and since the ^'s are uniquely specified by j, m, and k and the last expression implies that cr ■ fy^m(f) has opposite parity from 3^m(r)> it follows that a ■ ryk]m{r) must be proportional to y^{f).

However, the relations (6.24) are readily proved by direct computation. For the first relation,

Hence [J\<r ■ f] = 0. The second relation follows from

{P,crf} = [Per -f + <r -f P] = [-<T-f + cr-f] P = 0

removed from the equation, leaving only two unknown functions of the radial coordinate. Rearranging terms and dropping the subscripts and superscripts give


V(r)]f(r) =

dg{r) dr


[.E + m-

V(r)]fl(r) =


This completes the task for this section. In the next section these equations are solved for a constant potential.


As an illustration of the modern use of the Dirac equation, we give a very simple introductory discussion of the structure of hadrons (strongly interacting particles).*

There is strong experimental evidence to support the view that mesons and baryons are composed of elementary spin \ particles called "quarks." (For a brief summary of the particles of modern physics, see Appendix D.) Mesons are believed to be composed of a "valence" quark (q) and antiquark (q) pair, surrounded by a "sea" of gluons and other qq pairs, and baryons composed of three valence quarks surrounded by a similar sea. Furthermore, quarks are believed to ¿xist only in the combinations of quarks and antiquarks which exist in baryons and mesons. If we attempt to remove a single quark from such a combination, the energy grows with the distance the quark is separated from its neighbors, until it becomes so large that it is energetically favorable to create a qq pair and break the "string" connecting the quark to its neighbors. The situation is similar to trying to isolate a single north or south pole of a magnet; if we cut the magnet apart, a new pair of poles is created, defeating our purpose. Because of this property of the forces which bind quarks together, they are said to be confined.

The MIT bag model is a very simple model for hadronic structured Suppose the hadron occupies a spherical volume of radius R. If a quark is inside this volume, we assume its mass is small, and it may be taken to be zero. If it gets outside, interactions with the neighboring quarks which make up the rest of the hadron are assumed to generate an infinite mass for the quark. Since this implies infinite energy, the quark will not penetrate outside of the hadronic volume, which is designated "Region I" in Fig. 6.1, and is assumed to be spherical.

*For a review of modern ideas about hadronic structure, see Bhaduri (1988).

Hwo early papers introducing the bag model are [CJ 74], Additional references can be found in

Bhaduri (1988).

6.2 HADRONIC STRUCTURE Recall that the spherical Bessel functions are solutions of the equation*

Hence if k% = E2 -m? > 0, the solutions of (6.27) are je(kor) regular as r —> 0 nt(kor) singular as r —► 0 while if fcg = —Kq = E2 — m2 < 0, the solutions are h^\iKor) regular as r —> oo ^/i^1' -

All of these functions satisfy the recursion relations

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