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.
