# Quantum Mechanics - PDF

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```					Notes on Quantum Mechanics                                                                                                   by Mr. Scott

The Schrodinger equation uses THREE QUANTUM NUMBERS to describe the orbitals and
locations of each electron in an atom. A FOURTH Q. N. was later added.

Principal quantum number, n, is an integer that describes the average distance of the electron
from the nucleus. (n = 1, 2, 3, …)

Azimuthal quantum number, , can have integral values between 0 and n-1, and defines the
SHAPE OF THE ORBITAL’s electron density. The values of are usually associated with letter
designations that are drawn from the descriptions of spectra that were developed in the past:

0        1               2                   3
Letter         s        p               d                   f
Magnetic quantum number, m , can have integral values between +                                                             and - , and describes the
ORIENTATION of the orbital space.

s        p                       d                                                                                    f
m    0 1      0 -1      2     1      0      -1       -2        3                   2                  1                   0           -1                  -2                -3
px p y pz     d yz d xz d xy d x2 − y 2 d z 2       fz −
3    3
zr
2   fx −
3    3
xr
2   fy −
3    3
yr
2   f xyz f y ( x − z
2      2
)
fx( z   2     2
−y )
fz( x − y
2      2
)
5                   5                   5

Spin quantum number, ms , introduced by Wolfgang Pauli in his exclusion principle, requires that
two electrons in the same orbital must have opposite polarity or SPIN.
These values are + ½ or - ½ .

Describing the electronic structure of an atom
There are several ways of writing the structures of the electrons of atoms. These are referred to as
NOTATION and they all follow two basic rules or principles:

Aufbau Principle = electrons occupy the lowest available energy positions first

Hund’s Rule = orbitals of equal energy each hold one electron before they get a second electron
Increasing
Energy                                                             Aufbau Diagram
1s                                                                  (diagonal rule)
2s      2p
3s      3p       3d
4s      4p       4d         4f
5s      5p       5d         5f
6s      6p       6d         6f
7s      7p       7d         7f
Orbital Notation
Uses a line for each orbital and an arrow for each electron. Each orbital represented is labeled
beneath the line. The progression is in the order of energy. Both AUFBAU and HUND are used.

Examples:
Carbon (z = 6) ground state                       Sodium (z = 11) ground state

↑↓    ↑↓        ↑      ↑                 ↑↓    ↑↓     ↑↓     ↑↓     ↑↓     ↑
1s     2s       2 px   2 py               1s    2s    2 px   2 py   2 pz   3s

This notation shows 2 unpaired electrons.        This notation shows 1 unpaired electron.

Electron Configuration Notation
Uses a number-letter designation for each orbital type and a superscript number for the number of
electrons that occupy each type of orbital. The progression is in the order of energy. The AUFBAU
DIAGRAM is best used to determine the correct order.

Carbon (z = 6) ground state                      Sodium (z = 11) ground state

1s22s22p2                                       1s22s22p63s1
The superscripts add up to the number of electrons for the atom (or ion).

2+2+2=6                                         2 + 2 + 6 + 1 = 11

The maximum amounts are:       s2 , p6 , d10 , f14
Dot Notation
Uses the symbol of the element with dots placed around it in a certain pattern that represent the
VALENCE electrons. The valence electrons are those in the highest occupied PRINCIPAL
ENERGY LEVEL (n = 1, 2, 3, …). The dot pattern followed is similar to the pattern followed in
orbital notation.
3 6                               px
4      1
The dot pattern is:
7
Xy2
representing      py Xy s
pz
5 8
Carbon (z = 6) ground state                   Sodium (z = 11) ground state
i
i   C:                                    Na     i
Why just four dots?                       Why just one dot?
Lets reconsider the electron configuration of each of these atoms.

1s22s22p2                                       1s22s22p63s1
The highest occupied level is n = 2             The highest occupied level is n = 3

2s22p2 = 4 electrons                                   3s1 = 1 electron
so 4 dots are shown                                     so 1 dot is shown

```
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