Total Spin States of a Two Particle System Measured at the Ground State | Quantum Science Philippines
Quantum Science Philippines

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                                                          f1

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

f2

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,

f3

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 s1 = 2, the multiplicity is 2s + 1 = 2(2) + 1 = 5 states.

For particle 2 where s2 = 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

f4

Given the same values (s1 = 2 and s2 = 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 |s1 m1 > and |s2 m2 >.

f5

f7

Suppose that the spin’s z-components for the first and second particle are given to be 0 and 1/2 respectively, then  m1 = 0 and m2 = 1/2. The possible total spin and the corresponding probabilities can be found with the help of the Clebsch − Gordan coefficients’ table below.

clebsh_4colored_not stretched

f9

Looking at the table under the category 2 x 3/2 and with m1 = 0 and m2 = 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

f10

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.

f11

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.

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