)()(),(

)(

txtx

eee tiikxtkxi

φψ

ωω

=Ψ⇒= −−

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.

“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?

“Laplacian”

( )2322

212

2

321 2: nnn

mL

hELLLif ++===

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

U=0 inside the box

…make the most sense when describing atoms.

⎥⎦

⎤⎢⎣

⎡

∂∂

+∂∂

+∂∂

+∂∂

⎟⎠

⎞⎜⎝

⎛+∂∂

=∇2

2

22

2

22

22

sin

1

sin

cos12

φθθθ

θ

θrrrr

φθφθ

θ

sinsin

cossin

cos

rz

ry

rx

=

=

=conversion from cartesian coordinates to

spherical polar coordinates

Laplacian in spherical polar coordinates:

ErUd

d

rmd

d

d

d

rmdr

dR

rdr

Rd

mR=+

ΦΘ

−⎥⎦

⎤⎢⎣

⎡ Θ+

ΘΘ

−⎥⎦

⎤⎢⎣

⎡+

−)(

sin1

2cot

12

22 2

2

22

2

2

2

2

2

2

22

φθθθ

θhhh

The Schrodinger equation in spherical polar coordinates:

)()()()(where φθψ ΦΘ= rRr

Θ+−=Θ−Θ

+Θ

)1(sin

1

sin

cos2

22

2

lll θθθθ

θm

dd

dd

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:

2 20 11 1( ) ( ) 2 ( )f P Pθ θ θ= +

01P

11P

10y

11y

2 21 10 1( ) ( ) 2 ( )f y yθ θ θ= +

12P

22P

02P

2 2 20 1 22 2 2( ) ( ) 2 ( ) 2 ( )f P P Pθ θ θ θ= + +

2 2 22 2 20 1 2( ) ( ) 2 ( ) 2 ( )f Y Y Yθ θ θ θ= + +

20y

21y

22y

z

+1+2+3+4m

-1-2-3-4

0

cos ( 1)m L L = +

L=4

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: