Spin
1925: Goudsmit and Uhlenbeck
Every electron has an intrinsic angular momentum (spin) of \(\hbar/2\) which corresponds to a magnetic moment of one Bohr magneton, \(\mu_B=\frac{q\hbar}{2m}\).
- Stern-Gerlach (1922) Used an atom in 1s state \((L=0)\) \(H = H_0 + H_1 + H_2\), \(H_0=\frac{p^2}{2m_e}-\frac{e^2}{r}\), Paramagnetic term: \(H_1 = -\vec{M}\cdot\vec{B}\), Diamagnetic term: \(H_2 = \frac{q^2B^2}{8m_e}(x^2+y^2)\), \(M=\frac{q}{2m_e}\vec{L}\). Diamagnetic term is typically ignored unless paramagnetic term is zero. The force is then expected to be: \(\vec{F} = -\nabla(-\vec{M}\cdot\vec{B})\approx M_z(\nabla B)_z\) This then raised the suspicion that \(\vec{M}=\frac{q}{2m_e}\vec{J} = \vec{M_S} + \vec{M_L}\) \(S_z=\pm\frac{\hbar}{2}\)
- Anomalous Zeeman Effect
- A given particle is characterized by a unique value of \(s\)
- \([\vec{L}^2,\vec{S}^2] = 0\) (They act on different spaces, different quantum numbers)
- General properties:
- \(e^-\Rightarrow s=\frac{1}{2}\Rightarrow 2s+1=2\)
- \(|+\rangle = \left|\frac{1}{2}\frac{1}{2}\right\rangle\)
- \(|-\rangle = \left|\frac{1}{2}\frac{-1}{2}\right\rangle\)
- \(\langle\pm|\mp\rangle = 0\)
- \(\langle\pm|\pm\rangle = 1\)
- \(P_+ + |+\rangle\langle +| + |-\rangle\langle -| = \mathbb{I}\)
- \(\vec{S}^2|\pm\rangle = \hbar^2\frac{3}{4}\hbar^2|\pm\rangle\)
- \(S_z|\pm\rangle=\pm\frac{\hbar}{2}|\pm\rangle\)
- \(S_\pm=S_x\pm iS_y\)
- \(S_\pm|\pm\rangle = 0\)
- \(S_+|-\rangle=c_-|+\rangle\)
- \(S_-|+\rangle=c_+|-\rangle\)
- \(J_{\pm}|jm\rangle = \hbar\sqrt{j(j+1)-m(m\pm1)}|j(m\pm1)\rangle\)
\([S_i,S_j] = i\hbar\varepsilon_{ijk}S_k\), \(\vec{S}^2|sm_s\rangle=\hbar^2s(s+1)|sm_s\rangle\), \(S_z|sm_s\rangle=\hbar m_s|sm_s\rangle\), \(|m_s|\leq s\)
Matrix Representation
- \(|+\rangle=\begin{pmatrix} 1 \\ 0 \end{pmatrix}\)
- \(|-\rangle=\begin{pmatrix} 0 \\ 1 \end{pmatrix}\)
- \(|\alpha\rangle = \langle+ | \alpha\rangle |+\rangle + \langle- | \alpha\rangle |-\rangle = \begin{pmatrix} \langle +|\alpha\rangle \\ \langle -|\alpha\rangle \end{pmatrix}\)
- \(S_z=\begin{pmatrix} \frac{\hbar}{2} & 0 \\ 0 & -\frac{\hbar}{2} \end{pmatrix}\).
- \(\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\)
- \(\sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}\)
- \(\sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\)
- They all have negative 1 determinants (parity operators?, not quite since they reside in a half integer space)
- \([\sigma_i,\sigma_j] = 2i\sigma_k\varepsilon_{ijk}\)
- \(\sigma_i\) Unitary
- \(\sigma_i\) Hermitian
- Zero trace
- \(\{\sigma_i,\sigma_j\}=0\)
Examples
Inverses
\((2\mathbb{I} + \sigma_x)^{-1} = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix} = \frac{1}{3}\begin{pmatrix} 2 & -1 \\ -1 & 2 \end{pmatrix}\)
\((2\mathbb{I} + \sigma_x)^{-1} = \frac{1}{2}\left(\mathbb{I}+\frac{\sigma_x}{2}\right)^{-1} \approx \frac{1}{2}\left(I - \frac{\sigma_x}{2}+\left(\frac{\sigma_x}{2}\right)^2+\cdots\right) = \frac{1}{2}\left(I(1+\frac{1}{2^2}+\frac{1}{2^4}+\cdots)\right)-\frac{\sigma_x}{2}\left(1+\frac{1}{2^2}+\frac{1}{2^4}+\cdots\right) = \frac{2}{3}I-\frac{1}{3}\sigma_x\). \(\sum_{n=0}^\infty q^n=\frac{1}{1-q}\). \(\sigma_x^2=I\).
