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

Electron Trap

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

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

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

Particle in a Box

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

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

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

Infinite Potential Well Solution using Schrödinger Wave Equation or

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?

Infinite Potential Well Solution using Schrödinger Wave Equation Energy level transitions

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

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.

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

2 D and 3 D rectangular corrals

Simple Harmonic Oscillator

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….

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…….

Hydrogen Line Spectra Johannes Balmer 1897: Balmer Series Spectrum

Atomic Line Spectra Rydberg Formula

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

Bohr Model of the Atom

Bohr Model of the Atom

The Bohr Model and Standing Electron Waves (Arthur Sommerfeld)

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

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

The Solution to the Schrödinger Equation for Hydrogen

The Solution to the Schrödinger Equation for Hydrogen

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