1. PHYS274: Quantum Mechanics VI
Max Born,
Quantum Dots
Further basics of Quantum Mechanics
Particle in a finite box and tunneling
Also review/reread QM harmonic oscillator
Today’s theme: Matching of boundary conditions in QM

2. Announcements
Next Midterm: Monday November 13th will cover waves as particles (Chap 38), particles as waves (Chap 39) and quantum mechanics (Chap 40).

3. The Born Interpretation of Quantum Mechanics
In 1926, age 34, Max Born proposed that |ψ|2 is the probability distribution function for a particle that is described by a wavefunction ψ. He also coined the term “Quantenmechanik”.
He received the 1954 Nobel Prize in Physics. He fled to the UK in 1933 when the Nazi’s came to power.
Olivia Newton John, singer and star of Grease is his grand-daughter
1981 Hit song: “Let’s get physical”

4. Review: Infinite Potential Well (“particle in a box”)
Probability of finding particle at a specific x-location?
Probability of finding particle at a specific x-location?
What is plotted here ?
Screwy rep…. Energy vs. space..!
Just the spatial part

5. Review: Infinite Potential Well (“particle in a box”)
Screwy rep…. Energy vs. space..!

6. Review: Infinite Potential Well (“particle in a box”)
Probability of finding particle at a specific x-location?
Screwy rep…. Energy vs. space..!
Review: What are the boundary conditions ?

7. Q25.4 Born Interpretation and Particle in a box
What is probability in these 3 cases of finding particle at
X = L/2 ?
n = 1 greatest
n = 3 greatest
Same (n = 1,3)
n = 2, for sure

8. Q25.4 Born Interpretation
What is probability in these 3 cases of finding particle at
X = L/2 ?
n = 1 greatest
n = 3 greatest
Same (n = 1,3)
n = 2, for sure
What about n=2 ?

9. Summary: Infinite Potential Well (“particle in a box”)
Quantized: k=np/L
Quantized:
energies are
“eigenvalues”
wave functions
“eigenfunctions”
Screwy rep…. Energy vs. space..!
Energy level diagram

10. Application of QM potentials: QM semiconductor dots
Nanometer-sized (nm) particles of a semiconductor such as CdSe
Discovered by Russian physicist Alexey Ekimov in St. Petersburg in1981
The fluorescence color under UV is determined by the size of the dot. (5-6 nm orange or red; 2-3 nm blue or green)
Can be injected into patients for medical research and treatment (tagging and treating cancer cells). Also solar cell and computing applications 10

11. Finite Potential Well (“particle in a box”)
What happens if the walls are no longer infinite?
Screwy rep…. Energy vs. space..!

12. Finite Potential Well (“particle in a box”)
What happens if the walls are no longer infinite?
Screwy rep…. Energy vs. space..!
Particle can “leak out” (‘tunnel’)
Wavelength is longer than infinite potential case

13. Finite Potential Well (“particle in a box”)
What happens if the walls are no longer infinite?
Screwy rep…. Energy vs. space..!
Energy levels?

14. Need to solve Schrodinger Eqn:
V(x) eV
Energy
Eelectron x L
Region I
Region II
Region III
In Region III … total energy E < potential energy V
α is real
Positive
What functional forms of y(x) work?
a. eiαx b. sin(αx) c. eαx d. more than one of these

15. In Region III … total energy E < potential energy V
Energy x L
V(x)
Region I
Region II
Region III
Eelectron
In Region III … total energy E < potential energy V
α is real
Positive
Why not eiax?
Answer is C: eαx … could also be e-αx.
Exponential decay or growth
LHS
RHS

16. Back to case with workfunction of XX eV
XX eV = V0
Energy
Eparticle
0 eV
0 L x
Positive number
Answer is (C) … could also be e-αx.
Exponential decay or growth ψ

17. (OK when Energy << work function)
Good Approximation:
Electrons never leave
ψ(x<0 or x>L) =0.
(OK when Energy << work function)
Exact Potential Energy curve (V):
Small chance electrons get out!
ψ(x<0 or x>L)~0, but not exactly 0!
V(x)
Energy x L
What happens if electron Energy bigger?
Or width to next well not so far away?
Then whether ψ leaks out a little or not is very important!
E(total)
Infinite Work Function!!!!

