The Clebsch-Gordan coefficients are mathematical quantities used in quantum mechanics to determine the composition of angular momentum states in composite quantum systems. They play a crucial role in the addition of angular momenta, which is fundamental in understanding the behaviour of systems such as atoms, nuclei, and particles with spin.

When two angular momenta are combined to form a total angular momentum, the Clebsch-Gordan coefficients describe the probability amplitudes of obtaining a particular total angular momentum state from the given individual angular momentum states.

We can express each of the new basis vectors as the linear combination of the old set of basis vectors as

The coefficients < j_1 m_1, j_2 m_2 l j_1 j_2 j m> are known as Clebsch Gordan coefficient or Wegner coefficient or vector addition coefficients.

Also each of the old set of basis vectors can be expressed as the linear combination of new basis vector.

Operating both sides Jz

Comparing the coefficient on both sides we get

m = m_1 + m_2

Now we calculate C- G coefficient considering the simple case

j_1 = 1/2 , j_2 = 1/2

m_1 = -1/2 , + 1/2

m_2 = -1/2 , + 1/2

Total number of old basis vectors (2j+1) x (2 j +1) = ( 2 x 1/2 + 1) (2 x 1/2 + 1) = 4

New set of basis vectors l j_1 j_2 j m >

First we consider

j= j_1 + j_2 = 1/2 + 1/2 = 1

and m= m_1 + m_2

Maximum value of m = 1/2 +1/2 = 1

The state

= < 1/2 + 1/2 , 1/2 + 1/2 ∣ 1/2 1/2 1, 1> ∣ 1/2 + 1/2 , 1/2 + 1/2 >+

< 1/2 + 1/2 , 1/2 - 1/2 ∣ 1/2 1/2 1, 1> ∣ 1/2 + 1/2 , 1/2 - 1/2 > +

< 1/2 - 1/2 , 1/2 + 1/2 ∣ 1/2 1/2 1, 1> ∣ 1/2 - 1/2 , 1/2 + 1/2 > +

< 1/2 - 1/2 , 1/2 - 1/2 ∣ 1/2 1/2 1, 1> ∣ 1/2 - 1/2 , 1/2 - 1/2 >

Note that the vectors l 1/2 1/2 , 1, 1> and l 1/2 1/2 , 1/2 , 1/2 > are same vector

So we conclude that , C- G coefficient < 1/2 + 1/2 , 1/2 + 1/2 l 1/2 1/2 1 1 > = 1

CG coefficients are highly significant as they provide fundamental insights into the behaviour of particles at the quantum level.

This note is a part of the Physics Repository.

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