What is quantum momentum: Definition and 1 Discussions

In quantum mechanics, the total angular momentum quantum number parametrises the total angular momentum of a given particle, by combining its orbital angular momentum and its intrinsic angular momentum (i.e., its spin).
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 azimuthal quantum number (parameterizing the orbital angular momentum) and s is the spin quantum number (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 angular momentum quantum number)




‖

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 reduced Planck constant. 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 Casimir invariant of the Lie algebra so(3) of the three-dimensional rotation group.

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