For the Crank Nicholson algorithm we again begin by defining TDSE:

.

Also, the same spatial expansion of (x) is used for d2/dx2 (x):

 

However, instead of separating the TDSE into real and imaginary parts, the Crank Nicholson Taylor expands the time evolution operator:

. 

Merging the time expansion of the Schrödinger equation and the spatial expansion of the Schrödinger Equation at this point produces an explicit equation that is neither unitary nor stable because of the estimations in the Taylor expansions. Substituting a finite difference equation for the time varying part creates an implicit, stable equation, but it is still not unitary. (A wave equation must be unitary by the properties of quantum mechanics meaning that the wave must normalize to one after each time step. Else, we risk creating or destroying matter/probability!)  Changing the time evolution operator to

results in an implicit, time varying SE

. 

Merging this new time varying Schrödinger equation into the spatial varying Schrödinger equation, results in a stable, unitary, but implicit equation:

,

where . 

To solve this massive, messy, and implicit equation, we use the spatially varying operator to step the original wave equation forward from x=j…J, then use the time evolution operator to step the new wave equation forward Δt from x=J… j. The wave function must be equal to zero at the endpoints so that Ψ(x+1) for x=0 and Ψ(x-1) for x=J remain constant when each step is taken. Below is a diagram of the algorithm:

 

  1. Define                                        

 

  1. Using the first boundary condition,  for all n, transverse the wave function from x=2 to x=J.

             

               x=1:     

                        

                          x=2..J:   use the above equations for and .

 

  1. Using the second boundary condition,  for all n, transverse the wave function backwards from x=J to x=1.

 

x=J-1:   

 

x=1:       

 

  1. Repeat steps 1-3, increasing t by Δt.

 

We set Δt =2dx2 from the example in Rubin H. Landau and Manuel J Paez’s book, Computational Physics with Computers. Theoretically, we could use any timestep, Δt, however, in we choose a small enough time step to see smooth progression of phase changes.