# Construction of a few Quantum mechanical Hamiltonians via L$\acute{e}$vi-Leblond type Linearization:Spinor states and Supersymmetry

## TL;DR

This paper develops new Levy-Leblond type equations with four-component spinor solutions, leading to Hamiltonians exhibiting features like L-S coupling and supersymmetry, including models such as Rashba effect and harmonic oscillators.

## Contribution

It introduces novel Levy-Leblond type equations that generate Hamiltonians with L-S coupling and supersymmetry, expanding the theoretical framework for quantum mechanical systems.

## Key findings

- Hamiltonians with L-S coupling derived from new Levy-Leblond equations
- Construction of Rashba effect and harmonic oscillator Hamiltonians
- Identification of supersymmetry in one-dimensional oscillator

## Abstract

A number of new L$\acute{e}$vi-Leblond type equations admitting four component spinor solutions have been proposed. The pair of linearized equations thus obtained in each case lead to Hamiltonians with characteristic features like L-S coupling and supersymmetry. The relevant momentum operators have often been understood in terms of Clifford algebraic bases producing Schr$\ddot{o}$dinger Hamiltonians with L-S coupling. As for example Hamiltonians representing Rashba effect or three dimensional harmonic oscillator have been constructed. The supersymmetric nature of one dimensional oscillator has also been appreciated.