Numerov Method for Solving Schrodinger's 1D Time Independent Wavefunction Equation

Numerov's Method

Below I summarize 'all I need to know'™ about the Numerov method for solving a family of second order differential equations, including the Schrodinger equation.

It's basically a sixth order error Taylor expansion in the wavefunction Psi, followed by a second order error central difference approximation for the fourth derivative of Psi. The combination of terms preserves a sixth order error in a single step.

Some sources say the accurate global error in this scheme is of fourth order, not fifth order as one might naively suppose, but I have not found or divined by my insufficient brainpower how one goes about proving this type of thing. So I'll duly report it's a fourth order scheme without any justification.

Typing this out in LaTex is beyond my current abilities with the language, and would require more hours, brain cells, and frustrations than I'm willing to invest right now. Also a liver transplant. So I'm just going to post the pictures of handwritten notes that I've carefully crafted.

Check the references if you want nicely formatted equations to steal using Copy & Paste. Cuz that's what I do.

Theory.  

(Derivation of the numerical formula)

Application. 

(Computing an explicit numerical formula to solve the dimensionless form of the 1D time independent harmonic oscillator)

References: 

(Where I learned/stole/plagiarized all this shit from)

• main guide to follow

- http://www.fisica.uniud.it/~giannozz/Corsi/MQ/LectureNotes/mq-cap1.pdf

• numerical difference formulas reference

- Numerical Methods Using Matlab, 4th Edition, 2004John H. Mathews and Kurtis K. FinkISBN: 0-13-065248-2

• dimensionless form of the schrodinger equation

- https://en.wikipedia.org/wiki/Nondimensionalization

• alternative explanation

- Generalized Matrix Numerov Solutions to the Schrodinger Equation http://www.physics.nus.edu.sg/student/Honours%20Projects%20Repository%202013-14/TANG_DONGJIAO%20thesis.pdf