A question Now can we figure out where everything comes from? Atoms, molecules, germs, bugs, you and me, the earth, the planets, the galaxy, the universe?  Lets Try  Start with the easiest – Then build  Lets build A hydrogen atom More complicated atoms Molecules

Our tools Schrödinger (a 3D version) U(x)=Ze2/r A “classical” picture (for our human brains” Electrons orbiting around a nucleus with charge Z

Thinking “classically” Electron going around nucleus of charge Z Planet going around sun n  Different planets have different orbits [ n ] N=1 = mercury, earth might be n=3 L  Planets have angular momentum [ L ] m L Planet orbits have a tilt [ m L ] S  Planets spin around their axis [ S ] m S Planet spin has a tilt! [ m S ]

Use a 3-D version of this U(x)=Ze 2 /r Rydberg’s Constant=R=13.6eV

We Get 3 Quantum numbers! n ~ Energy level L ~ Orbital Angular momentum m L ~ z-component of L Does the electron have spin?? Here it is: and its sort of like planets

Stern-Gerlach Experiment A beam of neutral silver atoms is split into two components by a nonuniform magnetic field The atoms experienced a force due to their magnetic moments The beam had two distinct components in contrast to the classical prediction If the electron had spin, it might explain this

Electron Spins Spin quantized  The electron can have spin S= up down In the presence of a magnetic field, the energy of the electron is slightly different for the two spin directions and this produces doublets in spectra of certain gases Intrinsic Spin Violates our intuition: How can an elementary particle such as the e¯ be point like, wavelike and have perpetual angular momentum? SZSZ SZSZ S Z =m s m s = +1/2 or -1/2

Electron Spins, cont The concept of a spinning electron is conceptually useful The electron is a point particle, without any spatial extent  Therefore the electron cannot be considered to be actually spinning The experimental evidence supports the electron having some intrinsic angular momentum that can be described by m s Sommerfeld and Dirac showed this results from the relativistic properties of the electron

Spin was first discovered in the context of the emission spectrum of alkali metals - "two-valued quantum degree of freedom" associated with th e electron in the outermost shell. In trying to understand splitting patterns and separations of line spectra, the concept of spin appeared "it is indeed very clever but of course has nothing to do with reality". W. Pauli A year later Goudsmit and Uhlenbeck, published a paper on this same idea. Pauli finally formalized the theory of Spin in 1927

The Exclusion Principle The four quantum numbers discussed so far can be used to describe all the electronic states of an atom regardless of the number of electrons in its structure The Exclusion Principle states that no two electrons in an atom can ever be in the same quantum state  Therefore, no two electrons in the same atom can have the same set of quantum numbers

Orbitals An orbital is defined as the atomic state characterized by the quantum numbers n, and m From the Exclusion Principle, it can be seen that only two electrons can be present in any orbital  One electron will have m s = ½ and one will have m s = -½

Allowed Quantum States, Example The arrows represent spin The n=1 shell can accommodate only two electrons, since only one orbital is allowed In general, each shell can accommodate up to 2n 2 electrons

The fact that two electrons could be in any energy level corresponding to a given set of quantum numbers (n,l,m) led eventually to the discovery of the spin of the electron. Many-electron atoms The energy levels corresponding to n = 1, 2, 3, … are called shells and each can hold 2n 2 electrons. The shells are labeled K, L, M, … for n = 1, 2, 3, ….

Fig 42-19, p.1376

Covalent bonds Model of H atom  particle in a box  E=-13.6 ev (n=1 state)  top of the box be at zero  width of the box is ~ 2a B ~0.1 nm 2 H atoms for a bond  Separation =.12 nm  match solutions at boundary Ge - κx outside solution Match solutions Asin(kx) inside solution

Now solve for the new energies E=-17.5 eV E=-9 eV n=1 n=2 prob density

Total energy – the covalent bond If the total energy is negative, then the particles are bound.  pp repulsive energy =  n=1 E=12 eV-17.5eV = -5.5eV covalent bond!!  n=2 E=12 eV – 9 eV=+3eV doesn’t bind

Lasers Excitation Emmission A laser Simulated Emmision

Lasers CD Telecommunications Surgery Etc etc Fig 42-27, p.1386 Optical Tweezers, DNA making an R

If the exclusion principle were not valid,  a. every electron in an atom would have a different mass.  b. every atom would be in its state of greatest ionization.  c. the quantum numbers and would not exist.  d. every pair of electrons in a state would have to have opposite spins.  e. every electron in an atom would end up in the atom’s lowest energy state.