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Verify eigenstates

  • #1

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!
 

Answers and Replies

  • #2
DrClaude
Mentor
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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##.
 
  • #3
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!
 

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