Own spinor

In quantum mechanics, intrinsic spinors are the basis vectors that describe the spin state of a particle. For a single spin 1/2 particle they can be viewed as the eigenvectors of the Pauli matrices . In terms of physics, spinors do not count as vectors, but as spinors . They form a complete orthonormal system .

background

The spin of a particle obeys the angular momentum algebra , so not all components of the spin can be measured at the same time . It is therefore not possible to specify all three components of the spin in all three spatial dimensions at the same time, but only at the same time its absolute value and its projection onto a coordinate axis . Due to the quantization of the spin in units of the (half) reduced Planckian quantum of action , various such settings exist . ${\displaystyle {\vec {S}}}$ ${\displaystyle S_{x},S_{y},S_{z}}$ ${\displaystyle |{\vec {S}}|=s(s+1)}$ ${\displaystyle \hbar }$ ${\displaystyle 2s+1}$

Spinors for particles with spin ½

For a particle with the spin 1/2, there are only two possible eigenstates for the spin: parallel or antiparallel to a coordinate axis. In Bra-Ket notation as state vectors , the spin is therefore noted as a two-component spinor. The parallel spin is referred to as , the anti-parallel as . Conventionally, the coordinate system is chosen in such a way that the projection onto the axis represents the marked direction. The following applies and for the other spatial directions: ${\displaystyle |0\rangle }$ ${\displaystyle |1\rangle }$ ${\displaystyle z}$ ${\displaystyle \chi _{\pm }^{z}=\chi _{\pm }}$

${\displaystyle S_{z}}$	${\displaystyle S_{x}}$	${\displaystyle S_{y}}$
${\displaystyle \chi _{+}^{z}=|0\rangle ={\begin{bmatrix}1\\0\end{bmatrix}}}$	${\displaystyle \chi _{+}^{x}={\frac {1}{\sqrt {2}}}\left(|0\rangle +|1\rangle \right)={\frac {1}{\sqrt {2}}}{\begin{bmatrix}1\\1\end{bmatrix}}}$	${\displaystyle \chi _{+}^{y}={\frac {1}{\sqrt {2}}}\left(|0\rangle +\mathrm {i} |1\rangle \right)={\frac {1}{\sqrt {2}}}{\begin{bmatrix}1\\\mathrm {i} \end{bmatrix}}}$
${\displaystyle \chi _{-}^{z}=|1\rangle ={\begin{bmatrix}0\\1\end{bmatrix}}}$	${\displaystyle \chi _{-}^{x}={\frac {1}{\sqrt {2}}}\left(|0\rangle -|1\rangle \right)={\frac {1}{\sqrt {2}}}{\begin{bmatrix}1\\-1\end{bmatrix}}}$	${\displaystyle \chi _{-}^{y}={\frac {1}{\sqrt {2}}}\left(|0\rangle -\mathrm {i} |1\rangle \right)={\frac {1}{\sqrt {2}}}{\begin{bmatrix}1\\-\mathrm {i} \end{bmatrix}}}$

Spherical coordinates ( ): radius , polar angle , and azimuthal angle ${\displaystyle r,\theta ,\varphi }$ ${\displaystyle r}$ ${\displaystyle \theta }$ ${\displaystyle \varphi }$

These results are special cases of the intrinsic spinors for the parameters specified by and in spherical coordinates - these intrinsic spinors are: ${\displaystyle \theta }$ ${\displaystyle \varphi }$

${\displaystyle \chi _{+}={\begin{bmatrix}\cos(\theta /2)\\e^{i\varphi }\sin(\theta /2)\\\end{bmatrix}}}$

${\displaystyle \chi _{-}={\begin{bmatrix}-e^{-i\varphi }\sin(\theta /2)\\\cos(\theta /2)\\\end{bmatrix}}}$

Spinors for particles with higher spin

Spinors for particles with spin can be represented as dyadic products of the basis spinors for particles with spin ½. ${\displaystyle \textstyle s>{\frac {1}{2}}}$

Individual evidence

• Griffiths, David J. (2005) Introduction to Quantum Mechanics (2nd ed.). Upper Saddle River, NJ: Pearson Prentice Hall. ISBN 0-13-111892-7 .