Tensor Products
\(|\Psi\rangle\in\mathcal{E}_x \leftrightarrow |\Psi\rangle\in\mathcal{E}_{\vec{r}}\)
\(\Psi(\vec{r})=\Psi_x(x)\Psi_y(y)\Psi_z(z)\rightarrow|\Psi\rangle=|\Psi_x\rangle\otimes|\Psi_y\rangle\otimes|\Psi_z\rangle\) \(\mathcal{E}_{\vec{r}}=\mathcal{E}_x\otimes\mathcal{E}_y\otimes\mathcal{E}_z\)
\(\otimes =\) tensor (direct) product.
The vector space \(\mathcal{E}\) is the tensor product of \(\mathcal{E}_1\) and \(\mathcal{E}_2\) if there is a vector \(|\phi_1\rangle\otimes|\chi_2\rangle\in\mathcal{E}\) associated with \(|\phi_1\rangle\in\mathcal{E}_1\) and \(|\chi_2\in\mathcal{E}_2\) which satisfies:
- linear with respect to multiplication by complex numbers \(\lambda|\phi_1\rangle\otimes|\chi_2\rangle = \lambda(|\phi_1\rangle\otimes|\chi_2\rangle)\)
- Distributive with respect to vector addition \(|\phi_1\rangle\otimes(|\chi_2\rangle+|\kappa_2\rangle) = |\phi_1\rangle\otimes|\chi_2\rangle + |\phi_1\rangle\otimes|\kappa_2\rangle\)
- Basis: \(\{|u_{i,1}\rangle\}\in\mathcal{E}_1\), \(\{|v_{i,2}\rangle\}\in\mathcal{E}_2\) gives an overall basis, \(\{|u_{i,1}\rangle\otimes|v_{j,2}\rangle\}=\{|u_{i,1}\}\times\{|v_{j,2}\rangle\}\). Behaves like a Cartesian product. Thus, if \(\mathcal{E}_i\) has dimension \(N_i\) then the overall space has dimension \(N=N_1N_2\).
- \(\vec{L}^2|n\ell m\rangle\otimes|sm_s\rangle\) operates on \(\mathcal{E}_\vec{r}\) hence acts on \(|n\ell m\rangle\) An operator that acts on the entire space would be \(\vec{L}^2\otimes I_s\in\mathcal{E}\). Or for the reverse case, \(I_{\vec{r}}\otimes \vec{S}^2\). Note this ordering is assumed by writing \(\vec{L}^2\) to be implicitly \(\vec{L}^2\otimes I_s\) or whatever identities are needed.
- Operators acting on different spaces commute \([A_1\otimes I_2,I_1\otimes B_2]|\Psi\rangle = 0, \Psi=u_1\otimes u_2\)
- . \(|u_1\rangle = (1\:0),|u_2\rangle=(0\:1)\) \(|v_1\rangle = (1\:0\:0),|v_2\rangle=(0\:1\:0),|v_3\rangle=(0\:0\:1)\) Multiply top number by each of the other basis’s elements, then go down one number in the first and repeat \(|u_i\rangle\otimes|v_j\rangle=\begin{pmatrix} u_{i,1}v_{j,1}\\u_{i,1}v_{j,2}\\u_{i,1}v_{j,3}\\u_{i,2}v_{j,1}\\u_{i,2}v_{j,2}\\u_{i,2}v_{j,3}\end{pmatrix}\) \(|u_1\rangle\otimes|v_1\rangle=\begin{pmatrix} 1\\0\\0\\0\\0\\0\end{pmatrix}\) \(|u_2\rangle\otimes|v_1\rangle=\begin{pmatrix} 0\\0\\0\\1\\0\\0\end{pmatrix}\) \(|u_2\rangle\otimes|v_3\rangle=\begin{pmatrix} 0\\0\\0\\0\\0\\1\end{pmatrix}\)