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Non-Relativistic Quantum Mechanics in Multidimensional Geometric Frameworks

This paper develops a generalized formulation of non-relativistic quantum mechanics within multidimensional geometric frameworks characterized by power-law dispersion relations, deriving a consistent $j$ -th order Schrödinger equation that modifies spectral scaling and eigenfunction structures while preserving translational invariance and the Heisenberg uncertainty principle.

Original authors: Dalaver H. Anjum, Shahid Nawaz, Muhammad Saleem

Published 2026-03-31

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CC BY 4.0

Original authors: Dalaver H. Anjum, Shahid Nawaz, Muhammad Saleem

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant video game. For the last century, the "physics engine" running our world has been based on a specific set of rules called Standard Quantum Mechanics. In this game, space is "Euclidean" (like a standard grid), and the rules for how particles move and have energy are based on a simple, quadratic formula (like the area of a square: side × side).

This paper asks a fascinating "What if?" question: What if the universe ran on a different geometry?

What if, instead of a square grid, space was shaped like a triangle, a star, or a complex 4D shape? The authors propose a new framework called Multidimensional Geometric (NG) Frameworks. They suggest that if the "shape" of space changes, the laws of physics (specifically how particles move and how much energy they have) must change to match that shape.

Here is a breakdown of their ideas using simple analogies:

1. The Shape of Space Changes the Rules of the Road

In our normal world (3D space), if you drive a car, the energy it takes to speed up depends on the square of your speed ( $v^2$ ). This is like rolling a ball on a flat, smooth floor.

The authors imagine a world where space is "curved" or "stretched" in a specific mathematical way (called an $L_j$ -norm).

In our world (3G): The energy rule is a square ( $v^2$ ).
In their new world (4G, 5G, etc.): The energy rule becomes a cube ( $v^3$ ), a fourth power ( $v^4$ ), and so on.

The Analogy: Imagine walking on a flat sidewalk (our world). It's easy to predict how fast you'll go. Now, imagine walking on a surface made of giant, bouncy springs or steep, jagged rocks (the new geometry). The way you move, how much energy you need to take a step, and how you bounce back are completely different. The authors wrote a new "rulebook" for how particles move on these weird surfaces.

2. The "Particle in a Box" Experiment

To test their new rules, they looked at a classic physics problem: a particle trapped inside a box (like a ping-pong ball in a sealed shoebox).

In our world (3G): The ball bounces back and forth. It can only have specific, distinct energy levels (like rungs on a ladder). The math describing its position is a simple wave (sine/cosine).
In the new worlds (4G, 5G):
- The Energy Ladder: The "rungs" on the ladder get spaced out differently. In 4D space, the energy levels grow much faster (cubic growth). In 5D, they grow even faster (quartic growth).
- The Wave Shape: In our world, the particle's wave looks like a smooth hill. In the new worlds, the wave gets "weird." It becomes a mix of smooth waves, sharp spikes, and exponential curves. It's like the wave is trying to fit into a box that has corners made of glass and rubber at the same time.

3. The "Ghost" Problem (2D Space)

They also looked at a 2D version of this world.

The Result: In this specific 2D geometry, the particle cannot be trapped.
The Analogy: Imagine trying to catch a ghost in a cage. In our world, the cage works. In this 2D world, the "cage" is made of a material the ghost can simply pass through. The particle just flows out. This suggests that in a 2D universe with these specific rules, matter couldn't form atoms or stable structures; everything would just fly apart.

4. The "Probability" Puzzle

In quantum mechanics, we don't know exactly where a particle is; we only know the probability of finding it.

In our world: To find the probability, we multiply the wave by its "mirror image" (complex conjugate). It's like checking a reflection in a mirror to make sure the image is real.
In the new worlds: The math gets complicated. To get a real, physical probability, you can't just use one mirror. You have to use multiple mirrors (or "conjugates") at once.
- In 4D space, you need 3 mirrors.
- In 5D space, you need 4 mirrors.
- Only when you combine all these reflections do you get a real, measurable number. It's like trying to see a 3D object by looking at it from four different angles simultaneously to get the full picture.

