# Solving the

Time-Dependent-Schroedinger Equation

## Evolution of particle in different potentials:

free, step, square, harmonic

*This research project was part of my MSc Physics at Imperial College London in 2021 - one week project.*

As part of my studies at Imerpial I embarked on a 1 week project.
I did chose to play around with the centrepiece of quantum mechnaics -
the *Time-Dependent Schroedinger Equation (TDSE)*:

### Motivation

The aim of the 1 week project built something cool with *Mathematica*.
Having dealt with the time-independent version of the Schroedinger Equation in
countless probelem sheets and exams, problems and exercises with the time-dependent
version always seemed very distant. I wanted to gain intition via a nice animation.

This was the goal: Gain Intition of the TDSE in the standard potential wells.

### Project Summary

In this project the time-dependent Schrödinger Equation has been solved using the 'Finite Domain Time Difference Method' for different simple static potential wells for a Gaussian wave packet.

The behaviour of the particle waves exhibits the inherently quantum mechanical behaviour that is expected. The animations generated allow to gain physical intuition.

### Theory

The equation to solve is the **time-dependent Schrödinger Equation** (TDSE):

with H(x) the static Hamiltonian. For simplicity here a static potential was chooses.

$$ H(x) = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V(x) $$The TDSE can be rewritten applying the standard "Finite Time Difference Method" (see here):

$$ \frac{\partial \Psi(x,t)}{\partial t} = \frac{\Psi(x, t + \Delta t) - \Psi(x,t)}{\Delta t} $$ $$ \frac{\partial^2 \Psi(x,t)}{\partial x^2} = \frac{\Psi(x, t + \Delta t) - 2 \Psi(x,t) + \Psi(x-\Delta x,t)}{\Delta x^2} $$ Substituting this into the TDSE yields: $$ \Psi(x, t+\Delta t) = \Psi(x,t) - \frac{\Delta t \hbar}{2 i (\Delta x)^2 m} \Bigg[ \Psi(x+\Delta x,t) -2 \Psi(x,t) + \Psi(x-\Delta x, t)\Bigg] + \frac{\Delta t}{i \hbar} V(x) \Psi(x,t) $$ where we define the constants as $$ c_1 = \frac{\Delta t}{2 (\Delta x)^2 m} $$ and $$ c_1 =\frac{\Delta t}{\hbar} $$ Then taking the real and imaginary parts of the wavefucntion one finally arrives at the equations used for the calculation: $$ \boxed{ \Phi_R(x,t+\Delta t) = \Phi_R(x,t) - c_1 \Bigg[\Phi_I(x+\Delta x, t)+ 2 \Phi_I(x,t) +\Phi_I(x-\Delta x, t) \Bigg] +c_2 V(x) \Phi_I(x,t)} $$ $$ \boxed{ \Phi_I(x,t+\Delta t) = \Phi_I(x,t) + c_1 \Bigg[\Phi_R(x+\Delta x, t)+ 2 \Phi_R(x,t) +\Phi_R(x-\Delta x, t) \Bigg] +c_2 V(x) \Phi_R(x,t)} $$ These two equations are the core of the simulation. For the purpose of generating a meaningful animation the constants c_{1}and c

_{2}have been slightly adapted.

^{[1]}The

**wavefunction**defining the particle to simulate is chosen to be an well defined electron. In order to define the particle well a Gaussian wave packet is chosen. For a Gaussian wavepacket a single wavevector can be defined as its Fourier transform frequencies are clustered around that single wave vector. This allows to define a somewhat localized particle. The Gaussian wavepacket is: $$ \Psi(x, 0) = \Phi_0(x) = \frac{1}{\sqrt{\sigma \sqrt{\pi}}} e^{- \frac{x^2}{2*\sigma^2}} $$ let us define the normalisation as: $$\sqrt{\sigma \sqrt{\pi}} = A$$ Then the modulated real and imaginary parts are \begin{equation} \Phi_R(x,0) = \frac{1}{A} e^{-\frac{1}{2} \big( \frac{x-x_0}{s} \big)^2} cos[k (x-x_0)] \end{equation} \begin{equation} \Phi_I(x,0) = \frac{1}{A} e^{-\frac{1}{2} \big( \frac{x-x_0}{s} \big)^2} sin[k (x-x_0)] \end{equation} where the normalization constant can be found by integration according to the Born Rule: \begin{equation} 1 = \int_{-\infty}^{\infty} |\Psi(x,t)|^2 \,dx \end{equation} The particles momentum is specified using the deBroglie relations: \begin{equation} p = \hbar k \end{equation} where k is the wavevector defines as: \begin{equation} k = \frac{2 \pi}{\lambda} \end{equation} The

**potentials**chosen to present here are all finite potentials:

- finite square well
- finite step well
- quadratic/harmonic

_{e}. Therefore the energy of the chosen electron particle has a kinetic energy as a free particle given by the deBroglie relation as: \begin{equation} E = \hbar \omega = \frac{h c}{\lambda} \approx 6.625*10^{-18} J\approx 41.35 eV \end{equation} Therefore to explore the interesting behaviour of a bound particle we choose the welldepth or wellheight to be 80eV.

### Aims & Objectives

#### Aims

- solve the time dependent Schrödinger equation using the 'Finite Domain Time Difference Method'
- demonstrate the working method on a few potentials
- comment on physical intuition gained

#### Objectives

- discretize space and time
- implement a physical potential parameters
- implement physical particle/wave parameters
- generate potential wells: free, square, harmonic, step
- demonstrate potential wells

### Results

See Sum-up video <here:The individual simulations are also shown here:

#### Finite Square Well

#### Finite Square Well

#### Quadratic Well

#### Step Well

Note the obvious quantum tunneling here.