Quantum Mechanics, part 3 Trapped electrons

1. “Confinement leads to quantization” Quantum Mechanics, part 3Trapped 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. 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 Equation or

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 • Predicts 3 quantum numbers –n,l,ml • Successfully describes atomic spectra Solution is the product of 3 functions 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