It is chosen to have maximal symmetry.^7
9.6.1 The orthogonal groupSO(N)
In the case ofN= 3, we simply have the isotropic 3-dimensional harmonic oscillator,
N= 3 H=
1
2 m
p^2 +
1
2
mω^2 x^2 (9.27)
It is of course well-known that this Hamiltonian is invariant under simultaneous rotations of
xandp, given byδx=~ω×xandδp=~ω×p. The associated conserved charges are the
three components of angular momentumL=x×p.
For general dimensionN, the harmonic oscillator is invariant under orthogonal transfor-
mations inN-dimensional space. A convenient way to see this is by organizing thedegrees
of freedom in column matrices,
Q=
q 1
q 2
·
qN
P=
p 1
p 2
·
pN
(9.28)
The Hamiltonian then takes the form,
H=
1
2 m
PtP+
1
2
mω^2 QtQ (9.29)
Orthogonal transformations inN-dim space are defined aslinear transformations which leave
theN-dimensional Euclidean normQtQinvariant. OnQ, we have,
Q→Q′=MQ such that (Q′)tQ′=QtQ (9.30)
for a realN×N matrixM, and for allQ. This requires thatMtM=I. The set of all real
matrices satisfyingMtM=Iforms a group under matrix multiplication, referred to as the
orthogonal groupO(N). Indeed, if we haveM 1 tM 1 =M 2 tM 2 =Ithen (M 1 M 2 )t(M 1 M 2 ) =
M 2 tM 1 tM 1 M 2 =I. The identity element is the unit matrixIand the inverse is the transpose
M−^1 =Mt. Clearly, if we letQ→MQandP →MP, then both the Hamiltonian and
the canonical commutation relations are invariant, and thus all orthogonal transformations,
which belong toO(N) are symmetries of the harmonic oscillator quantum system.
Rotations inN dimensional space are all orthogonal transformations, but all orthogonal
transformations are not rotations. A rotation is continuously connected to the identity or-
thogonal transformation, since one may think of a rotation as obtained by making a sequence
(^7) A more general form of theN-dimensional harmonic oscillator would be obtained by replacing the
potential term by^12 m
∑N
i,j=1ωij^2 qiqjfor some positive real matrixω^2.