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Changes to this chapter were last made on 11 April 2005. |

A spin vector space and the Hilbert space of a Hamiltonian operator can be combined by writing a
*base ket*, using Dirac's *bra-ket* notation,
as a direct product [Shankar, pp.249; Sakurai, p.203]

from which we can define the operator

It will also prove useful to define the following *ladder operators*

For spin one-half systems
(

which are sometimes also expressed as

respectively.
The eigenvalue equations for

The ladder operators have the following effect on the spin eigenstates

and, in particular, for

The two-component spin states are also orthonormal

and the spin operators also have matrix representations, for example,

and the individual components are

The spin eigenstates,
α and β, can also be written as functions,

This is a form completely equivalent to the matrix representations [Eq. (2.18)], and one which could, for example, be represented by the following two simple functions

The orthonormality conditions, expressed before in Eq. (2.17) using Dirac's *bra-ket*
notation, take here the equivalent form

The following notation will also be used to denote the sum over spin states

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where

The effect spin angular momentum *ladder operators* have on spin eigenstates was shown in Eqs. (2.15) and (2.16).
In terms of the direct product states of these become

The spin-orbitals that correspond to the kets in Equations (2.29) and (2.30) are given by

respectively. This can be summarized more concisely as

or, as in Eq. (2.32), a spin-down electron can be labeled with a horizontal bar,
*atomic orbital* (AO) for many-electron atoms and
as a *molecular orbital* (MO) when applied to molecules.

The total wavefunction of a *one-particle system* can be written as a two-component spinor

which can also be expressed in terms of the spin eigenvectors as

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to emphasize that
*direct product* of the orbital-angular momentum operator
(**1**)
of a two-dimensional spin vector space. Similarly,
*direct product* of the identity operator (**1**) of a Hilbert space
spanned by position kets and the spin angular momentum operator
(*electronic* angular momentum. For our purposes, individual nuclei can be treated as spherically symmetric point particles, but for molecules the rotational angular momentum of the nuclear framework must also
be considered. In this case, the total angular momentum operator, excluding nuclear spin, is given by
*electronic* angular momentum, and, of course, for atoms

All the operator relationships described previously for spin angular momentum can be generalized to the total angular momentum.

from which we can define the operator

It will also prove useful to define the following *ladder operators*

The operator

where

and

The ladder operators have the following effect on the eigenstates

*decoupled representation* described by Eqs. (2.89)-(2.91).
An eigenket in this representation can be written as
*N*-electron atoms, where η represents all the other quantum numbers of the state.

The commutators

so
*coupled representation* described by Eqs. (2.92)-(2.95).
A base ket in this representation can be written as
*N*-electron atoms it will be written as

To show that the commutators

because all other relevant commutators are equal to zero, as was stated in Eq. (2.57).
Equation (2.50), with the substitution

and for the three coordinates, *i = x, y, z*, this leads to

Equations (2.61), (2.62), and (2.63) can be added to obtain

By exchanging

and adding the two commutators one finds

the eigenvalues of Eq. (2.67) are given by

On the other hand, we can use Eq. (2.47), with the substitutions

then

We can use Eq. (2.71) to show explicity what happens when

The spin operators for a *N*-particle system are
given by

The total angular momentum operators of a *N*-particle system are given by

where in the last equation [Eq. (2.80)]
*single* particle ladder operators:

The angular momentum commutation relations, shown in Eqs. (2.49) to (2.52),
are satisfied by both the *N*-particle operators,
*single* particle operators

**(2.95)**

The *Wigner symbols* and the *Racah coefficients*
[47][48] are defined in terms of the transformation coefficients,
and therefore can also be used to perform the
transformation between the uncoupled representation and the coupled representation.
In the case of the addition of two angular momenta
(i.e.,
*Wigner 3j-symbols* and the
*Racah V-coefficients*, respectively;
they are related to the Clebsch-Gordan coefficients as follows,

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