Numerical Solutions of Schrödinger Equation

• In quantum mechanics, the Schrödinger equation is used to find the eigenvalues and eigenfunctions of the particle moving in one dimension at the potential $V(x)$:

  $$-\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2}+V(x)\psi(x)=E\psi(x)$$

• An analytical solution to this equation is available for only very few potential functions such as harmonic oscillator, infinite well, hydrogen atom, ....
• Therefore, numerical solutions of the Schrödinger equation is an indispensable tool in physics research.
• The numerical solution of the Schrödinger equation is complicated for general solutions of the problem.
• However, if we assume the potential function to be as symmetric, the problem can be solved in a much easier way.

• For a symmetric potential,

  $$V(-x)=V(x)$$

• Therefore, the solutions of the Schrödinger equation also fall into two groups:

  Symmetrical wave functions:	$\psi(-x)=\psi(x)$
  Antisymmetric wave functions:	$\psi(-x)=-\psi(x)$

• This property allows us to determine exactly the initial conditions necessary to start the eigenvalue problem.

  Symmetric (even) wave functions:	$\psi(0)=1$	&	$\psi'(0)=0$
  Antisymmetric (odd) wave functions:	$\psi(0)=0$	&	$\psi'(0)=1$