5. The Uncertainty Principle Still Holds

One of the most famous rules in physics is the Heisenberg Uncertainty Principle: You can't know a particle's position and speed perfectly at the same time.

The authors proved that even in these weird, high-dimensional worlds, this rule still works.
However, the "fuzziness" (uncertainty) is different. In the new geometries, the fuzziness is generally larger than in our world. It's as if the "fog" in these universes is thicker, making it even harder to pin down exactly where things are.

The Big Picture Takeaway

The authors are essentially saying: "Quantum mechanics isn't a fixed set of laws written in stone; it's a set of laws that emerge from the shape of space itself."

If space is shaped like a square (our world), we get the physics we know.
If space is shaped like a cube or a star, the physics changes: energy scales differently, waves look different, and matter might not be able to stick together.

Why does this matter?
It helps us understand that our universe's specific rules might just be one possibility among many. It suggests that if we were observers living in a different geometric reality, we would see a completely different kind of quantum world, where particles behave like they are dancing to a different, more complex rhythm.

In short, they built a new mathematical engine to simulate how the universe would work if the "grid" of space was built differently, and they found that while the core rules of quantum mechanics survive, the details become much stranger and more complex.

1. Problem Statement

Standard non-relativistic quantum mechanics (NRQM) is formulated within a 3-dimensional Euclidean (3G) geometric framework, characterized by a quadratic kinetic energy term ( $E \propto p^2$ ) and a second-order spatial derivative in the Schrödinger equation. This formulation assumes the underlying space follows the standard Euclidean metric ( $L_2$ -norm).

The paper addresses the question: How does quantum mechanics change if the underlying spatial geometry is generalized to $N$ -dimensions (NG) characterized by a power-law dispersion relation $E \propto |p|^j$ , where $j = N-1$ ?

Existing extensions, such as the fractional Schrödinger equation, often introduce modified kinetic operators phenomenologically or via stochastic processes without a direct link to an underlying geometric principle. This work seeks to derive the dynamical laws of quantum mechanics directly from the metric structure of generalized $L_j$ -normed spaces.

2. Methodology

The authors develop a geometry-driven formulation by generalizing the Minkowski distance and kinetic energy operators based on the $L_j$ -norm.

Geometric Foundation: The distance between two points in an $N$ -dimensional space is defined via the $L_j$ -norm: $\Delta s_{NG} = (\sum |x'_i - x_i|^j)^{1/j}$ , where $j = N-1$ .
Generalized Kinetic Energy: The non-relativistic kinetic energy is derived from this metric as $(K.E.)_{non-rel.} = \frac{1}{j} (\frac{v}{c})^j mc^2 = \frac{p^j}{j m^{j-1} c^{j-2}}$ .
Operator Construction:
- The momentum operator $\hat{p}$ and energy operator $\hat{E}$ are constructed using the $j$ -th roots of negative unity, $(-1)^{1/j}$ .
- For $j \geq 2$ , the operators involve complex phases derived from these roots (e.g., for $j=3$ , cube roots of $-1$ are used).
- This leads to a generalized Schrödinger equation (SE) involving $j$ -th order spatial derivatives.
Generalized Probability Framework:
- To ensure real-valued probability densities in $L_j$ spaces, the authors introduce a $j$ -fold conjugation scheme.
- The probability density is defined as the product of the wavefunction and its $j-1$ conjugates (associated with the $j$ roots of $-1 $):$ P = \int \psi^_1 \psi^2 \dots \psi^*{j-1} \psi_j dx$.
- Expectation values and uncertainty measures are redefined using this multi-phase inner product structure.
Application: The formalism is applied to two specific systems:
1. Free Particles: To analyze dispersion relations and translational invariance.
2. Infinite Potential Well (Particle in a Box): To study bound states, quantization, and spectral scaling for $j=1$ (2G), $j=2$ (3G), $j=3$ (4G), and $j=4$ (5G).

