Read Martin&Shaw chapter 4.
Symmetries and conservation laws play an important role in particle physics. A symmetry is simply a statement that the results of an experiment must be independent of our choice of coordinate system, our definition of up and down, our the assignment of positive and negative charge, and the like. It results from an invariance of the equations of physics under a transformation of one of the above.
Some symmetries we look at are fundamental while others are only approximate. The fundamental symmetries have associated with them a conserved quantity. Conserved quantities play a vital role in the advance of our understanding of particle physics, but we must be constantly vigilant for signs of a violation of their conservation.
Symmetries can be either continuous (coordinate transformations) or discrete (charge conjugation). The conserved quantity of a continuous symmetry is additive (momentum or energy). The conserved quantity of a discrete symmetry is multiplicative (parity).
Recall the QM equation for a particle:
If we are dealing with an isolated system, the physical law represented by this equation is independent of any coordinate system, and therefore the equation itself is independent of the particular choice of coordinate system. Hence if ψ(x) is a solution in one coordinate system, then ψ'(x') must be a solution in a primed coordinate system, translated by Δx from the unprimed system. In particular, this must be true for an infinitisimal translation δx.
We build up a finite translation by taking the limit as n goes to infinity of n infinitisimal tranlations:
In a similar way, we can construct an operator that generates infinitisimal rotations about an axis, for instance, the z axis:
As they should be, the operators D and R are unitary. That is, they leave the normalization of ψ' unchanged since D*D = exp(-ipΔx/hbar)exp(ipΔx/hbar) = 1:
We'll come back to the rotation operator. Now let's move to a discrete symmetry, parity.
The parity operation, denoted P, is a reversal of sign on all coordinates.
A wavefunction will have a defined parity if and only if it is an even or odd function. For example:
For any system bound by a central potential, V(r), the wave function can be decomposed into radial and angular parts, with the angular parts described by spherical harmonics: