38
2 One Magnetic Center
p x =
1
√
2
−Y 1,1 + Y 1,−1
p y =
+i
√
2
Y 1,1 + Y 1,−1
p z = Y 1,0
(2.7)
The first two functions are no longer eigenfunctions of ˆ l z , because linear combinations
are made of functions with different m quantum number. The action of the ˆ l x,y,z on
these functions is worked out for one case and then summarized in Eq. 2.9.
ˆ l z p x =
1
√
2
ˆ l z
−Y 1,1 + Y 1,−1
=
1
√
2
−Y 1,1 − Y 1,−1
= ip y
(2.8)
ψ
ˆ l x |ψ
ˆ l y |ψ
ˆ l z |ψ
p x
0
−ip z
ip y
p y
ip z
0
−ip x
p z
−ip y
ip x
0
(2.9)
The matrix elements of ˆ l = ˆ l x + ˆ l y + ˆ l z in the basis of the p x,y,z functions are easily
obtained
ˆ l
p x
p y
p z
p x
0
−ii
p y
i
0
−i
p z
−ii
0
(2.10)
and the subsequent diagonalization results in two non-zero expectation values of
ˆ l. In analogy to the spin operators the N-electron angular moment operators
ˆ
L are obtained by summing over the corresponding one-electron operators. The
N-electron eigenfunctions are products of the one-electron spherical harmonics Y l,m .
Notice that the orbital moment of the p 5 electron configuration can be treated as if it
were a one-electron system because of the hole-electron analogy.
2.3 Confirm that p x | ˆ l z |p y =−−p y | ˆ l z |p x =−i.
The situation changes drastically when an external potential removes the degeneracy of the three functions. This can be caused by the crystal field exerted by the ions
in an extended lattice for solid state compounds or by the ligands in the coordination
sphere of an ion in a coordination complex. Although magnetic phenomena are more
common for systems with incomplete d-shells, we will continue with our example
concerning the p 1 configuration for simplicity. Imagine an external potential that
makes, for example, the state with one electron in the p x orbital lowest in energy
and places the other two orbitals at slightly higher energy. Then, the orbital angular
moment of the ground state is defined from the matrix element p x | ˆ l|p x only, which
is zero as can be seen in the matrix 2.10. In many physics textbooks, this effect is
2 One Magnetic Center
p x =
1
√
2
−Y 1,1 + Y 1,−1
p y =
+i
√
2
Y 1,1 + Y 1,−1
p z = Y 1,0
(2.7)
The first two functions are no longer eigenfunctions of ˆ l z , because linear combinations
are made of functions with different m quantum number. The action of the ˆ l x,y,z on
these functions is worked out for one case and then summarized in Eq. 2.9.
ˆ l z p x =
1
√
2
ˆ l z
−Y 1,1 + Y 1,−1
=
1
√
2
−Y 1,1 − Y 1,−1
= ip y
(2.8)
ψ
ˆ l x |ψ
ˆ l y |ψ
ˆ l z |ψ
p x
0
−ip z
ip y
p y
ip z
0
−ip x
p z
−ip y
ip x
0
(2.9)
The matrix elements of ˆ l = ˆ l x + ˆ l y + ˆ l z in the basis of the p x,y,z functions are easily
obtained
ˆ l
p x
p y
p z
p x
0
−ii
p y
i
0
−i
p z
−ii
0
(2.10)
and the subsequent diagonalization results in two non-zero expectation values of
ˆ l. In analogy to the spin operators the N-electron angular moment operators
ˆ
L are obtained by summing over the corresponding one-electron operators. The
N-electron eigenfunctions are products of the one-electron spherical harmonics Y l,m .
Notice that the orbital moment of the p 5 electron configuration can be treated as if it
were a one-electron system because of the hole-electron analogy.
2.3 Confirm that p x | ˆ l z |p y =−−p y | ˆ l z |p x =−i.
The situation changes drastically when an external potential removes the degeneracy of the three functions. This can be caused by the crystal field exerted by the ions
in an extended lattice for solid state compounds or by the ligands in the coordination
sphere of an ion in a coordination complex. Although magnetic phenomena are more
common for systems with incomplete d-shells, we will continue with our example
concerning the p 1 configuration for simplicity. Imagine an external potential that
makes, for example, the state with one electron in the p x orbital lowest in energy
and places the other two orbitals at slightly higher energy. Then, the orbital angular
moment of the ground state is defined from the matrix element p x | ˆ l|p x only, which
is zero as can be seen in the matrix 2.10. In many physics textbooks, this effect is