Particle A with angular momentum $j=\frac{3}{2}$ decays into two particles B and C with angular momenta $j_1$ and $j_2$, respectively. If $|\frac{3}{2}, \frac{3}{2}\rangle_A = \alpha |1,1\rangle_B ×|\frac{1}{2}, \frac{1}{2}\rangle_C$, the value of $\alpha$ is_____.
Understanding Angular Momentum Coupling
The problem involves the decay of particle A, characterized by angular momentum $j_A = \frac{3}{2}$, into particles B and C with angular momenta $j_B = 1$ and $j_C = \frac{1}{2}$, respectively. The relationship between the initial state and the final coupled state is given:
$|\frac{3}{2}, \frac{3}{2}\rangle_A = \alpha |1,1\rangle_B \times |\frac{1}{2}, \frac{1}{2}\rangle_C$
We need to find the value of the coefficient $\alpha$.
Identifying Key Quantum Numbers
The notation $|j, m\rangle$ represents a state with angular momentum $j$ and magnetic quantum number $m$. Here:
- For particle A: $j_A = \frac{3}{2}$, $m_A = \frac{3}{2}$.
- For particle B: $j_B = 1$, $m_B = 1$.
- For particle C: $j_C = \frac{1}{2}$, $m_C = \frac{1}{2}$.
The total magnetic quantum number $m_A$ in the coupled state is the sum of the individual magnetic quantum numbers: $m_A = m_B + m_C$. In this case, $\frac{3}{2} = 1 + \frac{1}{2}$, which holds true.
Applying Clebsch-Gordan Coefficients
The coefficient $\alpha$ represents the Clebsch-Gordan coefficient for coupling the specific states $|j_B, m_B\rangle$ and $|j_C, m_C\rangle$ to form the state $|j_A, m_A\rangle$. The general form is:
$|j_A, m_A\rangle = \sum_{m_B+m_C=m_A} \langle j_B, m_B; j_C, m_C | j_A, m_A \rangle |j_B, m_B\rangle \times |j_C, m_C\rangle$
In our specific case, only one combination of $(m_B, m_C)$ yields $m_A = \frac{3}{2}$, which is $m_B = 1$ and $m_C = \frac{1}{2}$. Thus, the equation simplifies to:
$|\frac{3}{2}, \frac{3}{2}\rangle_A = \langle 1, 1; \frac{1}{2}, \frac{1}{2} | \frac{3}{2}, \frac{3}{2} \rangle |1, 1\rangle_B \times |\frac{1}{2}, \frac{1}{2}\rangle_C$
Therefore, $\alpha = \langle 1, 1; \frac{1}{2}, \frac{1}{2} | \frac{3}{2}, \frac{3}{2} \rangle$.
Determining the Coefficient Value
A key property of Clebsch-Gordan coefficients is that the coefficient for coupling the states with the maximum possible magnetic quantum numbers ($m_1 = j_1, m_2 = j_2$) to form the state with the maximum possible total angular momentum ($J = j_1+j_2, M = J$) is always 1 (by standard convention).
Here, $j_A = j_B + j_C$ ($ \frac{3}{2} = 1 + \frac{1}{2} $), and we are considering the states where $m_A = j_A$, $m_B = j_B$, and $m_C = j_C$.
Thus, the coefficient $\alpha = \langle 1, 1; \frac{1}{2}, \frac{1}{2} | \frac{3}{2}, \frac{3}{2} \rangle = 1$.
This value of 1 lies within the specified range.
Important Questions from Angular Momentum Operators Eigenvalues Clebsch Gordan
- Consider two particles with angular momenta $j_1 = 2\hbar$ and $j_2 = \hbar/2$. If the expression
$$|j = 5/2, m = 3/2\rangle = \begin{cases} c_1|j_1 = 2, m_1 = 1\rangle|j_2 = 1/2, m_2 = 1/2\rangle + \\ c_2|j_1 = 2, m_1 = 2\rangle|j_2 = 1/2, m_2 = -1/2\rangle \end{cases}$$
gives an eigenstate of the total angular momentum of the two particles, using standard notation. Which of the following is true?
(Hint: $\hat{J}_{\pm}|j, m\rangle = \sqrt{j(j + 1) - m(m \pm 1)} |j, m \pm 1\rangle$) - A system of three non-identical spin $\frac{1}{2}$ particles has the Hamiltonian $H = \frac{A}{\hbar^2} (\vec{S}_1 + \vec{S}_2) \cdot \vec{S}_3$, where $\vec{S}_1, \vec{S}_2$ and $\vec{S}_3$ are the spin operators of particles labelled $1,2$ and $3$ respectively and $A$ is a constant with appropriate dimensions. The set of possible energy eigenvalues of the system is
- $H$ is the Hamiltonian, $\vec{L}$ the orbital angular momentum and $L_z$ is the $z$-component of $\vec{L}$. The $1s$ state of the hydrogen atom in the non-relativistic formalism is an eigen function of which one of the following sets of operators?
- An atom with non-zero magnetic moment has an angular momentum of magnitude $\sqrt{12}\hbar$. When a beam of such atoms is passed through a Stern-Gerlach apparatus, how many beams does it split into?
- In the vector model of angular momentum applied to atoms, what is the minimum angle in degrees (in integer) made by the orbital angular momentum vector and the positive $z$ axis for a $2p$ electron?
