## ABSTRACT

Thisisanimportantchapter,butnotbecausethethreedimensionalharmonic oscillatorisaparticularlyimportantphysicalsystem.Itis,however,abeautifulillustrationofhowSU(N)symmetriesariseinquantummechanics.

TheHamiltonian(hereaselsewherewehavesetn=1)is

where

H1-2mw2-2 =2mp+-2-r

=w(aLak+3/2)

fork=1to3.Theakanda!areloweringandraisingoperatorssatisfying thecommutationrelationsofannihilationandcreationoperatorsforbosons,

[aklal]=Ok£ [ak,a£]=[ak,al]=0

(14.3)

If the ground state is I 0), satisfying akiO) = 0 (14.4)

then the energy eigenstates are

(14.5)

with energy w(n+3/2) (14.6)

(n + l)(n + 2) 2

(14.7)

Just the dimension of the (n, 0) representation of SU(3). This suggests that the model has an SU(3) symmetry.