Question

Three S = 1 atoms are placed on the corners of an equi-lateral triangle and can be described by a Hamiltonian hat{mathcal{H}} = -2J(hat{mathbf{S}}_1 cdot hat{mathbf{S}}_2 + hat{mathbf{S}}_2 cdot hat{mathbf{S}}_3 + hat{mathbf{S}}_3 cdot hat{mathbf{S}}_1). Show that the energy eigenvalues of the system are 0, 2J and 6J.

          Three S = 1 atoms are placed on the corners of an equi-lateral triangle and can be described by a Hamiltonian
hat{mathcal{H}} = -2J(hat{mathbf{S}}_1 cdot hat{mathbf{S}}_2 + hat{mathbf{S}}_2 cdot hat{mathbf{S}}_3 + hat{mathbf{S}}_3 cdot hat{mathbf{S}}_1).
Show that the energy eigenvalues of the system are 0, 2J and 6J.

Three S = 1 atoms are placed on the corners of an equi-lateral triangle and can be described by a Hamiltonian
hatmathcalH = -2J(hatmathbfS1 cdot hatmathbfS2 + hatmathbfS2 cdot hatmathbfS3 + hatmathbfS3 cdot hatmathbfS1).
Show that the energy eigenvalues of the system are 0, 2J and 6J.

Added by Mary C.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Sam Stansfield

02/17/2024

Step 1

The interaction between the spins is given by the dot product of the spin operators, and the strength of the interaction is characterized by the coupling constant J. The Hamiltonian is: H = -2J(S₁·S₂ + S₂·S₃ + S₃·S₁) This Hamiltonian represents the total energy Show more…

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Three S=1 atoms are placed on the corners of an equi-lateral triangle and can be described by a Hamiltonian hat(H)=-2J(hat(S)_(1)*hat(S)_(2)+hat(S)_(2)*hat(S)_(3)+hat(S)_(3)*hat(S)_(1)). Show that the energy eigenvalues of the system are 0, 2J and 6J.