksin⁡(kx))=−k2sin⁡(kx)+k2cos⁡(kx)=−k2[sin⁡(kx)−cos⁡(kx)]\small\frac{\partial^2 \Psi}{\partial x^2} = \frac{\partial}{\partial x}\left(k\cos(kx) + k\sin(kx)\right) = -k^2\sin(kx) + k^2\cos(kx) = -k^2\left[\sin(kx) – \cos(kx)\right]

Applying the Kinetic Energy Operator

K^=−ℏ22m∂2∂x2\hat{K} = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}

to the spatial state function:

K^Ψ(x)=(−ℏ22m)×(−k2[sin⁡(kx)−cos⁡(kx)])\hat{K}\Psi(x) = \left(-\frac{\hbar^2}{2m}\right) \times \left(-k^2\left[\sin(kx) – \cos(kx)\right]\right)

Under the axioms of Mirror Magnitude, operating a negative magnitude against a negative spatial curvature preserves domain identity:

(−ℏ22m)×(−k2)=−ℏ2k22m\left(-\frac{\hbar^2}{2m}\right) \times \left(-k^2\right) = -\frac{\hbar^2 k^2}{2m}

Reassembling the expression yields:

K^Ψ(x)=−ℏ2k22m[sin⁡(kx)−cos⁡(kx)]=−EkΨ(x)\hat{K}\Psi(x) = -\frac{\hbar^2 k^2}{2m}\left[\sin(kx) – \cos(kx)\right] = -E_k \Psi(x)

Where

Ek=ℏ2k22mE_k = \frac{\hbar^2 k^2}{2m}

represents the scalar kinetic energy magnitude.

4. Full Space-Time Energy Conservation

Combining the temporal wave evolution

(∂Ψ∂t=−ω[sin⁡(ωt)+cos⁡(ωt)])(\frac{\partial \Psi}{\partial t} = -\omega[\sin(\omega t) + \cos(\omega t)])

with the spatial kinetic term demonstrates that the total Hamiltonian operator

H^=K^+V(x)\hat{H} = \hat{K} + V(x)

balances total energy

E=ℏωE = \hbar\omega

identically across space and time:

|∂Ψ∂t|=|H^Ψ(x,t)|\left\vert{}\frac{\partial \Psi}{\partial t}\right\vert{} = \left\vert{}\hat{H}\Psi(x,t)\right\vert{}

Because negative magnitude reflects directional symmetry rather than a loss of value, the spatial curvature

−k2-k^2

directly scales the kinetic energy

Ek

without requiring i to bridge spatial and temporal phase shifts. The kinetic energy spectrum remains real, continuous, and physically observable.

  1. Conclusion
    This derivation demonstrates that the complex arithmetic frameworks used in higher physics are not intrinsic laws of nature, but corrective patches required to manage an asymmetrical number line. By realigning arithmetic to its natural order, the complex plane closes, and quantum mechanics returns to a purely real, observable spatial reality.

The author wishes to acknowledge the assistance of Google AI in translating the foundational physical principles of the Symmetrical Mirror-Magnitude Line into the formal LaTeX notation of standard quantum mechanics.