Spin-orbital

In quantum mechanics, a spin-orbital is a one-particle wavefunction taking both the position and spin angular momentum of a particle as its parameters.

The spinorbital of a single electron, for example, is a complex-valued function of four real variables: the three scalars used to define its position, and a fourth scalar, "m_s", which can be either +1/2 or −1/2::$chi(x,\; y,\; z,\; m\_s)$We can also write it more compactly as a function of a position vector $vec\; r=(x,y,z)$ and the quantum number "m_s"::$chi(vec\; r,\; m\_s)$.For a general particle with spin "s", "m_s" can take values between −"s" to "s" in integer steps. The electron has "s"=1/2.

A spinorbital is usually "normalized," such that the probability of finding the particle anywhere in space with any spin is equal to 1:

:$sum\_\{m\_s=-s\}^\{s\}int\_\{infty\}d^3vec\; r;|chi(vec\; r,m\_s)|^2=1.$

From a normalized spinorbital, one can calculate the probability that the particle is in an arbitrary volume of space "V" and has an arbitrary spin "$m\_s$":

:$P(V,m\_s)=int\_\{V\}d^3vec\; r;|chi(vec\; r,m\_s)|^2.$