Theory that describes the physical properties of smallest particles (atoms, protons, electrons, photons)
Max Planck
"A scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die and a new generation grows up that is familiar with it."
Erwin Schrödinger
"I don't like it and I'm sorry I ever had anything to do with it."
Werner Heisenberg
"An expert is someone who knows some of the worst mistakes that can be made in his subject, and how to avoid them"
Max Born
"It is true that many scientists are not philosophically minded and have hitherto shown much skill and ingenuity but little wisdom."

Niels Bohr (1885-1962)
- electron orbits around the nucleus like a wave
- orbit is described by wavefunction
- wavefunction is discrete solution of wave equation
- only certain orbits are allowed
- orbits correspond to energy levels of atom

In the Bohr model of the atom, the hydrogen atom is like a planetary system with the electron in certain allowed circular orbits.
The Bohr model does not work for more complicated systems!

6
( r )


Each orbital is characterized by a set of quantum numbers.
 n , l , m l
( r )
Principal quantum number (n): integral values
(1,2,3). Related to the size and energy of the orbital.
 l ): integral values from 0 to (n-1) for each value of n.
Magnetic quantum number (m l
l to l for each value of n.
): integral values from

How many orbitals are there for each principle quantum number n = 2 and n = 3?
For each n, there are n different l -levels and (2l+1) different m l levels for each l .
n=2: n = 2 different l -levels l = 0, 1
(2l+1) = 2 x 0 + 1 = 1 m l
-levels for l = 0
(2l+1) = 2 x 1 + 1 = 3 m l
-levels for l = 1
Total: 1 + 3 = 4 levels for n = 2

How many orbitals are there for each principle quantum number n = 2 and n = 3?
For each n, there are n different l -levels and (2l+1) different m l levels for each l .
n=3: n = 3 different l -levels l = 0, 1,2
(2l+1) = 2 x 0 + 1 = 1 m l
-levels for l = 0
(2l+1) = 2 x 1 + 1 = 3 m l
-levels for l = 1
(2l+1) = 2 x 2 + 1 = 5 m l
-levels for l = 2
Total: 1 + 3 + 5 = 9 levels for n = 3
The total number of levels for each n is n 2

Names of atomic orbitals are derived from value of l :

Quantum numbers for the first four levels in the hydrogen atom.

Wavefunction itself is not an observable!
proportional to probability density
“I cannot but confess that I attach only a transitory importance to this interpretation. I still believe in the possibility of a model of reality
- that is to say, of a theory which represents things themselves and not merely the probability of their occurrence. On the other hand, it seems to me certain that we must give up the idea of complete localization of the particle in a theoretical model. This seems to me the permanent upshot of Heisenberg's principle of uncertainty.
(Albert Einstein , on Quantum Theory, 1934”


 n , l , m l
 n , l , m l
‘function’
 n , l , m l
2
‘probability’
 r


A subshell is a set of orbitals with the same value of l
. They have a number for n and a letter indicating the value of l
.
 n , l , m l
2 l
= 0 (s) l
= 1 (p) l
= 2 (d) l
= 3 (f) l
= 4 (g)

Where’s the electron?
That’s quite uncertain!
Werner Heisenberg
Life is uncertain!

It is not possible to know both the position and momentum of an electron at the same time with infinite precision.
 x      h
4 

 x is the uncertainty in position.

(mv) is the uncertainty in momentum.
h is Planck’s constant.

s
The orbital is defined as the surface that contains 90% or the
2 total electron probability ( ).
n , l , m l probability distributions

The higher energy orbitals have nodes , or regions of zero electron density.
s-orbitals have n-1 nodes.
The 1s orbital is the ground state for hydrogen.
orbital surfaces

How many electrons fit into 1 orbital?

2,1,0
2 m s
= +1/2

2,1,0
2 m s
= -1/2
Only 2 electrons fit into 1 orbital: 1 spin up
1 spin down

Electrons are fermions. There are also bosons
As the temperature is lowered, bosons pack much closer together, while fermions remain spread out.


E
 
R
H


Z
 n 2
2



E n =1
R
H
= 2.178 x 10 -18 J
Z = atomic number n = energy level n =∞ n =5 n =4 n =3 n =2

For the energy change when moving from one level to another:
 E   R
H


Z
 n 2 f
2

Z 2 n i
2



E
 n =∞ n =5 n =4 n =3 n =2 transition n =1

Change in energy corresponds to a photon of a certain wavelength:

E
 h
  hc

Change in energy
Frequency of emitted light
Wavelength of light emitted



What is the wavelength of the photon that is emitted when the hydrogen atom falls from n=3 into n=2?
 E   R
H


Z
 n 2 f
2

Z 2 n i
2




E
 

2.178

10

18
J




1
2
2
2


1
2
3
2



3.03

10

19
J
3.03

10

19
J

 hc

 
6.626
 10

34 
3.00

10
8
3.03

10

19

656 nm



E
hydrogen transition n =∞ n =5 n =4 n =3 n =2
Rhodamine
532 nm 570 nm n =1
‘Fluorescence’

Orbital energy levels for the hydrogen atom.

Hydrogen is the simplest element of the periodic table.
Exact solutions to the wave equations for other elements do not exist!

What do the orbitals of non-hydrogen atoms look like?
Multiple electrons: electron correlation
Due to electron correlation, the orbitals in non-hydrogen atoms have slightly different energies

Screening: due to electron repulsion, electrons in different orbits ‘feel’ a different attractive force from the nucleus e e e -
11 + e e e e e -
Sees a different effective charge!
Screening changes the energy of the electron orbital; the electron is less tightly bound.

Penetration: within a subshell ( n ), the orbital with the lower quantum number l will have higher probability closer to the nucleus n =2 orbital n=3 orbital

Hydrogen
Polyelectric atom
Orbitals with the same quantum number n are degenerate
Degeneracy is gone:
E ns
< E np
< E nd
< E nf

Due to lifting of degeneracy, many more lines are possible in the spectra of polyelectric atoms