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Angular Momentum in Quantum Mechanics 

Angular momentum is an important concept in both classical and quantum mechanics. Its value is preserved throughout the motion of most relevant systems.The angular momentum L of a particle of mass m with linear momentum p with a radial vector r with respect to a center of rotation is defined as
where the display of a symbol in red denotes that it is a vector. The above vector product can be represented in the form of a determinant; i.e.,
 u_{x}  u_{y}  u_{z}  
 r_{x}  r_{y}  r_{z}  
 p_{x}  p_{y}  p_{x}  
where u_{x}, u_{y} and u_{z} are unit vectors in the directions of the x, y and z axes, respectively.
For further developments it greatly simplifies things if the components of the vector r are represented as (x, y, z). The components of L are then given by
The operators associated with these components are generated by replacing a momentum component
p_{w} with −ih(∂/∂w), where
w can be x, y or z and h is Planck's constant divided by 2π. Thus
Before proceding further it is necessary to cover the algebra of operators.
An operator is a mapping that takes a function as an input and gives a function as an output. Partial derivatives are operators and so is the operation of multiplying a function by a specific function.
The commutator of two operators P and Q is denoted as [ P, Q] and is given by
for any function ψ.
Some of the elementary properties of the commutator are
[Q, P] = −[P, Q]
and hence
[P, P] = 0
[P, Q+R] = [P, Q] + [P, R]
and
[P+Q, R] = [P, R] + [Q, R]
[P, QR] = [P, Q]R + Q[P, R]
Consider now [w, ∂/∂u] where w and u can be any of the coordinates x, y or z. Then
In quantum mechanics a momentum component p_{u} is replaced by −ih(∂/∂u). Thus
for any u and w in the set {x, y, z}
These operator relations may now be applied to evaluating the commutators of the components of angular momentum. Consider first [L_{x}, L_{y}].
With an application of the product property and the result that [P, P]=0 the term [yp_{z}, xp_{z}] evaluates to the zero operator. Likewise for the term [zp_{y}, zp_{x}]. Thus
With a cyclic permutation of the indices the commutators of the three components of angular momentum are found to be
Two operators, P and Q, commute; i.e., PQ=QP, if [P, Q]=0.
Let
Now consider
The last term, involving repeated applications of L_{z} reduces to the zero operator. Thus
Likewise [L_{x}, L²]=0 and [L_{y}, L²]=0.
Although it is not proven here, the condition [L_{z}, L²]=0 means that L_{z} and L²
the same eigenfunctions.
The eigenvalues of L² are of the form hl(l+1) for l an integer. The eigenvalues of
L_{z} are of the form hm where m is an integer −l≤m≤l.
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