Eigenvalues of Kinetic Moments Calculation

We now would like to calculate the kinetic moment eigenvalues. We know that one can measure only the squared kinetic moment L² and one of its components (we're talking about L_z) simultaneously. We will try to find the eigenvalues of both measured quantities and the eigenfunction Y. One needs to understand the Commutativity relations for the following derivation. First, we know that the squared kinetic moment eigenvalues h²λ have been given. This is not a restriction since λ can be any possible number taking h² in a way that λ would be dimensionless. The corresponding derivation is true for the z-component of of kinetic moment, the eigenvalue of which we will assign as mh.

Since the total kinetic moment should be greater than its component, we have the following:

l - m²  ≥  0               m²  ≤  λ

Similar to the harmonic oscillator, when factorizing L_x² + L_y² we can introduce two operators:

It follows from the well-known Commutativity relations for L², L_x, L_y, L_z:

As for the harmonic oscillator operators L₊ and L₋ are producing and annihilating operators, correspondingly. By demonstrating this, we can produce the following table:
 

L₊ increases and L₋ decreases the eigenvalue of  L_z on 1.

From the relation l-m² ≥ 0  and correspondingly  m² ≤ λ it follows that it should be maximum m_max and minimum m_min for which it should be the following:

Multiplication of  L₊Ym_max by L₋to the left:

where we have used the Commutativity relation L_xL_y- L_yL_x =  i h L_z. We can write the following using the corresponding eigenvalues:

The transformation gives finally (m_max + m_min)(m_max− m_min) + (m_max + m_min)  =  0 

 The second term (m_max− m_min + 1) ¹ 0 can never be 0 in this product because m_max  ≥  m_min ! And finally:
 

since L₊ increases the eigenvalue on 1 the difference m_max − m_min should be integer:  m_min= − m_max  and  m_max− m_min = integer number:
 

We have found out from this quantum mechanical solution that the lowest kinetic moment can have zero meaning (along z-axis) m = ½. This kinetic moment is the electron spin or intrinsic kinetic moment of electron. We have come to this solution by applying only operators and the commutativity relation.

If we assign the maximum value of m by l, m_max = l then we will obtain since λ = m_max(m_max+1) and l ≥ 0 the following relations:
 

We characterize the electron spin by s, i.e. for spin s = ¹/₂  m_max = +¹/₂ and m_min = -¹/₂ assign two possible alignments along z-axis (it's often said: m_s = +¹/₂, -¹/₂) (or: Spin up ­ and Spin down ¯). If one wants to find the coordinate image of the eigenfunction then he (or she) will have troubles. The eigenfunctions is likely to be represented by matrixes. We will talk about it a bit later. Now we would like to leave half-integer kinetic moment and talk a bit about integer kinetic moments for which there is the eigenfunction Y. We would also like to find the eigenfunctions of the following:

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