University of Zagreb. Faculty of Science. Department of Physics.
Abstract
Polazeći sa stajališta funkcionalne analize, valne funkcije obično shvaćamo kao vektore u Hilbertovom prostoru kvantnih stanja. Kad je broj identičnih čestica veći od jedan, pojavi se specifična algebarska struktura, tako da je isti Hilbertov prostor graduirana algebra nad prstenom simetričnih polinoma (bozonskih pobuđenja). Generatori algebre (vakuumi) su od prije poznati u Kartezijevoj bazi, a u ovom radu ih prevodimo u bazu dobrog zakretnog impulsa za poseban slučaj dvije čestice. Tada tri generatora (Ψ1; Ψ2; Ψ3) čine vektor osnovnog stanja, a četvrti Ψ4 = Ψ1Ψ2Ψ3 je pseudoskalar i nalazi se medu dvostruko pobuđenim stanjima, kojih ima 28. Kad se sva ta stanja napišu u obliku bozonskih pobuđenja vakuuma i u bazi dobrog zakretnog impulsa, otkrije se da je Ψ4 komponenta određenog pobuđenog stanja zakretnog impulsa l=3 i projekcije m = ±2. Time se to stanje identificira kao glava vrpce pobuđenih stanja drugačije simetrije od vrpce koja nastaje pobuđivanjem osnovnog stanja.From the point of view of functional analysis, wave functions are understood as vector in Hilbert’s space of quantum states. When the number of identical particles is greater than one, there appears a specific algebraic structure, where the same Hilbert space is a graded algebra over the ring of symmetric polynomials (bosonic excitations). Generators of the algebra (vacuums) are known in the Cartesian basis. In this work we translate them to the basis of good angular momentum for the specific case of two particles. Then three generators (Ψ1; Ψ2; Ψ3) make up the ground-state vector and the fourth Ψ4 = Ψ1Ψ2Ψ3 is a pseudoscalar found among doubly excited states, of which there are 28. When all those states are written in the form of bosonic excitations of the vacuums and in the basis of good angular momentum, it is revealed that Ψ4 is a component of a specific excited state of angular momentum l=3 and projection m = ±2. Therefore this state is identified as a band head of excited states with different symmetry then the band which is generated by excitations of the ground state
