To solve this quantum mechanics problem, we need to find the energy eigenvalues of the given Hamiltonian for a system of three spin-\(\frac{1}{2}\) particles. The Hamiltonian is given by:
\(H = \frac{A}{\hbar^2} (\vec{S}_1 + \vec{S}_2) \cdot \vec{S}_3\)
The problem involves understanding the interaction between the spins of three particles. The key to solving this is recognizing how spin operators interact:
- The term \((\vec{S}_1 + \vec{S}_2)\) represents the combined spin of particles 1 and 2.
- The combined spin states of two spin-\(\frac{1}{2}\) particles can be either in a triplet state (with total spin 1: \(S=1\)) or a singlet state (with total spin 0: \(S=0\)).
- The triplet state (\(S=1\)) is symmetric and has three possible projections: \(m=1, 0, -1\).
- The singlet state (\(S=0\)) is antisymmetric and has only one projection: \(m=0\).
Now let's calculate the possible eigenvalues:
- If \((\vec{S}_1 + \vec{S}_2)\) is in the singlet state, the total spin is 0, contributing zero energy from the product with \(\vec{S}_3\):
- \(E = 0\)
- If \((\vec{S}_1 + \vec{S}_2)\) is in the triplet state, the possible total spins for \((\vec{S}_1 + \vec{S}_2)\) and \(\vec{S}_3\) combined can be 3/2 or 1/2, considering spin addition rules:
- For combined spin \(S_{\text{total}} = \frac{3}{2}\), the energy is:
- \(E = \frac{A}{2}\)
- For combined spin \(S_{\text{total}} = \frac{1}{2}\), the energy is:
- \(E = -A\)
Thus, the possible energy eigenvalues for this system are \(0, \frac{A}{2}, -A\). Therefore, the correct answer is:
$0, \frac{A}{2}, -A$
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: $\hat{J}_{\pm}|j, m\rangle = \sqrt{j(j + 1) - m(m \pm 1)} |j, m \pm 1\rangle$) - $H$ 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:
$Y_0^0 = \sqrt{\frac{1}{4\pi}}$, $Y_1^0 = \sqrt{\frac{3}{4\pi}} \cos\theta$, $Y_1^{\pm 1} = \mp \sqrt{\frac{3}{8\pi}} \sin\theta e^{\pm i\phi}$
