Total angular momentum quantum number

In <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a>, the total angular momentum quantum number parametrises the total <a href="/facts/Angular_momentum/7TuSVFxq">angular momentum</a> of a given <a href="/facts/Subatomic_particle/tMEzcTWQ">particle</a>, by combining its <a href="/facts/Angular_momentum_operator/d55sXn5u">orbital angular momentum</a> and its intrinsic angular momentum (i.e., its <a href="/facts/Spin_(physics)/4Jvp7e8P">spin</a>).
If s is the particle's spin angular momentum and ℓ its orbital angular momentum vector, the total angular momentum j is

j
        
        =
        
          s
        
        +
        
          ℓ
        
         
        .
      
    
    {\displaystyle \mathbf {j} =\mathbf {s} +{\boldsymbol {\ell }}~.}

The associated quantum number is the main total angular momentum quantum number j. It can take the following range of values, jumping only in integer steps:

|
        ℓ
        −
        s
        |
        ≤
        j
        ≤
        ℓ
        +
        s
      
    
    {\displaystyle \vert \ell -s\vert \leq j\leq \ell +s}

where ℓ is the <a href="/facts/Azimuthal_quantum_number/vHSIzbFC">azimuthal quantum number</a> (parameterizing the orbital angular momentum) and s is the <a href="/facts/Spin_quantum_number/5Nh2H8gr">spin quantum number</a> (parameterizing the spin).
The relation between the total angular momentum vector j and the total angular momentum quantum number j is given by the usual relation (see <a href="/facts/Angular_momentum_quantum_number/vHSIzbFC">angular momentum quantum number</a>)

‖
        
          j
        
        ‖
        =
        
          
            j
            
            (
            j
            +
            1
            )
          
        
        
        ℏ
      
    
    {\displaystyle \Vert \mathbf {j} \Vert ={\sqrt {j\,(j+1)}}\,\hbar }

The vector's z-projection is given by

j
          
            z
          
        
        =
        
          m
          
            j
          
        
        
        ℏ
      
    
    {\displaystyle j_{z}=m_{j}\,\hbar }

where mj is the secondary total angular momentum quantum number, and the 
 
 
 
 ℏ
 
 
 {\displaystyle \hbar }
 
 is the <a href="/facts/Reduced_Planck_constant/qGHAe3qq">reduced Planck constant</a>. It ranges from −j to +j in steps of one. This generates 2j + 1 different values of mj.
The total angular momentum corresponds to the <a href="/facts/Casimir_invariant/rG71qcFb">Casimir invariant</a> of the <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebra</a> <a href="/facts/Infinitesimal_rotation/c2K2Ftl8">so(3)</a> of the three-dimensional <a href="/facts/Rotation_group_SO(3)/yXbKPzhT">rotation group</a>.

Total angular momentum quantum number open-in-new

Total angular momentum quantum number