Quantum Mechanics basics: see the transparency here.

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Lattices: translational symmetry.
Crystals have translational symmetry, not e.g. rotational.

(if you jump to the next point of the lattice,
 the world looks exactly the same to you. Then
 you are on a lattice ... translationally symmetric world.)

(in a 'spherically symmetric world the world is the same
 if you TURN in any direction. But then you must be in
 the centre. If you are anywhere else, the world might
 look pretty disorganized .... )

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Why Group Theory:  Transformations

What is a group:
a set where we only have 'multiplication'

Shortest group theory:

if  A is from G there is A^(-1)   A . A^(-1) = E
    E is like  1
if  A, B are from G    A . B is from G
                       A . B is different from B . A
we can easily understand that transformations of most of
types (linear in any case) will form a group.

The next step are group representations:

Let us say that the transformations are an abstract group.
A set of matrices can be such that for each of the 
abstract transformations there is a matrix in the set
of matrices. Then the matrices are a representation.
(they would make an effect on some set of vectors.)

There might be sevaral different representations for 
a transformation group. OF INTEREST for  physics are 
those basis vectors ....

This is why Group theory is important in particle physics,
quantum mechanics, and some parts of solid state theory.
We shall not study this in detail, but it is good to know ...


Crystal Symmetry:  Lattice, NOT ROTATIONAL SPHERICAL

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Coupled linear oscillations (=harmonic)

Eigenmodes

we just mentioned eigenmodes of 
a system of coupled oscillators