Quantum Mechanics: Concepts and Applications · Problem 5.32

Three-Dimensional Problems — Nouredine Zettili

Physics — Quantum Mechanics · spin-five-halves-energy-levels

Problem

Find the energy levels of a spin 52\frac{5}{2} particle whose Hamiltonian is given by H^=ε0ℏ2(S^x2+S^y2)+ε0ℏS^z,\hat{H} = \frac{\varepsilon_0}{\hbar^2}(\hat{S}_x^2 + \hat{S}_y^2) + \frac{\varepsilon_0}{\hbar}\hat{S}_z, where ε0\varepsilon_0 is a constant having the dimensions of energy. Are the energy levels degenerate?
Intuitive explanation

The key is to rewrite the quadratic spin operators in terms of the ladder operators S^±=S^x±iS^y\hat{S}_\pm = \hat{S}_x \pm i\hat{S}_y, which lets you express S^x2+S^y2\hat{S}_x^2 + \hat{S}_y^2 in a form that acts simply on eigenstates of S^z\hat{S}_z. Once you have the Hamiltonian in this basis, you can directly compute eigenvalues for each msm_s value, then check whether different msm_s states share the same energy.

Intuitive explanation

The key is to rewrite the quadratic spin operators in terms of the ladder operators S^±=S^x±iS^y\hat{S}_\pm = \hat{S}_x \pm i\hat{S}_y, which lets you express S^x2+S^y2\hat{S}_x^2 + \hat{S}_y^2 in a form that acts simply on eigenstates of S^z\hat{S}_z. Once you have the Hamiltonian in this basis, you can directly compute eigenvalues for each msm_s value, then check whether different msm_s states share the same energy.

Related theory

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