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()

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

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!

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

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:

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

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

Sample Problem Electrons are incident on a step potential V0 = - 12.3 eV. Exactly ¼ of the electrons are reflected. What is the velocity of the electrons? Must have E > 0

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 . . .

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

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?

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