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3.3 Three-body oscillator with equal masses

We start from the symmetric Hamiltonian

2 2 2 H = p-1-+ -p2-+ p3-+ 1K (r 2 + r2 + r2 ) , r ≡ r − r , 2m 2m 2m 2 12 23 31 ij j i
(3.17)

and introduce the Jacobi coordinates

-- ρ = r − r , λ = (2r − r − r )∕√ 3 , R = (r + r + r )∕3 , 2 1 3 1 2 1 2 3
(3.18)

as well as their conjugate momenta pρ, pλ, and P . This results in

P 2 p 2ρ p 2λ 3 2 2 H = ----+ --- + ---+ -K (ρ + λ ) . 6m m m 4
(3.19)

We now disregard the centre-of-mass kinetic energy which vanishes in the rest frame of the baryon. The Hamiltonian appears as the sum of two three-dimensional harmonic oscillators. We thus find the eigenvalues

∘ ----- 3-K-- EN = 4 m (6 + 2N ) , N = 2nρ + lρ + 2n λ + lλ .
(3.20)

Using, again, the scaling laws of Section 2.3, which are immediately generalized to more than two bodies, one can assume that 3∕4K = m = 1.

The associated degeneracy is

d (6)= (N-+--1)(N--+-2)(N--+-3)(N-+--4)(N--+-5)-= (N-+-1)5-⋅ N 120 5!
(3.21)

The index (6) reminds us that H can be viewed as a 6-dimensional oscillator. One can easily check the generalization dN(q+1) = (N + 1) q∕q! A basis for the eigenspaces is provided by states of the type

| nρ,lρ,m ρ;n λ,lλ,m λ⟩ = | n ρ,lρ,m ρ⟩⊗ | n λ,lλ,m λ⟩ ,
(3.22)

which can be arranged by Clebsch–Gordan coupling into states of given total angular momentum l = lρ + lλ with the notations

∑ | nρ,lρ;nλ,lλ;l,m ⟩ = (| nρ,lρ⟩⊗ | nλ,lλ⟩)l,m = CG | nρ,lρ,m ρ;nλ,lλ,m λ⟩ .
(3.23)

For N > 1, these states, however, do not exhibit simple permutation properties. To impose the restrictions of the Pauli principle, one should recombine the states (3.23) having the same N, l, and m into states of well-defined permutation symmetry. This will be done shortly, after some basic reminders about the permutation group S3.

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