Subatomic Physics

15.7. QCD and Quark Models of the Hadrons 487
For each value of l, the magnetic quantum number m can assume the 2l +1values
from −l to l. The parity of each state is given by Eq. (9.10) as
π =(−1) l .
States of even and odd parity exist, and consequently the possible orbital angular
momenta for a state with quantum number N are given by
Neven πeven l =0, 2,...,N
Nodd πodd l =1, 3,...,N.
(15.29)
The degeneracy of each level N can now be obtained by counting: The possible angular
momenta are determined by Eq. (15.29); each angular momentum contributes
2l + 1 substates, and the total degeneracy becomes
1
(N +1)(N +2). (15.30)
2
The radial wave function R(r) =(2/r) 1/2 Λ is characterized by the number, nr, of
its nodes. It is customary to exclude nodes at r = 0 and include nodes at r = ∞ in
counting. The examples in Fig. 15.8 then show that
nr =1+k =1+ 1
(N − l). (15.31)
2
This relation is valid for all radial wave functions R(r).
After this long preparation we return to our goal, connecting the properties of
the harmonic oscillator to particle models. A state of a particle can be characterized
by its mass (energy) and its angular momentum. In Fig. 15.9, we show the lowest
few levels of the harmonic oscillator, labeled by the quantum numbers N, the radial
quantum numbers nr, and the angular momenta in units of �, and the corresponding
levels of the e + e − system without the magnetic force effects.
We expect the qq bound states to lie somewhere between these two extremes,
as shown in the figure; we have also included the effect of the short-range spin–spin
force, Eq. (15.16), for the lowest several states. The first two states correspond to
the 0 − and 1 − multiplets, the next two to 3 P and 1 P1 states. The 3 P state is split
by spin–orbit forces into 0 + , 1 + ,and2 + meson multiplets, most of the members
of which have masses above 1 GeV/c 2 . The center of the 1 P1 state is at about
1240 MeV/c 2 , as is that of the 3 P0 state. The 3 P1 state lies at about 1350 MeV/c 2
and the 3 P2 state at 1400 MeV/c 2 .
The harmonic oscillator shows another feature of the particle spectrum, namely,
some general relationships between particles of different spins but the same parity.
In some cases these particles appear to be rotationally excited states of the particle of
lowest mass; we then expect to find a multiplet with the same number of components
as in the lowest mass state.