18. Need to solve for exact Potential Energy curve:
V(x): small chance electrons get out of wire
ψ(x<0 or x>L)~0, but not exactly 0!
Finite
Square Well
V(x)
Energy
Work function x L
Target.
Important for thinking about “Quantum tunneling”:
Scanning tunneling microscope to study surfaces
Radioactive decay with alpha emission.

19. Need to solve Schrodinger Eqn:
V(x) eV
Energy
Eelectron x L
Region I
Region II
Region III
In Region II … total energy E > potential energy V
k is real
Negative number
When E>V: Solutions = sin(kx), cos(kx), eikx.
Always expect sinusoidal (oscillatory) functions

20. What will wave function in Region III look like?
Energy x L
V(x)
Region I
Region II
Region III
Eelectron
What will wave function in Region III look like?
What makes sense for constants A and B?
a. A must be b. B must be c. A and B must be equal
d. A=0 and B= e. A and B can be anything, need more info.

21. What will wave function in Region III look like?
Energy x L
V(x)
Region I
Region II
Region III
Eelectron
What will wave function in Region III look like?
What makes sense for constants A and B?
Answer is a. A must be 0 .. otherwise ψ blows up as x gets bigger.
This doesn’t make physical sense!
ψ and probability should  0 at large x!
Need to be able to normalize ψ

22. Outside well (E<V):
Inside well (E>V):
(Region II)
Outside well (E<V):
(Region III)
Outside well
(E<V):
(Region I)
Energy
Eelectron
V=0 eV L x
Boundary
Conditions:

23. Outside well (E<V):
Inside well (E>V):
Energy
Eelectron
“Classically forbidden” region.
V=0 eV L x
Electron is delocalized … spread out.
Some small part of wave is where the
total Energy is less than potential energy!

24. wire
Eelectron L
wire
If a long wire gets closer and closer, what will happen?
a. electron is “shared” between wires, with fraction in each constant over time
b. the electron will flow away through wire 2
c. electron will jump back and forth between wire 1 and wire 2
d. electron stays in wire 1.
e. something else happens.

25. wire
Eelectron L
wire
If very very long wire gets closer and closer, what will happen?
a. electron is “shared” between wires, with fraction in each constant over time
b. the electron will flow away through wire 2
c. electron will jump back and forth between wire 1 and wire 2
d. electron stays in wire 1.
e. something else happens.

26. If a very long wire gets closer and closer, what will happen?
Eelectron L
wire
If a very long wire gets closer and closer, what will happen?
a. electron is “shared” between wires, with fraction in each constant over time
b. the electron will flow away through wire 2
c. electron will jump back and forth between wire 1 and wire 2
d. electron stays in wire 1.
e. something else happens.
Recall long wire E ~ 1/L^2

27. Particle in a box (infinite potential well).
Q26.1
Compare a particle confined inside a box (infinite potential well) and a particle in a finite potential well, which one has a lower ground state energy?
Particle in a box (infinite potential well).
Particle in a finite potential well.
Both have the same ground state energy
B (wavefunction can leak out) 27

28. Particle in a box (infinite potential well).
Q26.1
Compare a particle confined inside a box (infinite potential well) and a particle in a finite potential well, which one has a lower ground state energy?
Particle in a box (infinite potential well).
Particle in a finite potential well.
Both have the same ground state energy
B (wavefunction can leak out)
Wavefunction can leak out 28

29. Q26.2 29

30. Q26.2 30

31. For next time More Quantum Mechanics Read material in advance
Concepts require wrestling with material