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4) Adding angular momenta Consider a system of two particles with spin. Particle 1 has spin $s_1 = \frac{1}{2}$ and particle 2 has spin $s_2 = 2$, where $\hbar^2 s_i(s_i + 1)$ stands for the eigenvalue of the single particle spin operator $\vec{S}_i^2$ ($i = 1, 2)$. Assume now that the combined system has spin $s = \frac{5}{2}$, where $\hbar^2 s(s + 1)$ is the eigenvalue of the spin operators $\vec{S}^2$ where $\vec{S} = \vec{S}_1 + \vec{S}_2$. 1 (a) Write down all states $|\frac{5}{2}, m\rangle$ in terms of the tensor product states $|\frac{1}{2} m_1; 2 m_2\rangle$. Now, let the spin in the z-direction of the combined system be $m = \frac{1}{2}$, where m is the eigenvalue of the spin operator $S_z = S_{1,z} + S_{2,z}$. Use the formulae derived in the lecture and cross check them with the tabulated Clebsch-Gordon coefficients on page 33 of the lecture notes. (b) Which values of $m_2$ (eigenvalues of spin operator $S_{2,z}$) are in this state possible for particle 2? Determine the probabilities that each of these values is obtained in a measurement.

          4) Adding angular momenta
Consider a system of two particles with spin. Particle 1 has spin $s_1 = \frac{1}{2}$ and particle 2 has spin
$s_2 = 2$, where $\hbar^2 s_i(s_i + 1)$ stands for the eigenvalue of the single particle spin operator $\vec{S}_i^2$ ($i = 1, 2)$.
Assume now that the combined system has spin $s = \frac{5}{2}$, where $\hbar^2 s(s + 1)$ is the eigenvalue of the
spin operators $\vec{S}^2$ where $\vec{S} = \vec{S}_1 + \vec{S}_2$.
1
(a) Write down all states $|\frac{5}{2}, m\rangle$ in terms of the tensor product states $|\frac{1}{2} m_1; 2 m_2\rangle$. Now, let the
spin in the z-direction of the combined system be $m = \frac{1}{2}$, where m is the eigenvalue of the spin
operator $S_z = S_{1,z} + S_{2,z}$. Use the formulae derived in the lecture and cross check them with
the tabulated Clebsch-Gordon coefficients on page 33 of the lecture notes.
(b) Which values of $m_2$ (eigenvalues of spin operator $S_{2,z}$) are in this state possible for particle 2?
Determine the probabilities that each of these values is obtained in a measurement.

4) Adding angular momenta
Consider a system of two particles with spin. Particle 1 has spin s1 = (1)/(2) and particle 2 has spin
s2 = 2, where ħ^2 si(si + 1) stands for the eigenvalue of the single particle spin operator S⃗i^2 (i = 1, 2).
Assume now that the combined system has spin s = (5)/(2), where ħ^2 s(s + 1) is the eigenvalue of the
spin operators S⃗^2 where S⃗ = S⃗1 + S⃗2.
1
(a) Write down all states |(5)/(2), m⟩ in terms of the tensor product states |(1)/(2) m1; 2 m2⟩. Now, let the
spin in the z-direction of the combined system be m = (1)/(2), where m is the eigenvalue of the spin
operator Sz = S1,z + S2,z. Use the formulae derived in the lecture and cross check them with
the tabulated Clebsch-Gordon coefficients on page 33 of the lecture notes.
(b) Which values of m2 (eigenvalues of spin operator S2,z) are in this state possible for particle 2?
Determine the probabilities that each of these values is obtained in a measurement.

Added by Brittany R.

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