Foundations of Quantum Mechanics

1. Foundations of Quantum Mechanics The Bizarre World of the Really (Really) Small, Part 1! Chapter 4

2. Strange Days • Max Planck (1894) studied black-body radiation • (when solids are heated to incandescence) • Results could not be explained by the physics of the day. • Based on his experiments, energy can only be transferred in discrete (quantized) amounts (1899) Remember this, E = hν? Called these small packets of energy “Quanta” (singular = quantum)

3. The Cat’s out of the Bag • Einstein proposed that electromagnetic radiation itself is quantized and can be thought of as a stream of “particles” or photons (1905). • E = h & c =   E = hc/ • Neils Bohr attempted to explain the stability of the atom and the Hydrogen Emission spectrum using the idea of quantized energy (1913). The Problem was: We didn’t know where the electrons were inside the atom – we needed an “address” for the little buggers!

4. Bohr’s Model This is equivalent to the State in your home address – a very large region where the electron can be found. • Electrons only in permitted circular orbits • The Ground State is the open orbit closest to nucleus (lowest available energy) • The Excited State is an orbit farther away from nucleus (it has higher energy) • Principal Quantum # (n) given to determine orbit • Small n means a small radius, closer to the nucleus • Value: n > 0 Nucleus n = 2 n = 1 n = 3

5. Light Emission occurs when • Electrons absorb energy and • Jump to a higher energy level (excited state) • Electrons then release energy • Fall to a lower energy level • Emit photon of light • Calculate the difference using E = h n = 3 n = 2

6. For every Line there is an associated electron jump! This means another energy level!

7. Something’s Rotten in Denmark • Bohr’s model works perfectly…for hydrogen. • More precise measurements of line spectra of higher elements reveal more lines. • Arnold Sommerfeld (1868-1951) suggests elliptical orbits. Remember, more lines = more Energy Levels!

8. Shape Matters • Azimuthal Quantum # (l) given to determine shape of the electron’s orbit • 0 l (n – 1) • l = 0  s (spectral) • l = 1  p (principal) • l = 2  d (diffuse) • l = 3  f (fine) • l = 4  g (but we don’t have enough ‘s) This is equivalent to the City in your home address – a smaller region inside the atom where the electron can be found. Old Spectroscopy Terms

9. More Lines Still – Zeeman Effect • When atoms were placed into a magnetic field, triplets were formed from singlets • Sommerfeld chimes in again, in 1916, saying orientation in space must matter Singlet Triplet More Energy Levels!?!

10. Orient This, Baby! • Magnetic Quantum # (ml) given to determine orbital’s orientation in space • - l ml  l • For s orbitals (l = 0), only 1 possibility • For p orbitals, 3 possible suborbitals • For d’s, 5 • For f’s, 7 This is equivalent to the Street in your home address – a very small region where the electron can be found. pz-orbital s-orbital 3 p-orbitals px-orbital py-orbital x z y

11. You Guessed it…still more lines I’ll Be Back! Just You Wait! • AZE – Anomalous Zeeman Effect = even more lines under certain circumstances • Wolfgang Pauli (1900 – 1958) proposed a hidden rotation • Unfortunately, Pauli is unable to visualize it, so Uhlenbeck & Goudsmit get credit for it (leading to a Nobel Prize) Wolfgang Pauli

12. You Spin Me Right ‘Round… • Spin Quantum # (ms) given to electrons to specify additional angular momentum • Nothing to do with the orbitals • Two values, +1/2 or -1/2 • Also called Up or Down • Not literally true – have to be spinning ~10x speed of light to account for extra momentum This is equivalent to your House Number in your home address – Combined with n, l, & ml, we have a very specific address for the electron.

13. Now have 4 Quantum Numbers • Specify location of electrons in atom • n = energy level (n > 0) • lower the number, closer to the nucleus • l = orbital shape (0  l  [n – 1]) • Shapes are abbreviated s p d f… • ml = suborbital orientation (- l  ml  l) • s  1 possible, p  3, d  5, f  7… • ms = spin (+1/2, -1/2) • Up & Down

14. Size Still Matters • Principal Quantum # (n) • Integer from 1 to 7 (theoretically more) • The larger n is, the larger the orbit Also, the larger the n, the more energy that is “stored” by the electron. n = 1 n = 2 n = 3 n = 4

15. Shapes Determined w/ Azimuthal (l) • “s” orbitals – spherical, • Larger n = bigger sphere • “p” orbitals – dumbbell shaped • Larger n = bigger dumbbell 1s 2s Also, the more complex the shape, the more energy that is “stored” by the electron. 3s 3p 4p 2p

16. Every Which Way (But Loose) • Each type of orbital has multiple orientations possible (except s) You Will Have to Draw These! n > 1

17. Crazy Orbital Shapes • “d”orbitals • Only possible when n > 2 • You don’t have to draw all of these!!! • Just this one!

18. Crazy Orbital Shapes (cont’d) Notice the Space? Only when n > 3

19. A Closer Look • An orbital is the probable location of the electron • 90% of time e- is in the orbital (other 10%?) • A node is the position within an orbital where the probability of finding an electron is 0. Node: Electron is Never There! 2px orbital

20. Odes to Nodes • In s-orbitals, the value of n tells you • The number of Anti-nodes • aka Peaks

21. So Far • What We Know • Electrons are in orbitals • Orbitals differ in: • Size (n) • Shape (l) • Orientation (ml) • Electrons have spins (ms) • Why do they matter?

22. Knock, Knock!!! I Told You I’d Be Back! And this time, I got my Nobel Prize!!! Mwah-ha-ha!!! • Pauli Exclusion Principle – no two electrons can have the same 4 quantum numbers! • Electrons cannot stack up on each other. • If an orbital is full, the next e- must go to a higher orbital. • This is what makes matter solid!!! Guess Who’s Back… (Photons do not obey this principle)