In Chapter 9 the spin operators of an electron were written as ,

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Symmetry, angular momentum and spin · Symmetry, angular momentum and spin

Quantum Mechanics: From the Photon to the Hydrogen Atom, Ch Orbital angular momentum, the spherical harmonics and rotations

Problem

In Chapter 9 the spin operators of an electron were written as S^x=ℏ2σx\hat S_x = \frac{\hbar}{2}\sigma_x, S^y=ℏ2σy\hat S_y = \frac{\hbar}{2}\sigma_y, S^z=ℏ2σz\hat S_z = \frac{\hbar}{2}\sigma_z withσx=(0110),σy=(0−ii0),σz=(100−1).\sigma_x = \begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad \sigma_y = \begin{pmatrix}0&-i\\i&0\end{pmatrix},\qquad \sigma_z = \begin{pmatrix}1&0\\0&-1\end{pmatrix}. ((a)) Multiply the matrices to show that [S^x,S^y]=iℏS^z[\hat S_x,\hat S_y] = i\hbar\hat S_z, the same relation the orbital components obey. ((b)) Because the algebra is the same, the ladder argument of this chapter applies unchanged. A measurement of S^2\hat S^2 gives 34ℏ2\frac34\hbar^2. Deduce the quantum number ss and the number of rungs. ((c)) Find the length ∣S∣=s(s+1) ℏ|\mathbf S| = \sqrt{s(s+1)}\,\hbar and the angle between S\mathbf S and the zz axis for the rung ms=+12m_s = +\frac12, in degrees. ((d)) The ladder allows l=12l = \frac12 as well. Explain why an orbital angular momentum can never have l=12l = \frac12, although the electron's spin does.

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