# How to plot numerically the wave functions according to the Hamiltonian?

It is often difficult to analytically solve the Schrodinger equation, and so we need to obtain a solution numerically. An example plot is shown below.

Here, the wave functions for a three junction flux qubit are plotted. The Hamiltonian is $$\frac{1}{2} \frac{\hat{P}_p^2}{M_p} + \frac{1}{2}\frac{\hat{P}_m}{M_m} + E_J \left[ 2 + \alpha - 2 \cos \hat{\phi_p} \cos \hat{\phi_m} - \alpha \cos \left( 2 \pi f_x^\text{DC} + 2 \hat{\phi}_m \right) \right]$$ where $$\hat{P}_p \equiv -i \hbar \frac{\partial}{\partial \hat{\phi}_p} \qquad \hat{P}_m \equiv -i \hbar \frac{\partial}{\partial \hat{\phi}_m}$$ and $M_p$ and $M_m$ are the effective masses of the two coordinates.

I don’t fully understand how to achieve from the equation to the figure. What method or software can be used to solve this problem?

• Can you give reference of the book. – hsinghal Jun 25 '16 at 3:14
• wmi.badw.de/publications/theses.htm《New Trends in Superconducting Circuit Quantum Electrodynamics: Two Amplifiers, Two Resonators, and Two Photons》 2009 PhD thesis, Mariantoni, Matteo – perlatex Jun 25 '16 at 3:33
• as mentioned on page 76 – perlatex Jun 25 '16 at 3:43
• This problem is slightly nontrivial because there are two degrees of freedom. Let's start at the beginning: do you know how to numerically solve a Hamiltonian with one degree of freedom? – DanielSank Jun 25 '16 at 4:24
• Do you have an idea how to solve partial differential equations? If not, I would suggest you start with the "simple" ones, like a diffusion equation. – CuriousOne Jun 25 '16 at 6:15