Interference and Fermions and Bosons Part II

Two particle product type wavefunctions may be created so as to be symmetric (bosons) or antisymmetric (fermions) on then interchange of the identical particles. As a result, spatial density at a point does not change upon interchange. This approach, however, appears to be mathematical in nature. In Part I of this note, we argued that exp(ipx) free particle momentum wavefunctions within a bound wavefunction W(x) interfere and that full wavefunctions Wn(x) in a linear combination also do the same. As a result,  probability interference is ultimately linked to the physical mechanism associated with the periodic exp(ipx) wave behind the Pauli rule that fermions may not occupy the same quantum state. In this note, we try to extend this idea and argue that one does not simply exchange identical particles in the wavefunction as a mathematical procedure, but that actual exchange may occur in the probabilistic quantum mechanical case. We note that a time-independent treatment of scattering from a target already includes the idea of “negative” or out of phase interference within a wavefunction W(x)= exp(ikz) + f(theta, phi) exp(ikr)/r  so as to remove some incoming flux as outgoing scattered flux now exists. We suggest that the “minus” sign used in the antisymmetric fermion wavefunction behaves in a similar manner. In (1), we argued that W(x)W(x) in a bound state includes all possible a(p1)exp(i p1 x) a(p2)exp(i p2 x) combinations and represented a product probability. In other words, a particle at x with p1  P(p1/x) may be knocked into p2 at x. Similarly, we argue that cross terms in a two particle fermion wavefunction represent flux W*b(x1)Wa(x1)W*a(x2)Wb(x2) flowing from out of states as an interchange occurs. Main terms such as W*a(x1)Wa(x1)W*b(x2)Wb(x2) represent the presence of particles in the a, b states. If a=b, the presence and flow probabilities cancel locally and there is a vanishing wavefunction i.e. the two fermions may not occupy the same state.

     We argue that one should be able to describe the Pauli exclusion rule in this manner and note that Feynman (2) has used probability arguments in boson scattering to argue the boson enhancement factor. In other words, the mathematical argument of a symmetric wavefunction is not used, but a more physical argument is given.