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4.3 Harmonic-oscillator expansion (equal masses)

The method, which can also be used for mesons, is the following. One expands the eigenstates of

pρ2 p2λ ∑ ^H = m--+ m--+ V (rij) , i<j
(4.8)

into the eigenstates of

2 2 H = p-ρ + pλ-+ K (ρ2 + λ 2) , 0 m m
(4.9)

i.e., 

∑ Φ = c (| n ,l ⟩⊗ | n ,l ⟩) , l,m nρ,lρ,nλ,lλ ρ ρ λ λ l,m nρ,lρ,nλ,lλ
(4.10)

and studies the convergence as a function of the value of the maximal number of quanta allowed in the expansion, N = 2nρ + lρ + 2nλ + lλ. In principle, one could optimize the variational parameter K for each N. For high N, this results into expensive calculations and numerical instabilities. So, in practice, one optimizes K for a small number of quanta N′ and then pushes the expansion up to larger N. For given N and K, the harmonic-oscillator (h.o.) expansion results in the diagonalization of a symmetric matrix, which is the restriction of H^ to the sub-space spanned by the first levels of H0.

As an illustration, let us compute the two first levels having angular momentum and parity lP = 0+, corresponding to quark masses m i = 1 and a linear confinement. The h.o. basis is made of these states with the label [56, 0+] in Section 3.6. We rename them as

| 1 ⟩ (N = 0), | 2⟩ (N = 2), | 3⟩,| 4⟩ (N = 4) | 5⟩,| 6 ⟩,| 7⟩ (N = 6 ), | 8⟩,| 9 ⟩,| 10⟩,| 11⟩ (N = 8),
(4.11)

The results shown in Table 4.1 correspond to different prescriptions for the variational parameter K, namely different levels of expansion N′ at which it is adjusted.

Table 4.1: Symmetric and scalar states ([56,0+]) for quark masses m i = 1 and linear confinement V = 1 2 ∑ rij using a harmonic-oscillator expansion up to N quanta. The oscillator parameter K is adjusted by minimizing either the first or second level with an expansion limited to N′ quanta (the minimized energy is underlined). The exact values are very close to: E0,0 = 3.8631, E1,0 = 5.3207, and E2,0 = 6.5953.

N N′α = K1∕2 E 0,0 E1,0 E2,0
0 3.8711
2 3.8711 5.3766
4 0 0.4301 3.8640 5.3468 6.8082
6 3.8635 5.3214 6.6831
8 3.8632 5.3215 6.6008
4 3.8636 5.3695 7.0149
6 4 0.4782 3.8634 5.3259 6.7671
8 3.8632 5.3224 6.6364
2
4
6
8
2 0.3811
3.8732
3.8651
3.8639
3.8634
5.3445
5.3388
5.3223
5.3217
 
6.6729
6.6367
6.5975

As pointed out by Moshinsky [49], minimizing the ground state at the N′ = 2 order does not provide any improvement with respect to N′ = 0. Otherwise, the value of the harmonic oscillator parameter K depends rather sensitively on which level and to which order the minimization is performed. However, the accuracy of the eventual energies depends less on the prescription adopted for K than on the number of quanta N introduced in the expansion.

Another output of such a calculation is the set of coefficients ci which describe the wave function. For instance, in the case where K is optimized at the N′ = 4 order and the expansion pushed up to N = 8, one obtains, for the wave function Ψ = ∑ ci∣ i⟩ of the ground state and its first radial excitation the coefficients given in Table 4.2.

Table 4.2: Coefficients of the harmonic-oscillator expansion for the ground state and first excitation with lP = 0+. It includes up to N = 8 quanta, but the oscillator strength is determined by minimizing the ground state energy when the expansion is truncated at the N′ = 4 level. Quark masses are mi = 1, and the interquark potential 1 2 ∑ rij.

State
c1
(N = 0)
c2
(N = 2)
c3,c4
(N = 4)
c5 − c7
(N = 6)
c8 − c11
(N = 8)
n = 0 0.99431 –0.08993
–0.01953
0.05287
–0.00494
–0.00232
–0.00223
0.00115
–0.00178
–0.00304
0.00554
n = 1 0.10169 0.94301
0.05479
–0.27911
0.01608
–0.02957
0.13209
–0.00454
0.00486
0.00577
–0.02859

Note that the convergence is rather clear for the ground state, which is dominated by the N = 0 component c1 ∣ 1⟩. There is more mixing of configurations for the excited states, as is usual in variational calculations.

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