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How the Wavefunction of the Δ(1232) Baryon

Can be Constructed

Before reading this page it is strongly recommended that readers examine the following short page. This page relies on fundamental laws of physics and proves that the state of the Δ(1232) baryon is not a single particle s-wave of the three u quarks.

For reading a proof showing that QCD has been constructed on erroneous arguments Click here.

Here a naive approach where three valence quarks attracted to a baryonic core is discussed. Thus, space is empty, except for the baryonic core and the three valence quarks. The relativistic properties of the system show that a jj coupling is better than an LS coupling. Using this approach, let us write down some antisymmetric 3-particle wave functions that contribute to the entire state of the Δ(1232) baryon.

Since the Δ++(1232) baryon has 3 valence quarks of the u flavor, the isospin of all four Δ baryons is fully symmetric. Therefore, each of the space-spin 3-particle functions must be antisymmetric. (In this sense, the state of the Δ(1232) baryon resembles the corresponding three electron state of an atom. However, unlike the relatively small spin effects in atoms, here spin dependent interactions are very strong.) Obviously, each of the 3-particle functions must have a total spin J=3/2 and an even parity. For writing down wave functions of this kind, single particle wave functions having a definite jπ, parity and appropriate radial functions are used. A product of three specific jπ functions is called a configuration and the total wave function takes the form of a sum of terms, each of which is associated with a configuration. Here only even parity configurations are used. Angular momentum algebra is applied to the single particle wave functions and yields an overall J=3/2 state. In each configuration, every pair of u quarks must be coupled to an antisymmetric state. ri denotes the radial coordinate of the ith quark.

For the simplicity of the discussion, the specific configurations written below do not account for a baryonic core that contains closed shells of quarks. For example, assume that a baryonic core contains a closed shell of uudd quarks. Thus, each configuration of the three valence quarks should comply with the appropriate orthogonality to the wave functions of the core's quarks. The configurations which are written below pertain to a simple quarkless core.

Each of the A-D cases described below contains one configuration and one or several antisymmetric 3-particle terms. The radial functions of these examples are adapted to each case.

Notation:   fi(rj), gi(rj), hi(rj), vi(rj) denote radial functions of Dirac single particle ½+, ½-, 3⁄2-, 3⁄2+ states, respectively. The index i denotes the excitation level of these functions.
  1. f0(r0)f1(r1)f2(r2) ½+ ½+ ½+

    Here the spin part is fully symmetric and yields a total spin of 3/2. The spatial state is fully antisymmetric. It is obtained from the 6 permutations of the three orthogonal fi(rj) functions divided by √6.

  2. f0(r0)g0(r1)g1(r2) ½+ ½- ½-

    Here, the two ½- are coupled symmetrically to j=1 and they have two orthogonal radial functions gi. The full expression can be antisymmetrized.

  3. f0(r0)f0(r1)v0(r2) ½+ ½+ 3⁄2 +

    Here we have two ½+ single particle functions having the same non-excited radial function. These spins are coupled antisymmetrically to a spin zero two particle state. The third particle yields the total J=3/2 state. The full expression can be antisymmetrized.

  4. f0(r0)g0(r1)h0(r2) ½+ ½- 3⁄2 -

    Here all single particle spins are different and antisymmetrization of the spin coordinates can easily be done. (The spins can be coupled to a total J=3/2 state in two different ways. Hence, two different terms belong to this configuration.)

These four configurations contribute to the lowest energy state of the Δ(1232) baryon, because the total spin of each one of them is J=3/2 and the state's parity is even. They belong to the basis of the Hilbert space of the Δ(1232) baryon. In principle, one may construct the Hamiltonian matrix and diagonalize it. The Hamiltonian's eigenfunction having the lowest eigenvalue represents the ground state of the Δ(1232) baryon. Using other values of single-particle angular momentum and parity, one may add more functions to the basis of the Hilbert space of the three uuu quarks. Adding pairs of quark-antiquark, one can build a Fock space for the particle. Like in atomic spectroscopy, one expects that as the basis increases, the lowest eigenvalue and its eigenfunction will approach the true ground state.