Verify Eigenstates: Solving s_i & m_i Equations

[Philip Land]

Homework Statement

verify that $|s=1,m_s=0> = \frac{1}{\sqrt{2}}(| \uparrow \downarrow> + |\downarrow \uparrow >$ is an eigenstate of $\hat{S^2}$

Relevant Equations

(Drop hats) $$S^2 = S_1^2 + S_2^2 + 2S_{1z}S_{2z} + S_{1+}S_{2-}+S_{1-}S_{2+}$$

I simply use the equation above, and the eigenvalus whish yield:
$\hbar^2 [ s_1(s_1+1) + s_2(s_2+1) + m_1m_2 + \sqrt{s_1(s_1+1) - m_1(m_1+1)}\sqrt{s_2(s_2+1) - m_2(m_2-1)} + \sqrt{s_2(s_2+1) - m_2(m_2+1)}\sqrt{s_1(s_1+1) - m_1(m_1-1)}$

Very straight forward. My issue is that I don't know what $s_i$ and $m_i$ for i=1,2 is? I only have "s" and "m" from the definition in the question.

I recently had the same problem in an exercise but with angular momentum. Please bring me some clarity on this, thanks so much in advance!

 

[DrClaude]

Single-particle states $|\uparrow\rangle$ and $|\downarrow\rangle$ are a basis for spin-1/2 particles, therefore $s_1 = s_2 = 1/2$ and, by convention, $|\uparrow\rangle$ corresponds to $m= +1/2$ and, conversely, $|\downarrow\rangle$ to $m= -1/2$.

 

[Philip Land]

Single-particle states $|\uparrow\rangle$ and $|\downarrow\rangle$ are a basis for spin-1/2 particles, therefore $s_1 = s_2 = 1/2$ and, by convention, $|\uparrow\rangle$ corresponds to $m= +1/2$ and, conversely, $|\downarrow\rangle$ to $m= -1/2$.

I see, thanks for clearing this up!