Question

We have 2 identical spin 1/2 fermions inside the infitie potential well ($\hat{H}$ $|\phi_j\rangle = E_j |\phi_j\rangle$). There is also spin interaction. Hamiltonian is given by $\hat{H} = \hat{H}_A + \hat{H}_B + J \left(\vec{\hat{S}}^A \cdot \vec{\hat{S}}^B\right) = \left[\frac{\hat{p}_A^2}{2m} + V(\hat{x}_A)\right] + \left[\frac{\hat{p}_B^2}{2m} + V(\hat{x}_B)\right] + J \left(\vec{\hat{S}}^A \cdot \vec{\hat{S}}^B\right)$ where $\vec{\hat{S}}^A \cdot \vec{\hat{S}}^B = \hat{S}_x^A \hat{S}_x^B + \hat{S}_y^A \hat{S}_y^B + \hat{S}_z^A \hat{S}_z^B$ and J is conatant. $|\psi\rangle = \frac{1}{\sqrt{2}} \left[|\phi_1\rangle_A \otimes |z \uparrow\rangle_A \otimes |\phi_1\rangle_B \otimes |z \uparrow\rangle_B - |\phi_1\rangle_A \otimes |z \downarrow\rangle_A \otimes |\phi_1\rangle_B \otimes |z \uparrow\rangle_B\right]$ What is the corresponding eigenvalue of the above state?

          We have 2 identical spin 1/2 fermions inside the infitie potential well ($\hat{H}$ $|\phi_j\rangle = E_j |\phi_j\rangle$). There is also spin interaction. Hamiltonian is given by
$\hat{H} = \hat{H}_A + \hat{H}_B + J \left(\vec{\hat{S}}^A \cdot \vec{\hat{S}}^B\right) = \left[\frac{\hat{p}_A^2}{2m} + V(\hat{x}_A)\right] + \left[\frac{\hat{p}_B^2}{2m} + V(\hat{x}_B)\right] + J \left(\vec{\hat{S}}^A \cdot \vec{\hat{S}}^B\right)$
where
$\vec{\hat{S}}^A \cdot \vec{\hat{S}}^B = \hat{S}_x^A \hat{S}_x^B + \hat{S}_y^A \hat{S}_y^B + \hat{S}_z^A \hat{S}_z^B$
and J is conatant.
$|\psi\rangle = \frac{1}{\sqrt{2}} \left[|\phi_1\rangle_A \otimes |z \uparrow\rangle_A \otimes |\phi_1\rangle_B \otimes |z \uparrow\rangle_B - |\phi_1\rangle_A \otimes |z \downarrow\rangle_A \otimes |\phi_1\rangle_B \otimes |z \uparrow\rangle_B\right]$
What is the corresponding eigenvalue of the above state?

We have 2 identical spin 1/2 fermions inside the infitie potential well (Ĥ |⟩ = Ej |⟩). There is also spin interaction. Hamiltonian is given by
Ĥ = ĤA + ĤB + J (Ŝ⃗^A ·Ŝ⃗^B) = [(p̂A^2)/(2m) + V(x̂A)] + [(p̂B^2)/(2m) + V(x̂B)] + J (Ŝ⃗^A ·Ŝ⃗^B)
where
Ŝ⃗^A ·Ŝ⃗^B = Ŝx^A Ŝx^B + Ŝy^A Ŝy^B + Ŝz^A Ŝz^B
and J is conatant.
|ψ⟩ = (1)/(√(2))[|ϕ1⊗ |z ↑⊗ |ϕ1⊗ |z ↑ - |ϕ1⊗ |z ↓⊗ |ϕ1⊗ |z ↑]
What is the corresponding eigenvalue of the above state?

Added by Ashley A.

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