Similarly, we may define thematrix elementof an operator in terms of a pair of those orthonormal
basis states
Oij≡〈ui|O|uj〉.
With these definitions, Quantum Mechanics problems can be solved using the matrix representation
operators and states. (See section 18.1). An operator acting ona state is a matrix times a vector.
(Oψ) 1
(Oψ) 2(Oψ)i
=
O 11 O 12 ... O 1 j ...
O 21 O 22 ... O 2 j ...Oi 1 Oi 2 ... Oij ...
ψ 1
ψ 2ψj
The product of operators is the product of matrices. Operatorswhich don’t commute are represented
by matrices that don’t commute.
1.24 A Study ofℓ= 1 Operators and Eigenfunctions
The set of states with the same total angular momentum and the angular momentum operators
which act on them are often represented by vectors and matrices. For example the differentm
states forℓ= 1 will be represented by a 3 component vector and the angular momentum operators
(See section 18.2) represented by 3X3 matrices. There are both practical and theoretical reasons
why this set of states is separated from the states with different total angular momentum quantum
numbers. The states are often (nearly) degenerate and therefore should be treated as a group for
practical reasons. Also, a rotation of the coordinate axes will notchange the total angular momentum
quantum number so the rotation operator works within this group of states.
We write our3 component vectorsas follows.
ψ=
ψ+
ψ 0
ψ−
The matrices representing the angular momentum operators forℓ= 1 are as follows.
Lx=̄h
√
2
0 1 0
1 0 1
0 1 0
Ly=√ ̄h
2 i
0 1 0
−1 0 1
0 −1 0
Lz= ̄h
1 0 0
0 0 0
0 0 − 1
The same matrices also represent spin 1,s= 1, but of course would act on a different vector space.
The rotation operators (See section 18.6) (symmetry operators) are given by
Rz(θz) =eiθzLz/ ̄h Rx(θx) =eiθxLx/h ̄ Ry(θy) =eiθyLy/ ̄hfor the differential form or the matrix form of the operators. Forℓ= 1 these are 3X3 (unitary)
matrices. We use them when we need to redefine the direction of ourcoordinate axes. Rotations of
the angular momentum states are not the same as rotations of vectors in 3 space. The components
of the vectors represent different quantities and hence transform quite differently. The “vectors” we
are using for angular momentum actually should be called spinors whenwe refer to their properties
under rotations and Lorentz boosts.