## Total Spin States of a Two Particle System Measured at the Ground State

Christine Marie Ceblano, Liza Marie Dangkulos, Lutchie Dyan Mendoza and Normie Jean Sajor

Spin in particle physics and quantum mechanics is a fundamental characteristic property of elementary particles, composite particles and atomic nuclei. All elementary particles of a given kind have the same spin quantum number which is an important part of a particle’s quantum state.

The principal spin quantum number* s* is given by

where* n = 0, 1, 2, 3*, … can be any non- negative integer.

Quantum mechanical spin also contains information about direction. Quantum mechanics states that the component of angular momentum measured along any direction (along * z *– axis) is

where *s* is the principal spin quantum number.

One can see that there are *2s + 1* possible values where *2s + 1* is the multiplicity of the spin system. This multiplicity corresponds to quantum states in which the spin is pointing in the *+z* or *−z* directions.

For an instance, there is a* 2*-particle system at the ground state. The first particle has spin* 2* and the second has a spin* 3/2* . That is,

In order to find the total spin states which are available to this system, the multiplicity of the spin system must be obtained.

For particle *1* where *s _{1}*

*= 2*, the multiplicity is

*2s + 1 = 2(2) + 1 = 5*states.

For particle* 2* where *s _{2}*

*= 3 /2*, the multiplicity is

*2s + 1 = 2(3/2) + 1 = 4*states.

Thus, the *5* states of the first particle are *−2,−1, 0, 1, 2* and the* 4* states of the second particle are *−3/2,−1/2 , 1/2 , 3/2 .*

To have the total spin states, combine all the obtained states above by pairing each spin’s* z*– component.

Summarizing all the states, we have

Given the same values (*s _{1}*

*= 2*and

*s*

_{2}*= 3/2*) and the particles spin’s

*z*-component, one can get the possible total spin values as well as the probabilities of getting such values. The particle state

*|s m >*with total spin’s and

*z*-component

*m*will be some linear combination of the composite states

*|s*

_{1}*m*

_{1}*>*and

*|s*

_{2}m_{2}*>.*

Suppose that the spin’s* z*-components for the first and second particle are given to be *0* and *1/2* respectively, then * m _{1}*

*= 0*and

*m*

_{2}*= 1/2*. The possible total spin and the corresponding probabilities can be found with the help of the

*Clebsch − Gordan*coefficients’ table below.

Looking at the table under the category *2 x 3/2* and with *m _{1}*

*= 0*and

*m*

_{2}*= 1/2*(the highlighted row), one could get the total spin of

*7/2*(with probability of

*18/35*),

*5/2*(with probability of

*3/35*),

*3/2*(with probability of

*1/5*) and

*1/2*(with probability of

*1/5*).

Using the form in equation *3*, we have

As always, the sum of probabilities is *1* (the sum of the squares of the row highlighted in the *Clebsch − Gordan* table is *1*), i.e.

As a summary, there are *20* possible total spin states for this 2-particle system at the ground state. Then by using *Clebsch−Gordan* coefficient table, the possible values of the total spin and probabilities of getting such values are obtained.