Draft:Ferhart's Series

Ferhart's Series

Ferhart's Series is a mathematical series introduced by student Nathan Ferhart in 2024. The series is defined by the function:

$y=a\sum _{n=1}^{b}\sin \left(\left(c+1.1\right)\operatorname {floor} \left(dnx^{2}\right)\right)$

where a, b, c, and d are constants, x is the variable of interest, and ⌊⋅⌋ denotes the floor function. All variables are initially set to 1 and are constrained such that they cannot equal zero.

Definition and Components

1. Variables and Constants:
   • a: Scaling factor applied to the entire sum.
   • b: Upper limit of the summation index n.
   • c: Base constant added to the factor inside the sine function.
   • d: Coefficient in the argument of the floor function.
   • x: Independent variable.
2. Function Analysis:
   • The floor function ⌊dnx2⌋ converts the product dnx2 to the greatest integer less than or equal to this product.
   • The sine function introduces periodic behavior dependent on the transformed argument (c+1.1)⌊dnx2⌋.
   • The summation runs from n=1 to n=b, aggregating the sine values over these intervals.

Physics-Based Use Cases

1. Modeling Quantum Mechanical Systems

Example: Quantum Harmonic Oscillator with Modulated Potential

In quantum mechanics, the behavior of particles in potential fields is described by wavefunctions. Suppose a quantum harmonic oscillator has a potential energy that varies periodically due to external factors. This modified potential can be represented by:

$V\left(x\right)=V_{0}+a\sum _{n=1}^{b}\sin \left(\left(c+1.1\right)\operatorname {floor} \left(dnx^{2}\right)\right)$

Case Study: For a=1, b=4, c=2, and d=1, the potential energy becomes:

$V\left(x\right)=V_{0}+\sum _{n=1}^{4}\sin \left(3.1\operatorname {floor} \left(nx^{2}\right)\right)$At x=1, this potential modifies the harmonic potential by adding periodic fluctuations. Solving the Schrödinger equation with this potential will yield wavefunctions that are influenced by the added periodic components, potentially resulting in new quantized energy levels and spatial distributions.

2. Condensed Matter Physics

Example: Artificial Lattices in Optical Systems

In condensed matter physics, researchers often study materials with periodic structures, such as optical lattices, where light creates a periodic potential for ultracold atoms. Using Ferhart's Series, one can model complex periodic potentials in these systems.

Case Study: Suppose an optical lattice is designed with a periodic potential influenced by:

$V\left(x\right)=V_{0}+2\sum _{n=1}^{6}\sin \left(\left(1+1.1\right)\operatorname {floor} \left(2nx^{2}\right)\right)$

This potential simulates complex periodic structures. Researchers can use this model to predict how particles will behave in such a lattice, including the formation of band structures and localization phenomena.

3. Experimental Physics

Example: Designing Periodic Potential Experiments

In experimental physics, the ability to create and manipulate periodic potentials allows for the exploration of new physical phenomena. Ferhart's Series can aid in designing experiments with non-standard potentials.

Case Study: To create a periodic potential for a specific experimental setup, one might use:

$V\left(x\right)=0.5\sum _{n=1}^{5}\sin \left(\left(0.5+1.1\right)\operatorname {floor} \left(3nx^{2}\right)\right)$

By adjusting a, b, c, and d, experimenters can explore how varying periodic potentials affect quantum states or classical particle motion, providing insights into phenomena such as resonance and non-linear dynamics.

Applications

• Quantum Mechanics: Analyzing wavefunctions and energy levels in systems with complex potentials.
• Condensed Matter Physics: Investigating materials with engineered periodic structures and their effects on particle behavior.
• Experimental Physics: Designing and interpreting experiments with custom periodic potentials to study various physical phenomena.

References

"Complex Potentials and Non-Hermitian Hamiltonians in Quantum Mechanics" by Carl M. Bender and Stefan Boettcher:

"Quantum Mechanics with Complex Potentials" by Carl M. Bender:

"Topological Insulators and Topological Superconductors" by M. Z. Hasan and C. L. Kane

"Cold Atoms in Optical Lattices: From Hubbard Models to Topological Quantum Matter" by Immanuel Bloch, Jean Dalibard, and Wilhelm Zwerger:

"Engineering Quantum Matter with Cold Atoms in Optical Lattices" by Immanuel Bloch

"Quantum Simulation of Many-Body Physics with Ultracold Atoms in Optical Lattices" by M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch: