9: Numerical Solutions for Schrödinger's Equation
- Page ID
- 127067
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)
Numerically solving the Schrödinger equation is a complex problem that stems from the large number of points needed on a grid and the requirement to satisfy boundary conditions.
- 9.1: Introduction to Numerical Solutions of Schödinger's Equation
- Solving Schrodinger's equation is the primary task of chemists in the field of quantum chemistry. However, exact solutions for Schrödinger's equation are available only for a small number of simple systems. Therefore the purpose of this tutorial is to illustrate one of the computational methods used to obtain approximate solutions.
- 9.7: Particle in a Slanted Well Potential
- Numerical Solutions for Schrödinger's Equation for the Particle in the Slanted Box.
- 9.8: Numerical Solutions for a Particle in a V-Shaped Potential Well
- Schrödinger's equation is integrated numerically for a particle in a V-shaped potential well.
- 9.10: Numerical Solutions for a Double-Minimum Potential Well
- Schrödinger's equation is integrated numerically for a double minimum potential well: V = bx⁴ - cx².
- 9.11: Numerical Solutions for the Quartic Oscillator
- Schrödinger's equation is integrated numerically for the first three energy states for the quartic oscillator.
- 9.12: Numerical Solutions for Morse Oscillator
- Schrödinger's equation is integrated numerically for the first three energy states for the Morse oscillator.
- 9.13: Numerical Solutions for the Lennard-Jones Potential
- Merrill (Am. J. Phys. 1972, 40, 138) showed that a Lennard-Jones 6-12 potential with these parameters had three bound states. This is verified by numerical integration of Schrödinger's equation.
- 9.16: Another Look at the in a Box with an Internal Barrier
- The purpose of this tutorial is to explore the impact of the presence of a large (100 Eh) thin (0.10 a0) internal barrier on the solutions to the particle-in-a-box (PIB) problem. Schrodinger's equation is integrated numerically for the first five energy states.