Question

2. Now consider two spin-1/2 particles, like a proton and an electron. Their spins commute and the joint Hilbert space for these two particles is also four-dimensional. Here we are going to start with the basis states $|\uparrow\uparrow\rangle$ $|\downarrow\uparrow\rangle$ $|\uparrow\downarrow\rangle$ $|\downarrow\downarrow\rangle$ Note that the first quantum number refers to the z-component of the spin of the first particle and the second quantum number refers to the z-component of the spin of the second particle. The total spin is $\vec{J} = \vec{S_1} + \vec{S_2}$ (a) Construct the 4 x 4 matrix for $J_z$ in this basis. (b) Now construct the matrices for the ladder operators $J_+ = S_1^+ + S_2^+$ and $J_- = S_1^- + S_2^-$. Remember the spin operators act only on their part of the ket, so for example $J_+|\downarrow\uparrow\rangle = |\uparrow\uparrow\rangle$ and $J_-|\uparrow\downarrow\rangle = |\downarrow\downarrow\rangle$ (c) Using these expressions for the ladder operators, construct $J_x$ and $J_y$ (d) Show that the matrix for $J^2$ is $\begin{pmatrix} 2 & 0 & 0 & 0\\0 & 1 & 1 & 0\\0 & 1 & 1 & 0\\0 & 0 & 0 & 2 \end{pmatrix}$ (e) Find the eigenvectors and eigenvalues of $J^2$ (Hint: you only need to diagonalize the inner 2 x 2 matrix. The states $|\uparrow\uparrow\rangle$ and $|\downarrow\downarrow\rangle$ are already eigenvectors of $J^2$ and $J_z$, with $j = 1, m_j = +1$ and $j = 1, m_j = -1.)

          2. Now consider two spin-1/2 particles, like a proton and an electron.
Their spins commute and the joint Hilbert space for these two particles
is also four-dimensional. Here we are going to start with the basis states
$|\uparrow\uparrow\rangle$
$|\downarrow\uparrow\rangle$
$|\uparrow\downarrow\rangle$
$|\downarrow\downarrow\rangle$

Note that the first quantum number refers to the z-component of the
spin of the first particle and the second quantum number refers to the
z-component of the spin of the second particle. The total spin is
$\vec{J} = \vec{S_1} + \vec{S_2}$
(a) Construct the 4 x 4 matrix for $J_z$ in this basis.
(b) Now construct the matrices for the ladder operators $J_+ = S_1^+ + S_2^+$ and $J_- = S_1^- + S_2^-$. Remember the spin operators act only
on their part of the ket, so for example
$J_+|\downarrow\uparrow\rangle = |\uparrow\uparrow\rangle$
and
$J_-|\uparrow\downarrow\rangle = |\downarrow\downarrow\rangle$
(c) Using these expressions for the ladder operators, construct $J_x$ and $J_y$
(d) Show that the matrix for $J^2$ is
$\begin{pmatrix}
2 & 0 & 0 & 0\\0 & 1 & 1 & 0\\0 & 1 & 1 & 0\\0 & 0 & 0 & 2
\end{pmatrix}$
(e) Find the eigenvectors and eigenvalues of $J^2$ (Hint: you only need
to diagonalize the inner 2 x 2 matrix. The states $|\uparrow\uparrow\rangle$ and $|\downarrow\downarrow\rangle$ are already eigenvectors of $J^2$ and $J_z$, with $j = 1, m_j = +1$ and $j = 1, m_j = -1.)

2. Now consider two spin-1/2 particles, like a proton and an electron.
Their spins commute and the joint Hilbert space for these two particles
is also four-dimensional. Here we are going to start with the basis states
|↑↑⟩
|↓↑⟩
|↑↓⟩
|↓↓⟩

Note that the first quantum number refers to the z-component of the
spin of the first particle and the second quantum number refers to the
z-component of the spin of the second particle. The total spin is
J⃗ = S⃗1⃗ + S⃗2⃗
(a) Construct the 4 x 4 matrix for Jz in this basis.
(b) Now construct the matrices for the ladder operators J+ = S1^+ + S2^+ and J- = S1^- + S2^-. Remember the spin operators act only
on their part of the ket, so for example
J+|↓↑⟩ = |↑↑⟩
and
J-|↑↓⟩ = |↓↓⟩
(c) Using these expressions for the ladder operators, construct Jx and Jy
(d) Show that the matrix for J^2 is
< p m a t r i x >
(e) Find the eigenvectors and eigenvalues of J^2 (Hint: you only need
to diagonalize the inner 2 x 2 matrix. The states |↑↑⟩ and |↓↓⟩ are already eigenvectors of J^2 and Jz, with j = 1, mj = +1 and j = 1, mj = -1.)

Added by John W.

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