Total angular momentum

The term total angular momentum describes the sum of several angular momentum . In classical mechanics, it generally refers to several bodies.

The term is also used when orbital angular momentum and intrinsic angular momentum of a body are added, such as the angular momentum of the earth's rotation around the sun and the earth's rotation around its own axis. The intrinsic angular momentum can be a classic angular momentum or the quantum mechanical spin .

In quantum mechanics , the total angular momentum of a particle is in particular the sum of orbital angular momentum and spin . An important example of this is the electron in the hydrogen atom , in which the spin and orbital angular momentum are linked by the spin-orbit coupling . As a quantum mechanical operator, the total angular momentum has the total angular momentum quantum number as a quantum number.

Addition of spin and orbital angular momentum

The total angular momentum is the sum of orbital angular momentum and spin . The formulas for adding two angular momentum operators are explained in more detail under Angular momentum operator . The sum satisfies corresponding commutation relations ${\displaystyle {\hat {\vec {J}}}=({\hat {J}}_{x},{\hat {J}}_{y},{\hat {J}}_{z})}$ ${\displaystyle {\hat {\vec {L}}}}$ ${\displaystyle {\hat {\vec {S}}}}$

${\displaystyle [{\hat {J}}_{a},{\hat {J}}_{b}]=\mathrm {i} \hbar \varepsilon _{abc}{\hat {J}}_{c}}$,

from which

${\displaystyle [{\hat {J}}_{a},{\hat {\vec {J}}}^{2}]=0}$

follows.

As for the spin operator and the angular momentum operator , consider the two quantum numbers and , which are given by ${\displaystyle j}$ ${\displaystyle m_{j}}$

${\displaystyle {\hat {\vec {J}}}^{2}\psi =\hbar ^{2}j(j+1)\psi }$

${\displaystyle {\hat {J}}_{z}\psi =\hbar m_{j}\psi }$

given are.

For a particle with spin 1/2 the spin quantum numbers are limited to and , but the orbital angular momentum quantum numbers are and . For the total angular momentum quantum numbers we get: ${\displaystyle s={\tfrac {1}{2}}}$ ${\displaystyle m_{s}=\pm {\tfrac {1}{2}}}$ ${\displaystyle l\in \mathbb {N} _{0}}$ ${\displaystyle m_{l}=-l,\dots ,l}$

${\displaystyle j=\left|l\pm {\frac {1}{2}}\right|}$

${\displaystyle m_{j}=-j,\dots ,j}$.

The two spin states are obtained for the ground state. There are four states for, of which the states with linear combinations of and , or of and are. The coefficients in these linear combinations are called Clebsch-Gordan coefficients . ${\displaystyle l=0}$ ${\displaystyle l=1}$ ${\displaystyle m_{j}=\pm {\tfrac {1}{2}}}$ ${\displaystyle \left|m_{l}=0,m_{s}=+{\tfrac {1}{2}}\right\rangle }$ ${\displaystyle \left|m_{l}=1,m_{s}=-{\tfrac {1}{2}}\right\rangle }$ ${\displaystyle \left|m_{l}=0,m_{s}=-{\tfrac {1}{2}}\right\rangle }$ ${\displaystyle \left|m_{l}=-1,m_{s}=+{\tfrac {1}{2}}\right\rangle }$

Individual evidence

1. ^ Daniel, Herbert: Physics - Atoms, solids, nuclei, particles . Walter de Gruyter, 1999, ISBN 978-3-11-080472-0 ( limited preview in the Google book search).
2. ^ Paul Allen Tipler, Ralph A. Llewellyn: Modern Physics . Oldenbourg Verlag, 2010, ISBN 978-3-486-58275-8 ( limited preview in Google book search).