Editing Quantum number (section)

==Spin-orbit coupled systems==
When one takes the [[spin–orbit interaction]] into consideration, the {{mvar|L}} and {{mvar|S}} operators no longer [[Commutativity|commute]] with the [[Hamiltonian (quantum mechanics)|Hamiltonian]], and the eigenstates of the system no longer have well-defined orbital angular momentum and spin. Thus another set of quantum numbers should be used. This set includes<ref>{{cite book|title=Molecular Quantum Mechanics Parts I and II: An Introduction to Quantum Chemistry |volume=1 |first=P. W. |last=Atkins |publisher=Oxford University Press |date=1977 |isbn=0-19-855129-0}}{{page needed|date=February 2019}}</ref><ref name="Atkins 1977">{{cite book|title=Molecular Quantum Mechanics Part III: An Introduction to Quantum Chemistry |volume=2 |first=P. W. |last=Atkins |publisher=Oxford University Press |date=1977}}{{ISBN missing}}{{page needed|date=February 2019}}</ref>

# The [[total angular momentum quantum number]]: <math display=block>
  j = |\ell \pm s|,
</math> which gives the total angular momentum through the relation <math display=block>
  J^2 = \hbar^2 j (j + 1).</math>
# The [[Azimuthal quantum number#Total angular momentum of an electron in the atom|projection of the total angular momentum]] along a specified axis:<math display=block>
  m_j = -j, -j + 1, -j + 2, \cdots, j - 2, j - 1, j
</math> analogous to the above and satisfies both <math display=block>
  m_j = m_\ell + m_s,
</math> and <math display=block>
  |m_\ell + m_s| \leq j.
</math>
# [[Parity (physics)|Parity]]<br>This is the [[eigenvalue]] under reflection: positive (+1) for states which came from even {{mvar|{{ell}}}} and negative (−1) for states which came from odd {{mvar|{{ell}}}}. The former is also known as '''even parity''' and the latter as '''odd parity''', and is given by<math display=block>
  P = (-1)^\ell .</math>

For example, consider the following 8 states, defined by their quantum numbers:
:{| style="border: none; border-spacing: 1em 0" class="wikitable"
! 
! {{mvar|n}}
! {{mvar|{{ell}}}}
! {{mvar|m<sub>{{ell}}</sub>}}
! {{mvar|m<sub>s</sub>}}
| rowspan=9  style="border:0px;" |
! {{math|''{{ell}}'' + ''s''}}
! {{math|''{{ell}}'' − ''s''}}
! {{math|''m<sub>{{ell}}</sub>'' + ''m<sub>s</sub>''}}
<!--                             N     L      ML      MS     L+S     L-S        ML+MS                          -->
|-align=right
!  (1) 
|  2 ||  1 ||   1  ||  +{{sfrac|1|2}} || {{sfrac|3|2}} || <s>{{sfrac|1|2}}</s>  ||  {{sfrac|3|2}}
|-align=right
!  (2) 
|  2 ||  1 ||   1  ||  −{{sfrac|1|2}} || {{sfrac|3|2}} || {{sfrac|1|2}}  ||  {{sfrac|1|2}}
|-align=right
!  (3) 
|  2 ||  1 ||   0  ||  +{{sfrac|1|2}} || {{sfrac|3|2}} || {{sfrac|1|2}}  ||  {{sfrac|1|2}}
|-align=right
!  (4) 
|  2 ||  1 ||   0  ||  −{{sfrac|1|2}} || {{sfrac|3|2}} || {{sfrac|1|2}}  ||  −{{sfrac|1|2}}
|-align=right
!  (5) 
|  2 ||  1 ||  −1  ||  +{{sfrac|1|2}} || {{sfrac|3|2}} || {{sfrac|1|2}}  ||  −{{sfrac|1|2}}
|-align=right
!  (6) 
|  2 ||  1 ||  −1  ||  −{{sfrac|1|2}} || {{sfrac|3|2}} || <s>{{sfrac|1|2}}</s>  ||  −{{sfrac|3|2}}
|-align=right
!  (7) 
|  2 ||  0 ||   0  ||  +{{sfrac|1|2}} || {{sfrac|1|2}} || −{{sfrac|1|2}} ||  {{sfrac|1|2}}
|-align=right
!  (8) 
|  2 ||  0 ||   0  ||  −{{sfrac|1|2}} || {{sfrac|1|2}} || −{{sfrac|1|2}} ||  −{{sfrac|1|2}}
|}

The [[quantum state]]s in the system can be described as linear combination of these 8 states. However, in the presence of [[spin–orbit interaction]], if one wants to describe the same system by 8 states that are [[eigenvector]]s of the [[Hamiltonian (quantum mechanics)|Hamiltonian]] (i.e. each represents a state that does not mix with others over time), we should consider the following 8 states:
:{| class="wikitable"
! {{math|''j''}} ||  {{math|1=''m<sub>j</sub>''}} || parity || 
|-
| {{sfrac|3|2}}|| align=right |   {{sfrac|3|2}}||  align=right | odd  || coming from state (1) above
|-
| {{sfrac|3|2}}|| align=right |   {{sfrac|1|2}}||  align=right | odd  || coming from states (2) and (3) above
|-
| {{sfrac|3|2}}|| align=right |  −{{sfrac|1|2}}||  align=right | odd  || coming from states (4) and (5) above
|-
| {{sfrac|3|2}}|| align=right |  −{{sfrac|3|2}}||  align=right | odd  || coming from state (6) above
|-
| {{sfrac|1|2}}|| align=right |   {{sfrac|1|2}}||  align=right | odd  || coming from states (2) and (3) above
|-
| {{sfrac|1|2}}|| align=right |  −{{sfrac|1|2}}||  align=right | odd  || coming from states (4) and (5) above
|-
| {{sfrac|1|2}}|| align=right |   {{sfrac|1|2}}||  align=right | even || coming from state (7) above
|-
| {{sfrac|1|2}}|| align=right |  −{{sfrac|1|2}}||  align=right | even || coming from state (8) above
|}