1. Free particle in 1D (1) 1D Unbound States
An unbound state occurs when the energy is sufficient to take the particle to infinity, E > V().
Free particle in 1D (1)
In this case, it is easiest to understand results if we use complex waves
Include the time dependence, and it is clear these are traveling waves
e+ikx e-ikx
Re() Im()

2. Wave packets are a pain, so we’ll use waves and ignore normalization
Free particle in 1D (2)
Comments:
Note we get two independent solutions, and solutions for every energy
Normalization is a problem
Wave packets solve the problem of normalization
The general solution of the time-dependant Schrödinger equation is a linear combination of many waves
Since k takes on continuous values, the sum becomes an integral
Re() Im()
Wave packets are a pain, so we’ll use waves and ignore normalization

3. The Step Potential (1) I II incident transmitted reflected
We want to send a particle in from the left to scatter off of the barrier
Does it continue to the right, or does it reflect?
Solve the equation in each of the regions
Assume E > V0
Region I
Region II
Most general solution is linear combinations of these
It remains to match boundary conditions
And to think about what we are doing!

4. The Step Potential (2) I II incident transmitted reflected
What do all these terms mean?
Wave A represents the incoming wave from the left
Wave B represents the reflected wave
Wave C represents the transmitted wave
Wave D represents an incoming wave from the right
We should set D = 0
Boundary conditions:
Function should be continuous at the boundary
Derivative should be continuous at the boundary

5. The Step Potential (3) I II incident transmitted reflected
The barrier both reflects and transmits
What is probability R of reflection?
The transmission probability is trickier to calculate because the speed changes
Can use group velocity, wave packets, probability current
Or do it the easy way:

6. The Step Potential (4) I II incident evanescent reflected
What if V0 > E?
Region I same as before
Region II: we have
Don’t want it to blow up at infinity, so e-x
Take linear combination of all of these
Match waves and their derivative at boundary
Calculate the reflection probability

7. The Step Potential (5) I II incident evanescent reflected
When V0 > E, wave totally reflects
But penetrates a little bit!
Reflection probability is non-zero unless V0 = 0
Even when V0 < 0! R T

8. Sample Problem
Electrons are incident on a step potential V0 = eV. Exactly ¼ of the electrons are reflected. What is the velocity of the electrons?
Must have E > 0

9. The Barrier Potential (1)
Assume V0 > E
Solve in three regions I II
III
Let’s find transmission probability
Match wave function and derivative at both boundaries
Work, work . . .

10. The Barrier Potential (2)II
III
Solve for :
Assume a thick barrier: L large
Exponentials beat everything

11. Sample Problem
An electron with 10.0 eV of kinetic energy is trying to leap across a barrier of V0 = 20.0 eV that is 0.20 nm wide. What is the barrier penetration probability?

12. The Scanning Tunneling Microscope
Electrons jump a tiny barrier between the tip and the sample
Barrier penetration is very sensitive to distance
Distance is adjusted to keep current constant
Tip is dragged around
Height of surface is then mapped out