[next] [prev] [prev-tail] [tail] [up]

3.6 Spatial wave functions of given permutation symmetry

Let us come back to the reduced and rescaled Hamiltonian

h = p2ρ + ρ 2 + p 2λ + λ2 .
(3.43)

The wave functions are often labelled with N, the number of quanta, with lP , the total angular momentum and parity, and with the dimension of the SU(6) representation. The Hamiltonian (3.43) has, indeed, a symmetry of structure U(6) ≡ SU(6) × U(1), where U(1) is associated with the number of quanta N [6]. In fact, SU(6) also denotes another symmetry combining spin and flavour. It becomes exact when one neglects hyperfine effects and takes the limit of equal quark masses mu = md = ms. We refer to the specialized literature [6] for these group theoretical aspects of the harmonic oscillator. For our purpose, it is sufficient to know the following correspondence between SU(6) representations and permutation properties:

[56] = symmetric [20] = antisymmetric [70] = mixed symmetry .
(3.44)

By superposition of the factorized states (3.23), one can form:

N = 0 | 56, 0+⟩ =| 0,0⟩⊗ | 0,0⟩ (3.45)

N = 1 | 70 λ,1− ⟩ =| 0,0⟩⊗ | 0, 1⟩ | 70 ,1− ⟩ =| 0,1⟩⊗ | 0, 0⟩, (3.46) ρ

N = 2 + | 20,1 ⟩ = (| 0,1 ⟩⊗ | 0,1⟩)1 + 1 | 70λ,2 ⟩ = √---(| 0,2⟩⊗ | 0,0⟩− | 0, 0⟩⊗ | 0,2 ⟩) 2 | 70ρ,2+⟩ = (| 0,1 ⟩⊗ | 0,1⟩)2 1 | 56,2+⟩ = √---(| 0,2⟩⊗ | 0,0⟩+ | 0, 0⟩⊗ | 0,2⟩) 2 (3.47) + -1-- | 70λ,0 ⟩ = √2--(| 1,0⟩⊗ | 0,0⟩− | 0, 0⟩⊗ | 1,0 ⟩) + | 70ρ,0 ⟩ = (| 0,1 ⟩⊗ | 0,1⟩)0 + -1-- | 56,0 ⟩ = √2--(| 1,0⟩⊗ | 0,0⟩+ | 0, 0⟩⊗ | 1,0⟩),

N = 3, lP = 3− 1 √ -- | 56,3− ⟩ =-[| 0,0⟩⊗ | 0,3⟩ − 3(| 0,2⟩⊗ | 0,1⟩)3] 2 √-- | 20,3− ⟩ = 1[ 3 (| 0,1⟩⊗ | 0,2⟩)3− | 0,3⟩⊗ | 0,0⟩] 2 (3.48) − 1-√-- | 70λ,3 ⟩ = 2[ 3 | 0,0⟩⊗ | 0,3 ⟩ + (| 0,2 ⟩⊗ | 0,1⟩)3] 1 √ -- | 70ρ,3− ⟩ =-[(| 0,1⟩⊗ | 0,2⟩)3 + 3 | 0,3⟩⊗ | 0,0⟩] 2

P − N = 3, l = 2 | 70λ,2− ⟩ = (| 0, 2⟩⊗ | 0,1⟩)2 | 70 ,2− ⟩ = (| 0, 1⟩⊗ | 0,2⟩) , (3.49) ρ 2

N = 3, lP = 1− 1 √ -- √ -- | 56,1− ⟩ = √---[− 3 | 0,0⟩⊗ | 1,1⟩ + 5 | 1,0⟩⊗ | 0,1⟩ + 2(| 0,2⟩⊗ | 0,1⟩)1] 12 − --1-- √ -- √ -- | 20,1 ⟩ = √12--[− 3 | 1,1⟩⊗ | 0,0⟩ + 5 | 0,1⟩⊗ | 1,0⟩ + 2(| 0,1⟩⊗ | 0,2⟩)1] 1 √ -- √ -- | 70λ,1− ⟩′ =-√--[ 5 | 0, 0⟩⊗ | 1,1⟩ + 3 | 0,1⟩⊗ | 1,0⟩] 2 2 − ′ --1--√ -- √ -- (3.50) | 70ρ,1 ⟩ = √--[ 5 | 1, 1⟩⊗ | 0,0⟩ + 3 | 0,1⟩⊗ | 1,0⟩] 2 2 √ -- √ -- | 70λ,1− ⟩′′ = √-1-[ 3 | 1, 1⟩⊗ | 0,0⟩ − 5 | 1,0⟩⊗ | 0,1⟩ + 4(| 0,2⟩⊗ | 0,1⟩)1] 24 − ′′ 1 √ -- √ -- | 70ρ,1 ⟩ = √----[ 3 | 1, 1⟩⊗ | 0,0⟩ − 5 | 0,1⟩⊗ | 1,0⟩ + 4(| 0,1⟩⊗ | 0,2⟩)1]. 24

Up to N = 2 or even N = 3, one can obtain these combinations of the factorized states ∣nρ,lρ⟩⊗∣nλ,lλ⟩ by empirical methods. For instance, dealing with a scalar (l = 0) polynomial of degree 2, one easily identify ρ2 + λ2 as being symmetric, whereas λ2 −ρ2 and −2λ⋅ρ form a pair of mixed symmetry. For larger N, however, one hardly avoids the use of more systematic methods. In each eigenspace labelled by N, with energy 6 + 2N, one can consider the subspaces (N,l) of given total angular momentum and in each subspace diagonalize the permutation operator P→ (P12 is already diagonalized by the even or odd character of lρ). The relevant matrix elements

l⟨n′,l′| ⊗ ⟨n′,l′| P → | nρ,lρ⟩⊗ | n λ,lλ⟩l = λ∫ λ ρ ρ ( √ -- ) ( √ --) 3 3 ∗ ∗ 1 3 1 3 d ρd λ ϕn′λ,l′λ(λ)ϕ n′ρ,l′ρ(ρ )ϕn ρ,lρ − 2ρ + -2--λ ϕn λ,lλ − 2λ − -2-ρ
(3.51)

(the Clebsch–Gordan summations are omitted for better reading in the integral) are called the

Brody–Moshinsky coefficients [42]. They can be computed by astute recursion relations [43, 44], or (still exactly) by brute force computer algebra. Other methods have been proposed, for instance by Horgan [45], to construct the harmonic-oscillator wave functions of given angular momentum, parity and permutation properties.

[next] [prev] [prev-tail] [front] [up]