V(x)=0for L>x>0

1
Particle in a 1-Dimensional Box
Time Dependent Schrödinger Equation
PE
KE
TE
Wave function is dependent on time and position
function
1
Time Independent Schrödinger Equation
V(x)0 for Lgtxgt0 V(x)8 for xL, x0
Applying boundary conditions
Classical Physics The particle can exist
anywhere in the box and follow a path in
accordance to Newtons Laws. Quantum Physics
The particle is expressed by a wave function and
there are certain areas more likely to contain
the particle within the box.
Region I and III
Region II
2
Finding the Wave Function
Our new wave function
But what is A?
This is similar to the general differential
equation
Normalizing wave function
So we can start applying boundary conditions
x0 ?0
xL ?0
where n
Calculating Energy Levels
Since n
Our normalized wave function is
3
Particle in a 1-Dimensional Box
Applying the Born Interpretation
n4
n4
n3
E
E
n3
n2
n2
n1
n1
x/L
x/L
4
Particle in a 2-Dimensional Box
A similar argument can be made
Doing the same thing do these differential
equations that we did in one dimension we get
Lots of Boring Math
In one dimension we needed only one n But in
two dimensions we need an n for the x and y
component.
Our Wave Equations
Since
For energy levels
5
Particle in a 2-Dimensional Equilateral Triangle
Types of Symmetry
Lets apply some Boundary Conditions
A
C23
Defining some more variables
w
v
u
So our new coordinate system
Our 2-Dimensional Schrödinger Equation
Solution
Substituting in our definitions of x and y in
terms of u and v gives
Where p and q are our nx and ny variables from
the 2-D box!
6
Finding the Wave Function
Substituting gives
So what is the wave equation?
It can be generated from a super position of all
of the symmetry operations!
So if
And we recall our original definitions
But what plugs into these?
If you recall
Substituting and simplifying gives
A1
Continuing with the others
A2
Energy Levels
7
Plotting in Mathematica
p1 q0
8
A1
9
A2