$$|j = 5/2, m = 3/2\rangle = \begin{cases} c_1|j_1 = 2, m_1 = 1\rangle|j_2 = 1/2, m_2 = 1/2\rangle + \\ c_2|j_1 = 2, m_1 = 2\rangle|j_2 = 1/2, m_2 = -1/2\rangle \end{cases}$$
gives an eigenstate of the total angular momentum of the two particles, using standard notation. Which of the following is true?
(Hint: $\hat{J}_{\pm}|j, m\rangle = \sqrt{j(j + 1) - m(m \pm 1)} |j, m \pm 1\rangle$)
Angular Momentum Coupling Coefficients
This solution addresses the calculation of coefficients for coupled angular momentum states. We consider two particles with angular momenta $j_1 = 2\hbar$ and $j_2 = \hbar/2$. The combined state, an eigenstate of total angular momentum, is given by:
$|j = 5/2, m = 3/2\rangle = c_1|j_1 = 2, m_1 = 1\rangle|j_2 = 1/2, m_2 = 1/2\rangle + c_2|j_1 = 2, m_1 = 2\rangle|j_2 = 1/2, m_2 = -1/2\rangle$
Ladder Operator Method for State $|j = 5/2, m = 3/2\rangle$
We utilize the angular momentum ladder operator $\hat{J}_-$ and the provided hint formula: $\hat{J}_{\pm}|j, m\rangle = \sqrt{j(j + 1) - m(m \pm 1)} |j, m \pm 1\rangle$. This method helps determine the coefficients $c_1$ and $c_2$ by relating the coupled state to the uncoupled basis states.
Highest Weight State Calculation
The highest weight state for $j = 5/2$ is $|5/2, 5/2\rangle$. This state corresponds to the combination of individual states with maximum magnetic quantum numbers ($m_1=2, m_2=1/2$):
$|5/2, 5/2\rangle = |j_1 = 2, m_1 = 2\rangle |j_2 = 1/2, m_2 = 1/2\rangle$
Ladder Operator Action on Highest Weight State
Applying the lowering operator $\hat{J}_-$ to the highest weight state $|5/2, 5/2\rangle$ produces:
$\hat{J}_- |5/2, 5/2\rangle = \sqrt{\frac{5}{2}\left(\frac{5}{2} + 1\right) - \frac{5}{2}\left(\frac{5}{2} - 1\right)} |5/2, 3/2\rangle$
$= \sqrt{\frac{5}{2} \cdot \frac{7}{2} - \frac{5}{2} \cdot \frac{3}{2}} |5/2, 3/2\rangle = \sqrt{\frac{35}{4} - \frac{15}{4}} |5/2, 3/2\rangle = \sqrt{\frac{20}{4}} |5/2, 3/2\rangle = \sqrt{5} |5/2, 3/2\rangle$
Ladder Operator Action on Uncoupled State
The total lowering operator is $\hat{J}_- = \hat{J}_{1-} + \hat{J}_{2-}$. Applying it to the uncoupled state $|2, 2\rangle|1/2, 1/2\rangle$ gives:
$\hat{J}_- (|2, 2\rangle|1/2, 1/2\rangle) = (\hat{J}_{1-} |2, 2\rangle)|1/2, 1/2\rangle + |2, 2\rangle(\hat{J}_{2-} |1/2, 1/2\rangle)$
Calculating individual actions:
- $\hat{J}_{1-} |2, 2\rangle = \sqrt{2(2+1) - 2(2-1)} |2, 1\rangle = \sqrt{6 - 2} |2, 1\rangle = 2|2, 1\rangle$
- $\hat{J}_{2-} |1/2, 1/2\rangle = \sqrt{\frac{1}{2}(\frac{1}{2}+1) - \frac{1}{2}(\frac{1}{2}-1)} |1/2, -1/2\rangle = \sqrt{\frac{3}{4} - (-\frac{1}{4})} |1/2, -1/2\rangle = \sqrt{1} |1/2, -1/2\rangle = |1/2, -1/2\rangle$
Substituting these results:
$\hat{J}_- (|2, 2\rangle|1/2, 1/2\rangle) = 2|2, 1\rangle|1/2, 1/2\rangle + |2, 2\rangle|1/2, -1/2\rangle$
Deriving Coefficients $c_1, c_2$
Equating the results from the coupled and uncoupled applications of $\hat{J}_-$:
$\sqrt{5} |5/2, 3/2\rangle = 2|2, 1\rangle|1/2, 1/2\rangle + |2, 2\rangle|1/2, -1/2\rangle$
To find $|5/2, 3/2\rangle$, we normalize the right side:
$|5/2, 3/2\rangle = \frac{2}{\sqrt{5}} |2, 1\rangle|1/2, 1/2\rangle + \frac{1}{\sqrt{5}} |2, 2\rangle|1/2, -1/2\rangle$
Comparing this normalized expression with the given form:
$|j = 5/2, m = 3/2\rangle = c_1|j_1 = 2, m_1 = 1\rangle|j_2 = 1/2, m_2 = 1/2\rangle + c_2|j_1 = 2, m_1 = 2\rangle|j_2 = 1/2, m_2 = -1/2\rangle$
The coefficients are identified as:
$c_1 = \frac{2}{\sqrt{5}}, \quad c_2 = \frac{1}{\sqrt{5}}$
Important Questions from Angular Momentum Operators Eigenvalues Clebsch Gordan
An electron in the Coulomb field of a proton is in the following state of coherent superposition of orthonormal states $\psi_{nlm}$
$\Psi = \frac{1}{3}\psi_{100} + \frac{1}{\sqrt{3}}\psi_{210} - \frac{\sqrt{5}}{3}\psi_{320}$
Let $E_1, E_2$, and $E_3$ represent the first three energy levels of the system. A sequence of measurements is done on the same system at different times. Energy is measured first at time $t_1$ and the outcome is $E_2$. Then total angular momentum is measured at time $t_2 > t_1$ and finally energy is measured again at $t_3 > t_2$. The probability of finding the system in a state with energy $E_2$ after the final measurement is $P/9$. The value of $P$ is ______________ (in integer).- $H$ 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?
A particle has wavefunction
$\psi(x,y,z) = N ze^{-\alpha(x^2+y^2+z^2)}$,
where $N$ is a normalization constant and $\alpha$ is a positive constant. In this state, which one of the following options represents the eigenvalues of $L^2$ and $L_z$ respectively?
Some values of $Y_l^m$ are:
$Y_0^0 = \sqrt{\frac{1}{4\pi}}$, $Y_1^0 = \sqrt{\frac{3}{4\pi}} \cos\theta$, $Y_1^{\pm 1} = \mp \sqrt{\frac{3}{8\pi}} \sin\theta e^{\pm i\phi}$
