There are two types of angular momentum: orbital and spin. If we define their operators as pseudo-vectors [itex]\vec{L}[/itex] and [itex]\vec{S}[/itex], then we can also define the total angular momentum operator [itex]\vec{J} = \vec{L}+\vec{S}[/itex].(adsbygoogle = window.adsbygoogle || []).push({});

Standard commutation relations will show that we can have simultaneous well defined values for [itex]J^2[/itex] and [itex]J_z[/itex] and etc. for [itex]\vec{L}[/itex] & [itex]\vec{S}[/itex]. i.e. We can have well defined total angular momentum and one component of it (usually z) for each type. The eigenvalues of these operators are then [itex]\hbar^2 j(j+1)[/itex] and [itex]\hbar m_j[/itex] respectively when we consider a simultaneous eigenstate [itex]|j,m_j>[/itex] of [itex]J^2[/itex] and [itex]J_z[/itex] only, and etc. for [itex]l[/itex] and [itex]s[/itex]

My question is really about how these different types combine.

Using these standard commutation relations:

[tex]

[J_i, J_j] = i \sum_k \epsilon_{ijk} J_k \hspace{10mm} [J_i, L_j] = i \sum_k \epsilon_{ijk} L_k \hspace{10mm} [J_i, S_j] = i \sum_k \epsilon_{ijk} S_k

[/tex]

It's very easy to show that we can have a simultaneous eigenstate [itex]|m_j,m_l,m_s>[/itex] of [itex]J_z, L_z, S_z[/itex] respectively, and thus the relation between the eigenvalues is

[tex]m_j=m_l+m_s[/tex]

We also have the commutation relations:

[tex]

[J^2, L^2] = 0 \hspace{10mm} [J^2, S^2] = 0 \hspace{10mm} [L^2, S^2] = 0

[/tex]

So we can have a simultaneous eigenstate [itex]|jls>[/itex] of [itex]J^2, L^2, S^2[/itex]. My question is then what is the relationship between [itex]j, l[/itex] and [itex]s[/itex]? So far as I can see, it is not straightforwards because:

[tex]

J^2 = L^2+S^2+2\vec{L}\cdot\vec{S} = L^2+S^2+2\sum_iL_iS_i \\

j(j+1) = l(l+1) + s(s+1) + \frac{2}{\hbar^2} \sum_i <jls|L_iS_i|jls>

[/tex]

Which presents a problem, since [itex][L_i, J^2], [S_i,J^2] \neq 0[/itex] so the state [itex]|jls>[/itex] cannot be an eigenstate of any of [itex]S_i, L_i[/itex] and so the relationship between the 3 numbers is not well defined.

How can we have a state in which [itex]j,l,s[/itex] are well defined and yet their relationship is not well defined?

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# I Adding types of angular momenta

Tags:

Have something to add?

Draft saved
Draft deleted

Loading...

Similar Threads - Adding types angular | Date |
---|---|

A Are there different "types" of renormalization | Feb 17, 2017 |

Adding expectation values to a CHSH animation | Jan 27, 2016 |

Mathematical conundrum when adding complex exponentials | May 22, 2015 |

Quantum imaging with undetected photons - adding of states | Jan 7, 2015 |

Exchange symmetry when adding angular momentum and in LS coupling? | Oct 24, 2014 |

**Physics Forums - The Fusion of Science and Community**