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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.

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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:

χ(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:

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

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 $m s$ :

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