# I Spin states for two identical 1/2 particles - Confused :s

1. Mar 10, 2016

### Sara Kennedy

Im having trouble with my thought process for spin states of a system of two electrons

Using Total Spin 'S' and Total spin mag quantum numbers 'MS' as state ket |S MS>

My textbook states....

" Three Symmetric Spin states
Triplet spin stats for twin identical spin -1/2 particles
• | Up Up> = |S MS> = |1, 1> "
My thought process, two half spin particles are involved so total spin S is 1/2 + 1/2=1 and two spin ups giving MS as +1/2 + 1/2 =1
• " 1/√2 ( |Up Down> + |Down Up> ) = |1,0> "
I see this, again two spin particles are involved so its 1 again. However this time MS we two zeros in the bracket, ( (1/2 - 1/2) + (1/2 - 1/2) ) giving overall zero.

• " | Up Up> = |S MS> = |1, -1> "
I understand, same reasoning as point 1. Total spin as 1 from two half spin particles, and two -1/2 for MS giving -1.

" One Antisymmetric spin state
Singlet spin state for two identical spin-1/2 particles
• 1/√2 ( |Up Down> - |Down Up> ) = |0,0> "
By my reasoning for the others, this has two spin particles of 1/2 so total spin should be 1 and MS zero again..... What is wrong with my though process for how the Total spin and total mag quantum numbers are worked... How are totals calculated?

2. Mar 11, 2016

### blue_leaf77

Unfortunately, the addition of angular momentum operators is not simple as it is had they been mere numbers. The eigenvalues of the addition of two angular momenta goes like this
$$s = |s_1-s_2|,|s_1-s_2|+1,\ldots,|s_1+s_2|-1,|s_1+s_2|$$
In your problem, $s_1=s_2=1/2$, putting these values into the above equation, you will get two possible values for the total angular momentum quantum number $s=0,1$. The first three points you put forth corresponds to $s=1$, while the last one to $s=0$.

EDIT: I have replaced the capital $S$ with small $s$, the former shall be used to denote the operator/matrix, while the latter for the eigenvalues.

Last edited: Mar 11, 2016
3. Mar 11, 2016

### Sara Kennedy

I've been sitting here 15minutes and still cant see that... I don't follow that equation you wrote or how its derived from adding spin of of two particles.
I get this ... S=|S1+S2| but where is the reasoning for the rest of the equation?

4. Mar 11, 2016

### PeroK

The square of the total angular momentum of a spin 1/2 particle is $S^2 = \frac{3 \hbar^2}{4}$. This is a positive scalar. So, how can two positive scalars of the same magnitude cancel out to give a total of $S^2 = 0$ for the composite system?

5. Mar 11, 2016

### blue_leaf77

You can find the values in the rightmost part of the equation in post #2 by first finding the eigenvalues of the operator $S^2$,
$$S^2 = S_x^2+S_y^2+S_z^2\\ S_i = S_{i1}\otimes \mathbf{1} + \mathbf{1}\otimes S_{i2}$$
where $i=x,y,z$ and $\mathbf{1}$ is an identity matrix with the same dimension as $S_{i1}$ (or $S_{i2}$). The symbol $\otimes$ is called "Kronecker product". It will be a tedious work, but if you are able to find the matrix form of $S^2$ for the composite system of two 1/2-particles and calculate its eigenvalues, you will conclude that $S$ has 4 possible values, three of them are unity, the other one is zero, or in short $s=0,1$.

Last edited: Mar 11, 2016
6. Mar 11, 2016