Anomalous Zeeman effect
When an atom is placed in a magnetic field, each of its fine structure lines further
splits into a series of equidistant lines with a spacing proportional to the magnetic
field strength.
Theoretically, this is explained by recogniz
Expansion of
about .
therefore, called the generator of the translation group.
By analogy and by similar reasoning, it can be shown that
rotations of vectors in the Hilbert space via the operator:
is,
is the generator of
which produces rotations of a vect
Representing states in the full Hilbert space
Given a representation of the states that span the spin Hilbert space, we now need to
consider the problem of representing the the states the span the full Hilbert space:
We will work with the following comple
Rotations in spin space
Given two types of angular momentum, orbital and spin, it is possible to define
a total angular momentum
plays a special role in quantum mechanics. Not only is it often a constant of the
motion even when is spin-dependent, but it i
Spin Operators
In the case of the spin operators, the commutation relations can be written
compactly in terms of a vector cross product as
which can also be written as
Here,
The
is called the Levi-Civita tensor and is defined by
are the structure constant
Experimental evidence for electron spin
Up to now, we have considered quantum particles to have three degrees of freedom,
, , and . This, then, leads to three quantum numbers that characterize the states. For
example, in a central potential, the three qua
M value orbitals
commutes with the total Hamiltonian for H ,i.e.,
Thus, the eigenvalue
can be used to characterize molecular orbitals. The following
nomenclature has been adopted for designating molecular orbitals from the
value:
orbital.
orbital.
orbital
Explicit description of spin-1/2
The Hilbert space of a spin-1/2 particle is two-dimensional. Moreover,
eigenvalue, can take on two values
and
. Therefore,
, the
should be
represented by a 2 2 matrix. If we choose to work in a basis in which
then as a mat
Change of spin state
It is also possible to change from one spin state to another by means of
the raising and lowering operators, which are defined by
which satisfy the following commutation rules:
and additionally:
where
be
denotes the anticommutator bet
5:
From the figure it is clear that
anti-bonding.
must be a bonding state while
must be
Finally, we can construct molecular orbitals from
or
orbitals. Since these will
be similar in shape and have the same energy, we only need to consider one case. Let
u
Uhlenbeck and Goudsmit proposal
In 1925, Uhlenbeck and Goudsmit postulated the existence of a new intrinsic property
of particles that behaved like an angular momentum as a means of explaining these
results. This intrinsic property was later termed spin b