This respository contains work done for my 4th year masters project in mathmemtatics at the University of Bristol: Exploring Qauntum Wave Propagation and Scattering. First we developed a method for evolving and animating an intial wavefunction in time, in the notebook titled ‘Wave_Animation’. We then developed code to plot the Wigner functions of said wave functions in the notebooks ‘Wigner_Plot’, and combined the evolution and Wigner plot to create animations of them in ‘Time_Evolving_Wigner’ and ‘Wigner_Animate’.
In the ‘Final_Results’ notebook, we detail the different scenarios and variables used in order to create them, including basic wave-barrier interactions, as well as more complicated situations in which we had more than one barrier. The implications of what we see are discussed in the ‘Animation Analyses’ section of my project, and we can see the animation in video format below. Finally, in ‘R_and_T_From_Schrodinger’, we use our wave animations to create plots of the reflection and transmission coefficients for varying energies in each scenario, using the integral of the modulus squared of the final reflected and transmitted waves.
The final masters project report can be found under Winterborne_J_Project_Report.pdf.
Videos
We link here the webpage containing all the animation scenarios set out below [https://jesswinterborne.github.io/Quantum_Wave_Propagation/].
Cat State
Wave function animation of a cat state
Wigner function animation of a cat state in a denisty plot
Wigner function animation of a cat state in a surface plot
Single Gaussian Potential
Wave function animation interacting with a single Gaussian potential with a height of $V_{0}$, where the energy of the wave is such that $E < V_{0}$.
Wigner function animation interacting with a single Gaussian potential with a height of $V_{0}$, where the energy of the wave is such that $E < V_{0}$.
Wave function animation interacting with a single Gaussian potential with a height of $V_{0}$, where the energy of the wave is such that $E = V_{0}$.
Wigner function animation interacting with a single Gaussian potential with a height of $V_{0}$, where the energy of the wave is such that $E < V_{0}$.
Wave function animation interacting with a single Gaussian potential with a height of $V_{0}$, where the energy of the wave is such that $E > V_{0}$.
Wigner function animation interacting with a single Gaussian potential with a height of $V_{0}$, where the energy of the wave is such that $E > V_{0}$.
Step Potential
Wave function animation interacting with a step potential with a height of $V_{0}$, where the energy of the wave is such that $E < V_{0}$.
Wigner function animation interacting with a step potential with a height of $V_{0}$, where the energy of the wave is such that $E < V_{0}$.
Wave function animation interacting with a step potential with a height of $V_{0}$, where the energy of the wave is such that $E = V_{0}$.
Wigner function animation interacting with a step potential with a height of $V_{0}$, where the energy of the wave is such that $E = V_{0}$.
Wave function animation interacting with a step potential with a height of $V_{0}$, where the energy of the wave is such that $E > V_{0}$.
Wigner function animation interacting with a step potential with a height of $V_{0}$, where the energy of the wave is such that $E > V_{0}$.
Rectangular Potential
Wave function animation interacting with a rectangular potential with a height of $V_{0}$, where the energy of the wave is such that $E < V_{0}$.
Wigner function animation interacting with a rectangular potential with a height of $V_{0}$, where the energy of the wave is such that $E < V_{0}$.
Wave function animation interacting with a rectangular potential with a height of $V_{0}$, where the energy of the wave is such that $E = V_{0}$.
Wigner function animation interacting with a rectangular potential with a height of $V_{0}$, where the energy of the wave is such that $E = V_{0}$.
Wave function animation interacting with a rectangular potential with a height of $V_{0}$, where the energy of the wave is such that $E > V_{0}$.
Wigner function animation interacting with a rectangular potential with a height of $V_{0}$, where the energy of the wave is such that $E > V_{0}$.
Double Gaussian Potential with Equal Heights
Wave function animation interacting with a double Gaussian with equal heights of $V_{0}$, where the energy of the wave is such that $E < V_{0}$.
Wigner function animation interacting with a double Gaussian with equal heights of $V_{0}$, where the energy of the wave is such that $E < V_{0}$.
Wave function animation interacting with a double Gaussian with equal heights of $V_{0}$, where the energy of the wave is such that $E = V_{0}$.
Wigner function animation interacting with a double Gaussian with equal heights of $V_{0}$, where the energy of the wave is such that $E = V_{0}$.
Wave function animation interacting with a double Gaussian with equal heights of $V_{0}$, where the energy of the wave is such that $E > V_{0}$.
Wigner function animation interacting with a double Gaussian with equal heights of $V_{0}$, where the energy of the wave is such that $E > V_{0}$.
Double Gaussian Potential with Unequal Heights
Wave function animation interacting with a double Gaussian with unequal heights of $V_{0}$ and $V_{1}$ respectively, where the energy of the wave is such that $E < V_{0}$.
Wigner function animation interacting with a double Gaussian with unequal heights of $V_{0}$ and $V_{1}$ respectively, where the energy of the wave is such that $E < V_{0}$.
Wave function animation interacting with a double Gaussian with unequal heights of $V_{0}$ and $V_{1}$ respectively, where the energy of the wave is such that $V_{0} < E < V_{1}$.
Wigner function animation interacting with a double Gaussian with unequal heights of $V_{0}$ and $V_{1}$ respectively, where the energy of the wave is such that $V_{0} < E < V_{1}$.
Wave function animation interacting with a double Gaussian with unequal heights of $V_{0}$ and $V_{1}$ respectively, where the energy of the wave is such that $E > V_{1}$.
Wigner function animation interacting with a double Gaussian with unequal heights of $V_{0}$ and $V_{1}$ respectively, where the energy of the wave is such that $E > V_{1}$.
Gaussian Potential Well with Equal Heights $V_{0}$
Wave function animation such that energy of the wave is such that $E < V_{0}$.
Wigner function animation such that energy of the wave is such that $E < V_{0}$.
Gaussian Potential Well with Unequal Heights $V_{0}$ and $V_{1}$ Respectively
Wave function animation such that energy of the wave is such that $E < V_{0}$.
Wigner function animation such that energy of the wave is such that $E < V_{0}$.