Definitions

# 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, ms, which can be either +1/2 or −1/2:

$chi\left(x, y, z, m_s\right)$
We can also write it more compactly as a function of a position vector $vec r=\left(x,y,z\right)$ and the quantum number ms:
$chi\left(vec r, m_s\right)$.
For a general particle with spin s, ms 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:

$sum_\left\{m_s=-s\right\}^\left\{s\right\}int_\left\{infty\right\}d^3vec r;|chi\left(vec r,m_s\right)|^2=1.$

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 $m_s$:

$P\left(V,m_s\right)=int_\left\{V\right\}d^3vec r;|chi\left(vec r,m_s\right)|^2.$

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