Chapter 9 built the spin operators , ,

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

Chapter 9 built the spin operators

\hat S_x=\tfrac{\hbar}{2}\sigma_x

\hat S_y=\tfrac{\hbar}{2}\sigma_y

\hat S_z=\tfrac{\hbar}{2}\sigma_z

, with

\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},

and found the eigenvectors of

\hat S_x

to be

\chi_\pm^{(x)}=\tfrac{1}{\sqrt2}(1,\pm1)^{\mathrm T}

, with eigenvalues

\pm\hbar/2

. Cycling

x\to y\to z\to x

through the relation

[\hat S_x,\hat S_y]=i\hbar\hat S_z

, which this chapter's argument gives for spin exactly as it does for the orbital operators, produces

[\hat S_y,\hat S_z]=i\hbar\hat S_x

; Chapter 8 supplies the generalised uncertainty relation

\Delta A\,\Delta B\ge\tfrac12|\langle[\hat A,\hat B]\rangle|

.An electron is prepared in

\chi_+^{(x)}=\tfrac{1}{\sqrt2}(1,1)^{\mathrm T}

, the eigenstate of

\hat S_x

with eigenvalue

+\hbar/2

. ((a)) Using the matrices

\sigma_y

and

\sigma_z

, compute

\langle S_y\rangle

and

\langle S_z\rangle

in this state. ((b)) Using the eigenvalue of

\hat S_x

in this state, give the bound the uncertainty relation places on

\Delta S_y\,\Delta S_z

, as a multiple of

\hbar^2

. ((c)) Using

\sigma_y^2=\sigma_z^2=\mathbb 1

(Chapter 9), compute

\Delta S_y

and

\Delta S_z

directly, as multiples of

\hbar

, and say whether the bound of (b) is reached. ((d))

\chi_+^{(x)}

plays, for the axis

x

, the role the top rung

m_l=l

plays for the axis

z

in the orbital ladder. In one sentence, say what property of a top rung this parallel rests on.