Example 10.3   The operator \(\hat n\cdot\vec\sigma\)

\(\hat A=\hat n\cdot\vec\sigma\). (a) Write its matrix in terms of \(\theta,\phi\). (b) For \(\hat n\) in the \(xz\)-plane find its eigenvalues.

Solution

• (a)  \(\hat A=\sin\theta\cos\phi\,\sigma_x+\sin\theta\sin\phi\,\sigma_y+\cos\theta\,\sigma_z=\begin{pmatrix}\cos\theta&\sin\theta\,e^{-i\phi}\\\sin\theta\,e^{i\phi}&-\cos\theta\end{pmatrix}\).

• (b)  \(\phi=0\): \(\begin{vmatrix}\cos\theta-\lambda&\sin\theta\\\sin\theta&-\cos\theta-\lambda\end{vmatrix}=\lambda^2-1=0\Rightarrow\lambda=\pm1\) (for any \(\hat n\)).

• Eigenvectors: \(\left|+\right\rangle_{\hat n}=\begin{pmatrix}\cos\frac\theta2\\e^{i\phi}\sin\frac\theta2\end{pmatrix}\),   \(\left|-\right\rangle_{\hat n}=\begin{pmatrix}-e^{-i\phi}\sin\frac\theta2\\\cos\frac\theta2\end{pmatrix}\).

Example 10.4   Time-evolution operator in a field

\(H=\mu\vec\sigma\cdot\vec B\), \(\omega=\mu B/\hbar\). Show \(e^{iHt/\hbar}=I\cos\omega t+\dfrac{i\vec\sigma\cdot\vec B}{B}\sin\omega t\).

Solution

• \(\dfrac{iHt}{\hbar}=i\,\dfrac{\mu Bt}{\hbar}\,(\hat n\cdot\vec\sigma)=i\omega t\,(\hat n\cdot\vec\sigma)\) with \(\hat n=\vec B/B\).

• Use \(e^{ia\hat n\cdot\vec\sigma}=I\cos a+i\hat n\cdot\vec\sigma\sin a\) with \(a=\omega t\).

Example 10.5   Rotation about \(x\)

Spins in the \(\sigma_z=+1\) state are rotated by \(60^\circ\) about \(x\). Find the fractions with \(\sigma_z=+1\) and \(-1\).

Solution

• \(R_x(\pi/3)\begin{pmatrix}1\\0\end{pmatrix}=\begin{pmatrix}\cos30^\circ\\-i\sin30^\circ\end{pmatrix}=\begin{pmatrix}\sqrt3/2\\-i/2\end{pmatrix}\).

• \(P(+1)=\tfrac34\),   \(P(-1)=\tfrac14\).

Example 10.6   Precession in an arbitrary field

An electron (\(g\)-factor \(g\), \(\vec S=\frac\hbar2\vec\sigma\)) is at rest in \(\vec B=(B_x,B_y,B_z)\), all components non-zero. At \(t=0\) it is in \(s_z=+\frac12\). Find the earliest time at which the spin state is orthogonal to the initial state.

Solution

• \(H=-\vec\mu\cdot\vec B=\dfrac{ge}{2m}\vec S\cdot\vec B=E_0|B|\,\hat n\cdot\vec\sigma\),   \(E_0=\dfrac{ge\hbar}{4m}\); eigenvalues \(\pm E_0|B|\).

• Eigenvectors \(\left|\phi_1\right\rangle=\left|+\right\rangle_{\hat n}\), \(\left|\phi_2\right\rangle=\left|-\right\rangle_{\hat n}\); initially \(\left|\uparrow\right\rangle=\cos\frac\theta2\,e^{-i\phi/2}\left|\phi_1\right\rangle-\sin\frac\theta2\,e^{i\phi/2}\left|\phi_2\right\rangle\) (up to phase).

• \(\left\langle \psi(0)\middle|\psi(t)\right\rangle =\cos^2\frac\theta2\,e^{-iE_0|B|t/\hbar}+\sin^2\frac\theta2\,e^{iE_0|B|t/\hbar}\).

• This vanishes only if \(\cos^2\frac\theta2=\sin^2\frac\theta2\), i.e. \(\theta=\pi/2\) (\(B_z=0\)). With \(B_z\ne0\) the spin precesses on a cone about \(\vec B\) and never becomes orthogonal.

