Quantum Mechanics in three dimensions

The Schrodinger Equation The time dependent Schrodinger equation: can be “separated” to get the time-independent Schrodinger equation which can be used to find the “stationary states” or standing waves in a potential.

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?

The Schrodinger Equation in Three Dimensions “Laplacian”

Particle in a 3-dimensional box U=0 inside the box Leads to “degenerate” states: unique states with the same energy!

A visualization: two dimensional box First Excited State Ground State Second Excited State

Spherical coordinates …make the most sense when describing atoms. f r q

The Schrodinger Equation in Spherical Coordinates conversion from cartesian coordinates to spherical polar coordinates Laplacian in spherical polar coordinates: The Schrodinger equation in spherical polar coordinates:

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:

The spherical harmonics

Quantization of Angular momentum +1 +2 +3 +4 m -1 -2 -3 -4

The Bohr Atom Revisited Classically: Bohr figured out that angular momentum was actually quantized: The Schrodinger equation in three dimensions gives us another insight as to why that is: