1. Quantum Mechanics
in three dimensions

2. 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.

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 box
Leads to “degenerate” states: unique states with the same energy!

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

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

8. 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:

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:

10. The spherical harmonics

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

17. 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: