QuantumPhysics.dvi

so thatUindeed correctly generates a rotation by̟. The rotation generatorsLij have the

following commutation relations,

[Lij,Lkl] =i ̄h(δikLjl−δjkLil−δilLjk+δjlLik) (9.38)

Antisymmetry of̟ij andLij under i ↔ j guarantees that the number of independent

rotations isN(N−1)/2.

9.6.2 The unitary groupsU(N) andSU(N)

TheN-dimensional harmonic oscillator actually has a symmetry larger thanSO(N). This

is not so easy to see directly inqi,picoordinates, but becomes apparent when we recast the

problem in terms of raising and lowering operators, defined by

ai =

1

√

2 m ̄hω

(

+ipi+mωqi

)

a†i =

1

√

2 m ̄hω

(

−ipi+mωqi

)

(9.39)

fori= 1,···,N. The Hamiltonian is then

H=

∑N

i=1

hω ̄

(

a†iai+

1

2

)

=

1

2

̄hωN+ ̄hω

∑N

i=1

a†iai (9.40)

while the canonical commutation relations are [ai,a†j] =δij. Again, it is convenient to arrange

theaianda†iinto matrices,

A=









a 1

a 2

·

aN









A†= (a† 1 a† 2 · a†N) (9.41)

The Hamiltonian then takes the form

H=

1

2

̄hωN+ ̄hωA†A (9.42)

Since the observablesaiare not self-adjoint, they are inherently complex variables, and linear

transformations between them should be allowed to take on complexvalues. Thus, we shall

now consider making linear transformations onAbut with complex coefficients,

A→A′=MA A†→(A′)†=A†M† (9.43)