# 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, $$\mu_B=\frac{q\hbar}{2m}$$.

1. 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 $$\vec{M}=\frac{q}{2m_e}\vec{J} = \vec{M_S} + \vec{M_L}$$ $$S_z=\pm\frac{\hbar}{2}$$
2. Anomalous Zeeman Effect
3. A given particle is characterized by a unique value of $$s$$
4. $$[\vec{L}^2,\vec{S}^2] = 0$$ (They act on different spaces, different quantum numbers)
5. 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 $$|u_i\rangle\otimes|v_j\rangle=\begin{pmatrix} u_{i,1}v_{j,1}\\u_{i,1}v_{j,2}\\u_{i,1}v_{j,3}\\u_{i,2}v_{j,1}\\u_{i,2}v_{j,2}\\u_{i,2}v_{j,3}\end{pmatrix}$$ $$|u_1\rangle\otimes|v_1\rangle=\begin{pmatrix} 1\\0\\0\\0\\0\\0\end{pmatrix}$$ $$|u_2\rangle\otimes|v_1\rangle=\begin{pmatrix} 0\\0\\0\\1\\0\\0\end{pmatrix}$$ $$|u_2\rangle\otimes|v_3\rangle=\begin{pmatrix} 0\\0\\0\\0\\0\\1\end{pmatrix}$$

Created: 2024-05-30 Thu 21:20

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