# qUANTUM zeno effect

### The **quantum Zeno effect** is a feature of quantum mechanical systems allowing a particle's time evolution to be arrested by measuring it frequently enough with respect to some chosen measurement setting.

Sometimes this effect is interpreted as:

"a system can't change while

you are watching it."

### MEASUREMENT

In quantum mechanics, the interaction mentioned is called "measurement" because its result can be interpreted in terms of classical mechanics.

Frequent measurement prohibits the transition. It can be a transition of a particle from one half-space to another, a transition of a photon in a waveguide from one mode to another, and it can be a transition of an atom from one quantum state to another.

### Dr. BenJamin Schumacher

I had the opportunity to perform an independent research study for completion of my capstone requirements for my scientific computing concentration from Kenyon. Working under Dr. Benjamin Schumacher, we set out to investigate this phenomena using numerical methods.

First we needed to develop a C++ program for time-evolving non-linear differential equations modeling both quantum mechanical particles and environments.

### time-dependent SCHRODINGER equation

This program numerically integrates the Schrodinger equation on finite complex scalar fields for simulating interactions of quantum particles under varied observation.

### Split-step Fourier Method

This version implements a second-order in time finite difference method known as the "split-step" Crank-Nicolson method. By calculating energy states using the hamiltonian in both position and momentum space, this program is able to achieve numerically stable integration, which is necessary for finite difference methods.

### Back and Forth

Each time-iteration, the program evolves the wave function in the position basis. Then we apply a Fourier transform the wave function to evolve the non-linear term of the Hamiltonian in the momentum basis/phase-space. The waveform is then reverse Fourier transformed back into position space in order to repeat this evolution of the waveform.