1. Modifying the Schrödinger Equation
Quantum Mechanics in
3-D
Modifying the Schrödinger Equation
The real universe has three space dimensions: x, y, and z
What needs to change about this formula?
Wave function needs to be more complicated
The momentum p is now a vector

2. Time Independent 3D Schrödinger
If the potential is independent of time
We have reduced our wave function from four variables to three (good)
But it’s still a partial differential equation (bad)
Need to find tricks to make this problem solvable
The interpretation of the wave function is pretty much the same
The amplitude squared is the probability density
To find probability in a region, integrate
Have to normalize like before

3. Separation of Variables
We need to solve this equation – try separation of variables?
Substitute it in, divide by wave function
none
???
x only
y only
z only
This method rarely works, because naturally occurring problems are rarely set up in Cartesian coordinates
Two problems we will solve:
Free particle
Particle in a 3D box

4. Free particle in 3D none x only y only z only
First term is pure function of x, but nothing else has x in it
Therefore, this term must be independent of x
Same argument applies to the other two terms
We can easily solve all these equations

5. Particle in a 3D box Lz We get same differential equations as before:
This time it is more useful to get real solutions:
Must vanish at x = 0 and x = Lx
Same type of solutions for y and z
Need to normalize wave functions Ly Lx

6. In math notation,  and  are swapped
Spherical Coordinates
Very few problems have “Cartesian symmetry”
Look at hydrogen-like atom
Many problems have spherical symmetry
Independent of angles z
r sin
Switch from Cartesian to Spherical Coordinates
r is the distance from the origin to the point
 is the angle compared to the z-axis
 is the angle of the projected “shadow” compared to the x-axis
r cos  r y 
r sincos
r sinsin x
Note that  = 0   = 2
In math notation,  and  are swapped

7. Derivatives in Spherical Coordinates
WARNING: This is nasty!
We need to write derivatives in terms of the new coordinates
Think of x, y, and z as functions of r, ,  and use the chain rule
Work, work, . . .
Let’s rewrite Schrödinger’s equation in spherical coordinates now

8. Schrödinger’s Eq. in Spherical Coords.
Assume potential depends only on r, and call the mass 
Change to spherical coordinates on the left
Multiply result by r2
Now we can try separation of variables, this time in new coordinates
Substitute in
Divide by the wave function, and bring first term on left over to the right

9. Schrödinger’s Eq. in Spherical Coords. (2)
Left side is independent of r
Right side is independent of  and 
Both sides must be constant – call them -L2
Multiply first equation by –Y
Multiply second equation by R/2r2

10. The Problem Broken in Two
No dependence on V(r)
Can be solved for all spherically symmetric problems
No more partial derivatives!
Looks like 1D-Schrödinger
The L2 term looks like an addition to the potential
The effective potential is just this term
Very similar to how classical mechanics solves this problem

11. Solving the angle equation
Strategy:
Guess some solutions
Rotate them and find some more

12. Solving the angle equation (2)
Will any l work?
We want Y to be finite
l  0
We want it to be continuous
l is an integer
l = 0, 1, 2, 3, …
Note that  = 0   = 2
Rotate them and find some more
Example: 180º rotation around x - axis
Work, work, until you find as many as you can
Normalize them (more on this later)

13. Spherical Harmonics
The functions you get this way are called spherical harmonics
They arise in any problem with spherical symmetry
The angular solutions are always the same
Look them up, don’t calculate them
What do l and m really mean?
Consider angular momentum operator
z-angular momentum is m
Total angular momentum squared is 2(l2+l)

14. Sample Problems
An electron in a spherically symmetric potential has a total angular momentum squared of 62. If we measure the angular momentum around the z –axis, what are the possible outcomes?
An electron in a spherically symmetric potential has a total angular momentum quantum number l < 3. How many possible pairs (l,m) are there?