1. Foundations of Quantum Mechanics The Bizarre World of the Really (Really) Small, Part 1! Chapter 11

2. 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) Called these small packets of energy “Quanta” (singular = quantum) Remember this, E = hν?

3. 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. 4 Bohr’s Model 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 n = 1 n = 2 n = 3 Nucleus This is equivalent to the State in your home address – a very large region where the electron can be found.

5. 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 = 2 n = 3

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

7. 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. 8 Old Spectroscopy Terms 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.

9. 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. 10 Orient This, Baby! Magnetic Quantum # (m l ) given to determine orbital’s orientation in space - l  m l  l For s orbitals ( l = 0), only 1 possibility For p orbitals, 3 possible suborbitals For d’s, 5 For f’s, 7 s-orbital p x -orbital p y -orbital p z -orbital 3 p-orbitals z x y This is equivalent to the Street in your home address – a very small region where the electron can be found.

11. 11 You Guessed it…still more lines 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 I’ll Be Back! Just You Wait!

12. 12 You Spin Me Right ‘Round… Spin Quantum # (m s ) 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, & m l, we have a very specific address for the electron.

13. 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…  m l = suborbital orientation (- l  m l  l ) s  1 possible, p  3, d  5, f  7…  m s = spin (+1/2, -1/2) Up & Down

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

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

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

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

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

19. 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. 2p x orbital Node: Electron is Never There!

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

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

22. (Photons do not obey this principle) 22 Knock, Knock!!! 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!!! I Told You I’d Be Back! And this time, I got my Nobel Prize!!! Mwah-ha-ha!!! Guess Who’s Back…

23. 23 Pauli’s Revenge Each e - must have a different set of Q#’s (0 ≤ l ≤ n-1) (- l ≤ m l ≤ l ) (+1/2 or -1/2) n (level) l (orbital) m l (suborbital) m s (spin) # e - s Possible 10 (s)0+½, -½2 2 0 (s)0+½, -½2 1 (p)-1,0,13(+½, -½)6 3 0 (s)0+½, -½2 1 (p)-1,0,13(+½, -½)6 2 (d)-2,-1,0,1,25(+½, -½)10

24. 24 Explain why each is incorrect n=1 l = 1 m l = 0 m s = +1/2 n=3 l = 0 m l = -2 m s = -1/2 n=3 l = 2 m l = 0 m s = +3/2 n=4.5 l = 0 m l = 0 m s = +1/2 0 ≤ l ≤ [n-1] - l ≤ m l ≤ l +1/2 or -1/2 n = integer