Question

A system of three non-identical spin-\(\frac{1}{2}\) particles has the Hamiltonian
\[ H = \frac{A \hbar^2}{2} (\vec{S}_1 + \vec{S}_2) \cdot \vec{S}_3, \] where \( \vec{S}_1, \vec{S}_2, \vec{S}_3 \) 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:

• A. \( 0, \frac{A}{2}, -A \)
• B. \( 0, \frac{A}{2}, -\frac{A}{2} \)
• C. \( 0, 3\frac{A}{2}, -\frac{A}{2} \)
• D. \( 0, -3\frac{A}{2}, \frac{A}{2} \)

Hint:

When dealing with spin-\(\frac{1}{2}\) particles, consider the possible total spin states for pairs of particles. The interaction Hamiltonian typically depends on the dot product of spin operators, which leads to different energy eigenvalues based on the total spin.

Answer 1

The Correct Option is A

1. Hamiltonian for the system:
The Hamiltonian for the system is given by: \[ H = \frac{A \hbar^2}{2} (\vec{S}_1 + \vec{S}_2) \cdot \vec{S}_3 \] where \( \vec{S}_1, \vec{S}_2, \vec{S}_3 \) are the spin operators for the three spin-\(\frac{1}{2}\) particles, and \( A \) is a constant. This Hamiltonian involves the interaction between the spin operators of particles 1, 2, and 3.

2. Total spin operators:
The total spin operator \( \vec{S}_{12} = \vec{S}_1 + \vec{S}_2 \) represents the combined spin of particles 1 and 2. Since both are spin-\(\frac{1}{2}\) particles, the possible total spin values are:

• \( S_{12} = 1 \) (triplet state)
• \( S_{12} = 0 \) (singlet state)

The total Hamiltonian can be rewritten using the identity: \[ (\vec{S}_1 + \vec{S}_2) \cdot \vec{S}_3 = \frac{1}{2} \left[ \vec{S}_{\text{tot}}^2 - \vec{S}_{12}^2 - \vec{S}_3^2 \right], \] where \( \vec{S}_{\text{tot}} = \vec{S}_1 + \vec{S}_2 + \vec{S}_3 \).

3. Energy eigenvalues for different spin configurations:
Each spin-\(\frac{1}{2}\) has \( \vec{S}^2 = \frac{3}{4} \hbar^2 \).
The possible total spin states and corresponding eigenvalues are:

• For \( S_{12} = 1 \) and \( S_{\text{tot}} = \frac{3}{2} \):
  \[ E = \frac{A \hbar^2}{2} \left( \frac{15}{4} - 2 - \frac{3}{4} \right) = \frac{A \hbar^2}{2} \]
• For \( S_{12} = 1 \) and \( S_{\text{tot}} = \frac{1}{2} \):
  \[ E = \frac{A \hbar^2}{2} \left( \frac{3}{4} - 2 - \frac{3}{4} \right) = -A \hbar^2 \]
• For \( S_{12} = 0 \), \( S_{\text{tot}} = \frac{1}{2} \):
  \[ E = \frac{A \hbar^2}{2} \left( \frac{3}{4} - 0 - \frac{3}{4} \right) = 0 \]

4. Possible energy eigenvalues:
Removing \( \hbar^2 \) as a constant factor (for normalized units), the possible energy eigenvalues of the system are:

• \( 0 \)
• \( \frac{A}{2} \)
• \( -A \)

Therefore, the set of possible energy eigenvalues of the system is: \[ \boxed{0, \frac{A}{2}, -A} \] which corresponds to option (A).