Question

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

Answer 1

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

12P^2pMp+12P^mMm+EJ[2+α−2cosϕp^cosϕm^−αcos(2πfDCx+2ϕ^m)]12P^p2Mp+12P^mMm+EJ[2+α−2cos⁡ϕp^cos⁡ϕm^−αcos⁡(2πfxDC+2ϕ^m)]

where

P^p≡−iℏ∂∂ϕ^pP^m≡−iℏ∂∂ϕ^mP^p≡−iℏ∂∂ϕ^pP^m≡−iℏ∂∂ϕ^m

and MpMp and MmMm 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

Answer 2

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