Thread: Quantum number.
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Dec1-03, 10:22 AM
P: 209
There are a whole lot of quantum numbers associated with different fields and particles. Its all about quantization; various properties can only be had in certain discreet values. The quantum numbers are actually just a description of how many of these discreet units are present in a given object.

For example, I deal alot with mesons (particles that consist of a quark and an antiquark in a bound system), and there are a number of quantum numbers to deal with. First of all, there are quantum numbers for angular momentum and spin momentum, called l and s respectively. There are also flavor quantum numbers, including isospin (I), strangeness (S), charm (C), bottom (B) and top (T). The isospin is a property of the lightest quarks (up and down), while the others are properties of the heavier quarks. There is also the baryon number (b). All quarks have an intrinsic baryon number of 1/3 and their antiquarks have baryon number -1/3. The result is that baryons have b = 1 and mesons have b = 0 (which is the natural result, after all).

The l and s quantum numbers can be combined through a process called "coupling", which is like addition;

j = l '+' s
= {[l+s], [l+s-1],..., [l-s]}

but it allows all the values in between the addition and subtraction of the two, as shown above. The result of coupling is the total momentum number j.

There are also quantum numbers associated with symmetries here. There is a parity number P which is either +1 or -1 based on the equation;

P = (-1)^l+1

a charge conjugation number based on the formula;

C = (-1)^l+s

and a G-parity number based on the formula;

G = (-1)^l+s+I

which includes the isospin in the symmetry. There is also a radial excitation quantum number N that is useful.

When we represent the quantum states that are occupied by mesons, we generally form the multiplets of mesons based on the quantum numbers N, l, s, j, P, and C. Within these multiplets are members with different values of I, G, S, C, B and T numbers as well. All mesons have b = 0. So we generally represent the mesons, in written form, by the statement IG(JPC). For example, the pion can be represented as 1-(0-+), the eta meson as 0+(0-+), the kaon as 1/2(0-). They all occur in the same multiplet, the ground state pseudoscalar multiplet with (0-+) being the key defining numbers there. *The kaon is a spin 1/2 particle, and hence not an eigenstate of C, thus the C and G numbers are ommited.

So there's some examples of how quantum numbers are used to keep track of which particles are which and how they are related to each other.