3. Key Contributions

Derivation of $j$ -th Order Schrödinger Equation: The paper establishes a consistent SE where the kinetic operator is a $j$ -th order differential operator derived strictly from the $L_j$ -norm geometry.
Multi-Phase Probability Theory: It proposes a novel probability definition based on $j$ -fold conjugation, ensuring physical observables remain real-valued in higher-order geometric spaces.
Generalized Uncertainty Principle: It demonstrates that the Heisenberg Uncertainty Principle ( $\Delta x \Delta p \geq \hbar/2$ ) holds for all $j \geq 2$ , despite the modification of the uncertainty measure to an $L_j$ -norm based variance.
Spectral Scaling Laws: The work derives explicit scaling laws for energy eigenvalues and wavefunction structures across different geometric dimensions.

4. Key Results

A. Free Particles

Translational Invariance: Plane-wave solutions persist in the NG framework.
Dispersion Relation: The energy scales as $E \propto k^j$ $E \propto k^{j}$ .
- 2G ( $j=1$ ): Linear dispersion ( $E \propto k$ ). The wavefunction is purely exponential (decaying), lacking oscillatory behavior.
- 3G ( $j=2$ ): Standard quadratic dispersion ( $E \propto k^2$ ).
- 4G/5G ( $j=3,4$ ): Cubic and quartic dispersion relations.

B. Bound States (Infinite Potential Well)

2G ( $j=1$ ): No bound states exist. The solution to the first-order differential equation with Dirichlet boundary conditions yields only a trivial solution ( $\psi=0$ ). The particle behaves as a massless, free-propagating excitation.
3G ( $j=2$ ): Recovers standard results: Eigenenergies scale as $(2n+1)^2$ , and eigenfunctions are sinusoidal (cosine/sine).
4G ( $j=3$ ):
- Eigenfunctions: Acquire a mixed form: $e^{-kx/2} \cos(\frac{\sqrt{3}}{2}kx)$ . They exhibit exponential modulation combined with oscillation.
- Eigenenergies: Scale as $(2n+1)^3$ .
5G ( $j=4$ ):
- Eigenfunctions: Acquire a mixed hyperbolic-trigonometric form: $\sinh(\dots)\cos(\dots)$ .
- Eigenenergies: Scale as $(2n+1)^4$ .
Energy Magnitude: For physically relevant velocities ( $v \ll c$ ), the absolute energy levels in 4G and 5G are significantly lower than in 3G due to the suppression factor $(v/c)^{j-2}$ inherent in the generalized kinetic energy.

C. Uncertainty Relations

The paper calculates the uncertainty product $\Delta x \Delta p$ for the ground state in 3G, 4G, and 5G.
Results:
- 3G: $\approx 0.568 \hbar$ (Standard).
- 4G: $\approx 1.36 \hbar$ .
- 5G: $\approx 1.07 \hbar$ .
Conclusion: The Heisenberg inequality holds for all cases, but the uncertainty product increases significantly in higher-order geometries, indicating a "blurring" of quantum states compared to the 3G case.

5. Significance

Geometry-Dependent Dynamics: The work fundamentally recasts quantum mechanics as a theory where dynamical laws (dispersion, spectral structure, probability) are not fixed a priori but emerge from the metric structure of space.
Unified Framework: It provides a systematic way to compare quantum phenomena as perceived by observers in different geometric settings (e.g., an observer in an $L_3$ -norm universe would perceive cubic energy scaling as "natural").
Physical Implications:
- The suppression of energy gaps in higher dimensions suggests that "sharp" quantum discreteness is a specific feature of our 3G Euclidean geometry.
- The framework bridges non-relativistic quantum mechanics with relativistic scales (via the Compton frequency $mc^2/\hbar$ ) even in a non-relativistic limit, as higher-order geometries introduce explicit dependence on the relativistic momentum scale $mc$.
Theoretical Extension: It offers a rigorous alternative to fractional quantum mechanics, grounding higher-order derivatives in geometric principles rather than stochastic path integrals.

In summary, the paper establishes that while the core principles of quantization and uncertainty persist in multidimensional geometric frameworks, the specific spectral properties, wavefunction morphologies, and statistical spreads are profoundly altered by the dimensionality and norm structure of the underlying space.