1. Quantum Mechanics, part 3 Trapped electrons
“Confinement leads to quantization”
Quantum Mechanics, part 3 Trapped electrons
Infinite Potential Well
Finite Potential Well
Quantum Traps
Nanocrystallites
Quantum Dots
Quantum corrals
2-D and 3-D Traps
Hydrogen Atom
Bohr Theory
Solution to Schrödinger Equation
A quantum corral of iron atoms

2. Electron Trap

3. Energy Level Diagrams---DISCRETE LEVELS NOT CONTINUOUS!!!!!!!!!!!11

4. Particle in a Box by analogy (Infinite Potential Well)
Standing waves in a string
Classically - any energy and momentum just like a free particle

5. Particle in a Box
QM - Boundary conditions for the matter wave

6. Particle in a Box

7. Introduction to Wave Mechanics (review)
The wave function
Interpretation - Probability function and density
Normalization
Probability of locating a particle
Expectation value

8. The general solution is that of SHM; i.e.,
Infinite Potential Well Boundary conditions are everything!!!!!!!!!!!!!! Solution using Schrödinger Wave Equation
The general solution is
that of SHM; i.e.,
U = 0 inside the well
and ¥ everywhere else,
so y = 0 if x < 0 or x > L.
Apply the boundary
Conditions at x = 0
Also
requires
where or

9. Determining the constant A in the Infinite Potential Well Solution using Schrödinger Wave Equation example prob 39-2
Normalize the probability

10. Infinite Potential Well Solution using Schrödinger Wave Equationor

11. Infinite Potential Well Solution using Schrödinger Wave Equation
Verify that the above is
a solution to the differential
equation.
Why isn’t n = 0 a valid
quantum number?

12. Infinite Potential Well Solution using Schrödinger Wave Equation
Energy level transitions

13. Particle Finite Potential Well
Regions of the potential well
Matter wave leaks into the walls. For any quantum state the wavelength is longer so the corresponding energy is less for the finite well than the infinite trap/well.
Wave function and
probability functions
Energy level diagram for
L = 100 pm and Uo = 450 eV

14. Finite Well Cont. Given U0=450 eV, L=100 pm
Remove the portion of the energy diagram of the infinite well above E=450 eV and shift the remaining levels (three in this case) down.

15. Examples of quantum electron traps
Nanocrystallites
A quantum corral of iron atoms
Quantum Dot

16. 2 D and 3 D rectangular corrals

17. Simple Harmonic Oscillator

18. The Nature of the Nuclear Atom
Rutherford 1911 (w/grad students Geiger and Marsden) Scattering
Scattering alpha particles from gold foil
Some alphas bounced back as if “a cannonball bouncing off tissue paper”
Established the nuclear atom
Electron outside a very small positive nucleus
Classical theory leads to contradiction
An electron would spiral into the nucleus in a time
AAargh….

19. Electrons are trapped by the Nucleus
Could the energy states be discrete?
Stability of the atom is due to quantization of energy much like the trapped electron in the finite well!!!!
Bohr postulates that angular momentum and thus energy is quantized in units of Planck's constant
There is a hint from the signature of atomic spectra…this week’s lab…….

20. Hydrogen Line Spectra
Johannes Balmer 1897: Balmer Series Spectrum

21. Atomic Line Spectra
Rydberg Formula

22. Bohr Model of the Atom (1913)
Semiclassical nuclear model
Assumes Electrostatic Forces
Stationary Orbits hypothesized
Note

23. Bohr Model of the Atom

24. Bohr Model of the Atom

25. The Bohr Model and Standing Electron Waves (Arthur Sommerfeld)

26. Results Consistent with basic Hydrogen spectrum
Explains origin of photons
Fails to explain more complex spectra and fine points of Hydrogen spectrum

27. The Solution to the Schrödinger Equation for Hydrogen
Solution is the product of 3 functions
Predicts 3 quantum numbers –n,l,ml
Successfully describes atomic spectra
Note

28. The Solution to the Schrödinger Equation for Hydrogen

29. The Solution to the Schrödinger Equation for Hydrogen

30. Correspondence Principle
For large quantum numbers, the results of quantum mechanical calculations approach those of classical mechanics