170
7 Spherical Symmetry and Spins
Typically, the first invariant multipole corresponds to the charge distribution of
the Platonic solids of the given symmetry: hence, the L = 4 term describes the
octahedron, and its dual the cube, while the next invariant, belonging to the L = 6
multipoles, represents the dominant term for the charge distribution in the 12-vertex
Archimedean solid, known as the cuboctahedron, with a ligand in every edge of
the octahedron. Entirely similar relations exist for the icosahedral group. The first
icosahedral invariant, with L = 6, is the dominant term for the icosahedron and its
dual dodecahedron. The next term, with L = 10, is the leading multipole for the
truncated dodecahedron, alias the “buckyball” [5, 6].
7.3 Interactions of a Two-Component Spinor
The standard vector space in spherical symmetry has three components. We now
explore the possibility of a function space with only two components. Such a space
will correspond to a spinor. The strategy is to set up a general Hamiltonian matrix in
a space with two components and verify that it has spherical symmetry. In order to
describe the general interaction Hamiltonian in a space of only two components, a
2 × 2 Hermitian matrix, H, is required. We can take this matrix to be traceless since
the trace will not introduce an interaction inside the spin space but will simply shift
the barycentre of the two levels with respect to an external reference. In its most
general form, such a traceless Hermitian matrix will thus contain three independent
real parameters, which we shall label as x,y,z:
H =
zx − iy
x + iy
−z
(7.21)
We have purposely chosen the Cartesian labels for the three independent parameters
since, later on, a connection will be established between the 2D complex interaction
space and the real 3D vector space. An example of such a Hamiltonian is the Zeeman
interaction of an isolated spin in a magnetic field:
H = 2.0023
μ B
B·S
(7.22)
Here, S is the operator for the spin momentum, expressed in units of .Thex,y,z
parameters in this case are proportional to the magnitude of the magnetic field, B,
in the three Cartesian directions.
However, for now, we shall not yet make use of this spatial connotation and
just continue to consider x,y,z as the general variables of the Hamiltonian matrix.
The two components of the space will be denoted as the spin functions |α, |β,
which together form a spinor. The corresponding interaction operator can then be
expressed as
H = z
|αα|−|ββ|
+ x
|αβ|+|βα|
− iy
|αβ|−|βα|
(7.23)
7 Spherical Symmetry and Spins
Typically, the first invariant multipole corresponds to the charge distribution of
the Platonic solids of the given symmetry: hence, the L = 4 term describes the
octahedron, and its dual the cube, while the next invariant, belonging to the L = 6
multipoles, represents the dominant term for the charge distribution in the 12-vertex
Archimedean solid, known as the cuboctahedron, with a ligand in every edge of
the octahedron. Entirely similar relations exist for the icosahedral group. The first
icosahedral invariant, with L = 6, is the dominant term for the icosahedron and its
dual dodecahedron. The next term, with L = 10, is the leading multipole for the
truncated dodecahedron, alias the “buckyball” [5, 6].
7.3 Interactions of a Two-Component Spinor
The standard vector space in spherical symmetry has three components. We now
explore the possibility of a function space with only two components. Such a space
will correspond to a spinor. The strategy is to set up a general Hamiltonian matrix in
a space with two components and verify that it has spherical symmetry. In order to
describe the general interaction Hamiltonian in a space of only two components, a
2 × 2 Hermitian matrix, H, is required. We can take this matrix to be traceless since
the trace will not introduce an interaction inside the spin space but will simply shift
the barycentre of the two levels with respect to an external reference. In its most
general form, such a traceless Hermitian matrix will thus contain three independent
real parameters, which we shall label as x,y,z:
H =
zx − iy
x + iy
−z
(7.21)
We have purposely chosen the Cartesian labels for the three independent parameters
since, later on, a connection will be established between the 2D complex interaction
space and the real 3D vector space. An example of such a Hamiltonian is the Zeeman
interaction of an isolated spin in a magnetic field:
H = 2.0023
μ B
B·S
(7.22)
Here, S is the operator for the spin momentum, expressed in units of .Thex,y,z
parameters in this case are proportional to the magnitude of the magnetic field, B,
in the three Cartesian directions.
However, for now, we shall not yet make use of this spatial connotation and
just continue to consider x,y,z as the general variables of the Hamiltonian matrix.
The two components of the space will be denoted as the spin functions |α, |β,
which together form a spinor. The corresponding interaction operator can then be
expressed as
H = z
|αα|−|ββ|
+ x
|αβ|+|βα|
− iy
|αβ|−|βα|
(7.23)