Foundations of Quantum Mechanics

The Bizarre World of the Really (Really) Small, Part 1!

Chapter 11

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)

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).

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 = 1n = 2

n = 3

Nucleus

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 levelEmit photon of lightCalculate the difference using E = h

n = 2

n = 3

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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

Old SpectroscopyTerms

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)

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 More Energy Levels!?!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 s-orbitalpx-orbitalpy-orbital

pz-orbital3 p-orbitals

z

x

y

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

You Spin Me Right ‘Round…

Spin Quantum # (ms) given to electrons to specify additional angular momentumNothing to do with the orbitals

Two values, +1/2 or -1/2Also called Up or DownNot literally true – have to be spinning ~10x

speed of light to account for extra momentum

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Now have 4 Quantum Numbers

Specify location of electrons in atomn = 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

n = 1

n = 2

n = 3

n = 4

Also, the larger the n, the more energy that is “stored” by the electron.

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Shapes Determined w/ Azimuthal (l)

“s” orbitals – spherical, Larger n = bigger sphere

“p” orbitals – dumbbell shapedLarger n = bigger dumbbell

1s2s

3s

2p 3p 4p

Also, the more complex the shape, the more energy that is “stored” by the electron.

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Every Which Way (But Loose)

n > 1

You Will Have toDraw These!

Each type of orbital has multiple orientations possible (except s)

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Crazy Orbital Shapes

“d”orbitals Only possible when

n > 2 You don’t have to

draw all of these!!! Just this one!

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Crazy Orbital Shapes (cont’d)

Only when n > 3Notice the

Space?

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A Closer Look

An orbital is the probable location of the electron90% 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.

2px orbital

Node:Electron is Never There!

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Odes to Nodes

In s-orbitals, the value of n tells you

The number of Anti-nodes

aka Peaks

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So Far What We Know

Electrons are in orbitalsOrbitals differ in:

Size (n) Shape (l) Orientation (ml)

Electrons have spins (ms) Why do they matter?

(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…

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Pauli’s Revenge Each e- must have a different set of Q#’s (0 ≤ l ≤ n-1) (-l ≤ ml ≤ l) (+1/2 or -1/2)

n(level)

l(orbital)

ml

(suborbital)

ms

(spin)

# e-s Possible

1 0 (s) 0 +½, -½ 2

20 (s) 0 +½, -½ 2

1 (p) -1,0,1 3(+½, -½) 6

3

0 (s) 0 +½, -½ 2

1 (p) -1,0,1 3(+½, -½) 6

2 (d) -2,-1,0,1,2 5(+½, -½) 10

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Explain why each is incorrect

n=1 l = 1 ml = 0 ms = +1/2

n=3 l = 0 ml = -2 ms = -1/2

n=3 l = 2 ml = 0 ms = +3/2

n=4.5 l = 0 ml = 0 ms = +1/2

0 ≤ 0 ≤ ll ≤ [n-1] ≤ [n-1]

--ll ≤ m ≤ mll ≤ ≤ ll

+1/2 or -1/2+1/2 or -1/2

n = integern = integer