So far, we have not explicitly considered the spin of the electrons.
For the next type of approximation we will consider, the
so-called *valene bond approximation*, we will need to consider
spin explicitly. As we have already seen in the case of the hydrogen
atom, the wave functions of an electron depend on the coordinates
or
*and* on the -component
of spin spin (the total spin is fixed).
While the coordinates can be anything, the spin can only
take on two values and . Recall that these
lead to the two values of the spin quantum number .
The value is what we call the ``spin-up'' state
and is what we call ``spin-down''.

We now define the spin wave functions. The spin-up wave function
is denoted

Since can take on only two values , the spin wave function only have two values:

The meaning of this wave function is that when the electron is in the spin-up state, the probability that a measurement of will yield the value is and the probability that is value will be is 0. Similarly, the spin-down wave function is

where

so that the probability that a measurement of yields the value is 1 and that its value is is 0. Note that the spin wave functions are normalized, meaning that they satisfy

and they are orthogonal, meaning that they satisfy

Now, for a hydrogen atom, there are four quantum numbers,
, and the wave function depends on
four coordinates,
. While
are continuous, takes on
only two values. The wave function can be expressed
as a simple product

As a shorthand notation, we can generally represent the complete set of spatial and spin coordinates with a vector . The spatial coordinates can be or or any other set of spatial coordinates useful for a given problem. For the hydrogen atom, using this notation, we would write