2 The Schrodinger Equation The time dependent Schrodinger equation:can be “separated” to get the time-independent Schrodinger equationwhich can be used to find the “stationary states” or standing waves in a potential.
3 The time-independent Schrodinger equation in 3 dimensions “Laplacian”Can we use our previous knowledge to guess some of the characteristics of a particle in a 3 dimensional “box”?What are the boundary conditions?What is the form of the wave function?Can you deduce anything about the ground state? Higher states?
4 The Schrodinger Equation in Three Dimensions “Laplacian”
5 Particle in a 3-dimensional box U=0 inside the boxLeads to “degenerate” states: unique states with the same energy!
6 A visualization: two dimensional box First Excited StateGround StateSecond Excited State
7 Spherical coordinates …make the most sense when describing atoms.frq
8 The Schrodinger Equation in Spherical Coordinates conversion from cartesian coordinates to spherical polar coordinatesLaplacian in spherical polar coordinates:The Schrodinger equation in spherical polar coordinates:
9 The polar solution The polar part of the Schrodinger equation is: With some rearrangement, this can be recognized as the associated Legendre equation:Luckily, someone has already solved this equation, so we don’t have to: