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

[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 'M_S' as state ket |S M_S>

My textbook states...

" Three Symmetric Spin states
Triplet spin stats for twin identical spin -1/2 particles

• | Up Up> = |S M_S> = |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 M_S 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 M_S we two zeros in the bracket, ( (1/2 - 1/2) + (1/2 - 1/2) ) giving overall zero.

• " | Up Up> = |S M_S> = |1, -1> "

I understand, same reasoning as point 1. Total spin as 1 from two half spin particles, and two -1/2 for M_S 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 M_S 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?

 

[blue_leaf77]

By my reasoning for the others, this has two spin particles of 1/2 so total spin should be 1

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.

 

[Sara Kennedy]

I've been sitting here 15minutes and still can't 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?

 

[PeroK]

I've been sitting here 15minutes and still can't 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?

Is this your question:

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?

 

[blue_leaf77]

I've been sitting here 15minutes and still can't 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?

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##.

 

[Nugatory]

You will find a pretty good explanation at http://www2.ph.ed.ac.uk/~ldeldebb/docs/QM/lect15.pdf