The issue of parity is a hazy issue that is starting to clear up for me. I now understand one has parity of a particle, (electron, nucleon and others), one has parity of the nucleus and one has parity conservation & violation from force interactions. The following references were a big help; but, I should note these references were not without questions and I hope you can help me with that.
Donald Perkins in his book “Particle Astrophysics”
http://books.google.com/books?id=WJnARsI5uywC&pg=PA72&sig=7PYKTuijnpzB5pH3aKxS-h0uacY#PPA66,M1
provides a general definition of particle parity on page 66 in the section 3.3 “The Parity Operation” where he indicates the inversion of spatial coordinates
(x, y, z) > (-x, -y, -z) is a discrete transformation of the wave amplitude brought about by the parity operator P:
P\psi \left(r\right) = \psi\left(-r\right)
And repeating the operation reverts to the original system
P^2 = 1
Thus the enginvalue of P must be +/- 1
Perkins indicates in a spherically symmetric potential that it has the property V(-r) = V(r) so the bound states by such potential can be parity eignestates. He goes on to indicate for the hydrogen atom the wavefunction in terms of the radial coordinate r and the polar and azimuthal angular coordinates \theta And \phi of the electron with respect to the proton is
\chi \left(r, \theta, \phi\right) = \eta\left(r\right)Y_l^m\left(\theta, \phi\right)
Under inversion
r \rightarrow -r,
\theta \rightarrow \left(\pi-\theta\right)
\phi \rightarrow \left(\pi+\phi\right),
produces the following results:
Y_l^m \left(\pi- \theta,\pi+ \phi\right) = \left(-1\right)^l \left(\theta, \phi\right)
Hence:
P\chi \left(r, \theta, \phi\right) = \left(-1\right)^l \chi \left(r, \theta, \phi\right)
Thus
P = \left(-1\right)^l
PERKINES DID NOT DEFINE \chi or \eta CAN YOU HELP WITH THIS?
ALSO, I HAVE NOT FOUND HOW \left(-1\right)^l WAS DERIVED. CAN YOU ALSO HELP HERE?
Perkins continues with an interesting discussion on parity conservations with strong (gluons) and electromagnetic (photon) interactions and parity violation with weak (heavy gauge boson) interactions.
In a different reference Sidney Yip in his MIT Open Courseware presents a discussion in his lecture “22.101 Applied Nuclear Physics, Fall 2004”
http://ocw.mit.edu/NR/rdonlyres/07F56F5C-3339-41E4-8F53-858388034EB1/0/lec10.pdf
where in his section ‘Prediction of Ground-State Spin and Parity’ on page 11 he presents the following rules which have a bearing on parity:
1- Angular momentum of odd-A nuclei is determined by the angular momentum of the last nucleon in the species (neutron or proton) that is odd.
2- Even-even nuclei have zero ground-state spin, because the net angular momentum associated with even N and even Z is zero, and even parity.
3- In odd-odd nuclei the last neutron couples to the last proton with their intrinsic spins in parallel orientation.
In the last rule Yip explains the determination of odd-odd nuclei spin but does not explain the parity associated with an odd-odd nucleus.
THUS I DON’T UNDERSTAND HOW TO DETERMINE THE ODD-ODD NUCLEI PARITY CAN YOU HELP WITH THIS ISSUE.
This last question may be answered by the parity formula posted by malawi_glenn where he indicates
<br />
(-1)^{l_1}*(-1)^{l_2} = (-1)^{l_1+l_2}<br />
If sub 1 and sub 2 represents the angular momentum of two different particles, specifically, a proton and a neutron then it would be a simple matter of doing the math to determine an odd-odd nuclei parity.
IS THIS A CORRECTION OBSERVATION?
