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Unformatted text preview: Introduction to Introduction to Nanoelectronics Nanoelectronics Prof. Supriyo Datta ECE 453 Purdue University Network for Computational Nanotechnology 09.24.2004 Lecture 13: Atomic Energy Levels Ref. Chapter 2.3 & 3.1 3D Schrödinger Equation 00:03 • 3D Schrödinger equation is: • Considering only one dimension: • We’ve learned how to turn this into a matrix equation using method of finite differences through which one can find the eigenvalues and eigenvectors. • In 3D, we’ve discussed that the size of matrices can get very large if one tries to turn the whole 3D equation into a matrix equation. But under certain conditions we can use the idea of separation of variables which is very powerful in making the problem more tractable. (discussed last day) • We can use Separation of variables if the potential can be written as: •If that is satisfied the one 3 dimensional problem can be written as 3 one dimensional problems which is easier to handle. • What we want to talk about today is how the energy levels look like for atoms. We start with the simplest of all atoms: hydrogen atom. • The potential that the electron sees in the hydrogen atom is: • The expression for “r” shows that this potential is not separable in Cartesian coordinates but it will be in spherical coordinates… Φ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ∇ − =...
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- Spherical Harmonics