# Ladder operators depending on all variables for a charged particle moving in a two-dimensional uniform magnetic field

## TL;DR

This paper extends the concept of ladder operators to a two-dimensional charged particle in a uniform magnetic field, allowing these operators to depend on all spatial variables and expressing the Hamiltonian via particle velocity.

## Contribution

It introduces a novel method for constructing variable-dependent ladder operators in a 2D magnetic field, generalizing previous 1D harmonic oscillator approaches.

## Key findings

- Ladder operators depending on all spatial variables are established.
- The Hamiltonian can be expressed in terms of particle velocity.
- The method broadens the applicability of ladder operators in quantum systems.

## Abstract

The ladder operators for one dimensional quantum harmonic oscillator were constructed by Schr\"odinger in 1940s. We extend this method to a two dimensional uniform magnetic field and establish the ladder operators which depend on all spatial variables of quantum system. The Hamiltonian of quantum system can also be written by the velocity of the particle.