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3.2 The spatial oscillator

In the chapter on mesons, we already mentioned that the three-dimensional oscillator is solvable and may serve as a convenient basis for other potentials. The scale-independent part h3 of the Hamiltonian

--- --- P 2 1 1∘ K 1 ∘ K H = ----+ -KR 2 = -- ---h3 = -- ---[− Δ + r 2] , 2M 2 2 M 2 M
(3.6)

can be written

h3 = h1(x) + h1(y) + h1(z) ,
(3.7)

resulting in eigenvalues

ϵ = 3 + 2N, N = 0,1... N
(3.8)

with eigenspaces of dimension

(3) (N--+-1)(N-+--2) dN = 2 ,
(3.9)

which have a possible basis

Ψnxnynz(r) = ϕnx (x )ϕny(y)ϕnz(z) .
(3.10)

An alternative approach consists of using spherical coordinates, as in Section 2.1. This provides a new labelling of the eigenvalues

ϵ = 3 + 4n + 2l . n,l
(3.11)

Here l is the orbital momentum and n is the number of nodes of the reduced radial wave function un,l(r), whose expression is

∘ ------------- 1 un,l(r) = -----2n!---- rl+1Ll+n1∕2(r2)e−2r2 , Γ (n + l + 32)
(3.12)

where L is a Laguerre polynomial

α ∑n -------Γ (n-+-α-+-1)-------(−-x)m- L n(x) = Γ (n − m + 1)Γ (α + m + 1 ) m! m=0
(3.13)

whose generating function is

( xy ) ∑∞ (1 − y)−α −1exp ------ = ynL αn(x ) . y − 1 n=0
(3.14)

One may notice the useful relations

[ ] √ ------- -d-− l +-1-+ r un+1,l(r ) = − 4n + 4 un,l+1(r) , dr r [ d l + 1 ] √ ------- − ---− -----+ r un,l+1 (r ) = − 4n + 4 un+1,l(r) , [ dr r ] -d- l +-1 √ ----------- dr − r − r un,l (r ) = − 4n + 4l + 6un,l+1(r) , [ ] − -d-− l +-1-− r u (r ) = − √4n-+-4l-+-6u (r) , dr r n,l+1 n,l+1
(3.15)

and the useful integral

∫∞ Γ ( 3+ l + β) u20,l(r )rβdr = ---2-3----2--. Γ (2 + l) 0
(3.16)

We have here a first example where levels are described in two different bases, namely ∣nx,ny,nz⟩ or ∣n,l,m⟩, with the restriction N = nx + ny + nz = 2n + l.

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