Remarks on the Proton's Wave Function

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(It is recommended that readers read the following introduction before reading this page - click here).

This page contains general remarks on the form of the proton's wave function. Analogous arguments hold for other baryons in general and for the neutron, in particular. The theory used is the Regular Charge-Monopole Theory (RCMT). (Click here for reading an overview on this theory and its applications to strong interactions.) Thus, we have three valence quarks of the uud flavor. They are ordinary spin-1/2 Dirac particles and the uu pair must be in an antisymmetric state. A result of this discussion is that one should not expect to find that the so called "naive quark model" is physically meaningful. This model uses just one configuration for describing the proton's state. Thus, the naive quark model assumes that the uu pair is in a state where L=S=J=0 and the d quark accounts for the proton's spin. This model certainly does not fit the data. Indeed, if it is correct then the proton's magnetic moment should depend on the single d quark whose electric charge is -e/3. In such a case the sign of the proton's magnetic moment should be negative with respect to its spin. This outcome contradicts the experimental value which is positive. Other inconsistencies of the naive quark model are seen in the neutron data and from a comparison of the proton-neutron magnetic moments.

The mathematical basis of the multiconfiguration structure of a quantum system that contain more than two particles is described here in few lines. The Hamiltonian is a hermitian operator. Since the overall angular momentum, parity and flavor are good quantum numbers of strong and electromagnetic interactions, one may find a nonvanishing off-diagonal matrix element which is related to two configurations that have the same total angular momentum, parity and flavor. Now, the following is a mathematical property of hermitian matrices: If the lowest diagonal matrix element is related to off-diagonal matrix elements then a diagonalization of such a matrix reduces the value of the lowest diagonal element. Therefore, the ground state of a particle is described by many configurations.

The points described below rely on theoretical principles and experimental data.
  1. As stated above, the theory used herein is RCMT and, in particular, quarks are ordinary Dirac particles.

  2. Hadronic data indicate that the elementary unit of the magnetic charge is much larger then its electric counterpart where e2=1/137. Hence, relativistic effects cannot be treated as a small perturbation. Thus, one concludes that:

    1. Spin-orbit and spin-spin interactions play a significant role and their values are expected to be of the same order of magnitude as the Coulomb-like interaction. Therefore, the magnetic monopole analog of the fine structure is expected to produce a very significant split.

    2. The overall wave function contains configurations having additional qq pairs. (Click here for observing a figure which shows that antiquarks are measured directly in a proton). Therefore, a Fock space should be considered. The relatively small mass of pions indicates that the energy price of adding a qq pair to a hadronic state is quite cheap.

  3. All terms of the proton must have the same overall angular momentum (J=1/2), parity (even) and flavor (uud). Its actual calculation looks like a very hard task. Thus, some qualitative arguments are presented below, in order to gain an insight into the problem. In particular, note that 35 terms are used in a calculation of the Jπ=0+ ground state of the helium atom [1]. Now, a ½+ state of a uud system can be created by many other states of the uu subsystem, besides the 0+ state. Thus, regarding all these arguments, one expects to find that a very large number of configurations are needed for a good description of the proton's ground state. (BTW, this very large number of configurations provides a straightforward explanation to the nagging problem called the proton spin crisis.)

  4. 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 uudd quark system and 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.

  5. The main purpose of this discussion is to show how a multi-configuration structure of a state arises. Therefore, the more intuitive L-S (also called Russell-Saunders) scheme is used. Here orbital angular momenta are coupled separately and the same is true for the spins. This scheme is mathematically equivalent to the alternative j-j coupling. An example of the j-j copling can be seen here.

  6. The configurations described below do not contain additional qq pairs. Evidently, this is the simplest set. Each term associated with this set can be characterized by the antisymmetric state of the uu quarks. Having the state of the uu quarks, one defines the state of the d quark so that the total angular momentum and parity take the required values. Let us examine three uu states of this kind. Other states can be constructed analogously. The functions fi(r), gj(r) and hk(r) denote radial functions associated with the s,p,d angular functions, respectively. The subscripts i,j,k denotes the order of the radial excitation. r1, r2 denote the radial coordinates of the two u quarks and r3 denote the radial coordinates of the d quark. As usual, L,S,J denote spatial, spin and total angular momentum.

    1. f0(r1) f0(r2); l1=l2=0; S=0.

      The spatial part of the two u quarks is symmetric and takes the lowest orbital. The spin part is antisymmetric. The d quark (not mentioned here) is in its lowest state and accounts for the overall spin and parity. This is the "naive" state.


    2. f0(r1) f1(r2); l1=l2=0; S=1.

      Here one radial function represents the first radial excitation. The spin state is symmetric and the radial functions are antisymmetrized. In this case, one sees a kind of a tradeoff between two energy effects. The radial excitation increases the kinetic energy of the state. On the other hand, the exchange integral reduces it. The latter effect is an analog of the Hund's rule. The role of the d quark is analogous to that of the previous case. However, here one can construct 2 configurations:

      1. [f0(r1) f1(r2); l1=l2=0; Juu=1] f0(r3); Jtotal=1/2.


      2. [f0(r1) f1(r2); l1=l2=0; Juu=1] h0(r3)j3=3/2; Jtotal=1/2.


      Note that in the second case the orbital of the d quark is l=2.


    3. g0(r1) g0(r2); l1=l2=1; L=1; S=1; J=0,1.

      Here we have two p-waves coupled to an antisymmetric spatial state L=1 and a symmetric spin state. g denotes the radial function of the p-waves. This part of the wave function produces two states, depending on the total angular momentum J of the uu subsystem. Thus, we have here j=0 and j=1. (The case of j=2 cannot produce a final spin-1/2 state with a ground state d quark.) The role of the d quark is analogous to that of the previous configurations. Like in the second case, the symmetric spin state of the uu quarks reduces the overall energy.

      The 2 couplings of the uu quarks together with the f0(r3) of the d quark produce 2 terms:

      1. [g0(r1) g0(r2); l1=l2=1; L=1; S=1; Juu=0] f0(r3)s=1/2;Jtotal=1/2.


      2. [g0(r1) g0(r2); l1=l2=1; L=1; S=1; Juu=1] f0(r3)s=1/2;Jtotal=1/2.




  7. Each of the wave functions constructed like the ones described above can be an element of the Hilbert space which contains the state of the uud quarks. Similarly, a Hilbert space for the system uuddd can be constructed, etc. Having a good approximation for the full space, one can calculate the Hamiltonian's matrix elements, diagonalize it and find the wave function which describes the proton's state.

  8. Now, some of the large number of terms that form a basis for the Hilbert space have spatial angular momentum l>0 (see e.g. cases B.b and C above). Therefore it makes sense to find that a part of the proton's overall spin is associated with spatial angular momentum. Hence, the "proton spin crisis" which was found nearly two decades ago should not be regarded as a surprise and vice versa: the fact that only a small portion of the proton's spin is associated with quarks' own spin is consistent with the analysis carried out above.

  9. Another issue is the magnetic moment of nucleons. Evidently, the experimental data are completely inconsistent with the "naive quark model" described above. However, the multitude of configurations and terms expected to be useful for a description of a nucleon's ground state, indicate that only a specific calculation will (hopefully) provide an explanation for the nucleons' magnetic moment.

[1] A. W. Weiss, Phys. Rev. 122, 1826 (1961).