• If \(B_z=0\): \(\cos(E_0|B|t/\hbar)=0\Rightarrow t=\dfrac{\pi\hbar}{2E_0|B|}=\dfrac{2\pi m}{ge|B|}\).

Example 10.7   Eigenvalues of \(\hat S_x+\hat S_y\)

Solution

• \(\hat S_x+\hat S_y=\dfrac\hbar2\begin{pmatrix}0&1-i\\1+i&0\end{pmatrix}\);   \(\lambda^2-\dfrac{\hbar^2}{4}\cdot2=0\Rightarrow\lambda=\pm\dfrac{\hbar}{\sqrt2}\).

Example 10.8   Measurements on \(\binom{3+4i}{3}\)

For \(\left|\psi\right\rangle=\binom{3+4i}{3}\) find the outcomes and probabilities of (a) \(\hat S^2\), (b) \(\hat S_z\), (c) \(\hat S_x\).

Solution

• \(\left\langle \psi\middle|\psi\right\rangle =25+9=34\).

• (a)  \(\tfrac34\hbar^2\) with probability 1.

• (b)  \(P(+\tfrac\hbar2)=\dfrac{25}{34}\),   \(P(-\tfrac\hbar2)=\dfrac{9}{34}\).

• (c)  \(\left\langle x_+\middle|\psi\right\rangle =\dfrac{6+4i}{\sqrt2}\), \(\left\langle x_-\middle|\psi\right\rangle =\dfrac{4i}{\sqrt2}\) \(\Rightarrow P(+\tfrac\hbar2)=\dfrac{26}{34}=\dfrac{13}{17}\),   \(P(-\tfrac\hbar2)=\dfrac{8}{34}=\dfrac{4}{17}\).

Example 10.9   Change of basis and time evolution

\(\left|\psi\right\rangle=\sqrt{\tfrac23}\left|\uparrow\right\rangle+\sqrt{\tfrac13}\left|\downarrow\right\rangle\). (a) Write \(\left|\psi\right\rangle\) in the \(\hat S_x\) eigenbasis \(\left|\phi_\pm\right\rangle\). (b) Find \(\left\langle \hat S_x\right\rangle\). (c) With \(H=-\mu_0B\hat S_x\) find \(\left|\psi(t)\right\rangle\).

Solution

• (a)  Insert \(I=\left|\phi_+\right\rangle\left\langle\phi_+\right|+\left|\phi_-\right\rangle\left\langle\phi_-\right|\): \[\left|\psi\right\rangle=\Big(\frac1{\sqrt3}+\frac1{\sqrt6}\Big)\left|\phi_+\right\rangle+\Big(\frac1{\sqrt3}-\frac1{\sqrt6}\Big)\left|\phi_-\right\rangle=\frac{\sqrt2+1}{\sqrt6}\left|\phi_+\right\rangle+\frac{\sqrt2-1}{\sqrt6}\left|\phi_-\right\rangle.\]

• (b)  \(\left\langle \hat S_x\right\rangle=\dfrac\hbar2\big(P_+-P_-\big)=\dfrac\hbar2\cdot\dfrac{4}{\sqrt{18}}=\dfrac{\sqrt2}{3}\hbar\).

• (c)  \(E_\pm=\mp\dfrac{\mu_0B\hbar}{2}\):   \(\left|\psi(t)\right\rangle=\dfrac{\sqrt2+1}{\sqrt6}e^{i\mu_0Bt/2}\left|\phi_+\right\rangle+\dfrac{\sqrt2-1}{\sqrt6}e^{-i\mu_0Bt/2}\left|\phi_-\right\rangle\).

Example 10.10   General spinor

\(\left|\psi\right\rangle=\cos\alpha\,\chi_{+}+\sin\alpha\,e^{i\beta}\chi_{-}\) (\(\alpha,\beta\) real). Find \(P(S_x=+\frac\hbar2)\).

Solution

• \(\left|\phi_+\right\rangle=\frac{1}{\sqrt2}(\chi_++\chi_-)\):   \(P=\tfrac12\big|\cos\alpha+\sin\alpha\,e^{i\beta}\big|^2=\tfrac12\big(1+\sin2\alpha\cos\beta\big)\)