Qunatum mechanics demands that we express states in terms of complex amplitudes.

 

Let A be the amplitude for something to happen and B be the amplitude for a different thing to happen.

 

Examine th total either A or B

 

P(A)= A* A      (hitting the wall through slit 1)

 

P(B)= B* B        (hitting the wall through slit 2)

 

P(A or B)= (A+B) * (A+B)= A* A+ B* B+ B*A+ A* B    (hitting the wall)

 

 

 

Convenient expression for amplitude:

 

 

To preserve the probability we are restricted to only changing the phase (not relative phases because this changes the interference, just overall phase)

 

Unitary operators preserve the probability.

 

Rotations are orthogonal and preserve length of a vector

 

 

Since our amplitudes are complex this adds an additional rule

 

 

The general format for a unitary transformation

 

 

 

hermetian operators have real eigenvalues. Observables are expressed as hermetian operators.

 

Rotations

 

The J+ J-  are the raising and lowering operators.

 

 

 

Choosing an eigenstate

Summary

 

General ideas of a group as a set of abstract operations

Operations can be expressed as transformations of QS

Requires unitarity

Unitary representations of a group take the form

The commutation relations between the generators determine the algebra or the overall characteristics of the group.

 

The unitary representations of the rotation group are:

 

 

This the same group as isospin that combine u,d quarks

 

 

Each group has a fundamental representation and then higher representations that are combinations of the fundamental. 

 

 

Appendix D