1. Wavefunctions and Quantum Wells
Quantum Mechanics
Wavefunctions and Quantum Wells

2. Quantum Mechanics (Wave Mechanics)
The probability of finding a particle in a particular region within a particular time interval is found by integrating the square of the wave function:
P (x,t) =  |Y(x,t)|2 dx =  |c(x)|2 dx
|c(x)|2 dx is called the “probability density; the area under a curve of probability density yields the probability the particle is in that region
When a measurement is made, we say the wave function “collapses” to a point, and a particle is detected at some particular location

3. Quantum Mechanics
Schroedinger’s Equation:
where

4. Application: particle in a box
If a particle is confined to a region by infinitely-high walls, the probability of finding it outside that region is zero.
Since nature is generally continuous (no instantaneous changes), the probability of finding it at the edges of the region is zero.
The position-dependent solution to the Schrödinger equation for this case has the form of a sine function:
c(x) = B sin (npx/a) (a=well width)

5. More particle in a box c(x) = B sin (npx/a) n=3 c(x) |c(x)|2 n=2
Only certain wavelengths l = 2a/n are allowed
Only certain momenta p = h/l = hn/2a are allowed
Only certain energies E = p2/2m = h2n2/8ma2 are allowed - energy is QUANTIZED
Allowed energies depend on well width

6. What about the real world?
So confinement yields quantized energies
In the real world, infinitely high wells don’t exist
Finite wells, however, are quite common
Schrödinger equation is slightly more complicated, since Ep is finite outside well
Solution has non-trivial form (“trust me”)
|c(x)|2
n=2
n=1

7. What about the real world?
Solution has non-trivial form, but only certain states (integer n) are solutions
Each state has one allowed energy, so energy is again quantized
Energy depends on well width a
Can pick energies for electron by adjusting a
|c(x)|2
n=2
n=1 x

8. Putting Several Wells Together
How does the number of energy bands compare with the number of energy levels in a single well?
As atom spacing decreases, what happens to energy bands?
What happens when impurities are added?

9. Quantum wells
An electron is trapped since no empty energy states exist on either side of the well

10. Escaping quantum wells
Classically, an electron could gain thermal energy and escape
For a deep well, this is not very probable

11. Escaping quantum wells
Thanks to quantum mechanics, an electron has a non-zero probability of appearing outside of the well
This happens more often than thermal escape

12. What have we learned today?
Quantum mechanics challenges our physical intuition but it is the way things really work.
Particles are described with a wave function Y(x,t) which describes the propagation through space and time (when unobserved).

13. What have we learned today?
Integrating the square of the wave function over a region gives us the probability of finding the object in that region
A “particle” confined to an infinitely-high box is described by a wave function of a sine.
“particles” in finite wells or in atoms are described by more complicated wave functions
All three situations result in quantized (only certain values allowed) energies

14. Before the Next Class Do Activity 23 Evaluation before next class
Finish Homework 24
Start Reading Chapters 9 and 10 (3rd hour exam will cover)
Do Reading Quiz 25 (due before class 25)