Spin

Stern-Gerlach Experiment

Bohr Magneton

Bohr magneton is a physical constant that is defined as: ${\mu }_B:=\frac{e\hslash }{2m_e}=9.27\times 10^{-24}J/T$

Def Stern-Gerlach Experiment The Stern-Gerlach experiment is the experiment showing that the

Intrinsic Spin

Definition

Def Spin Spin is an intrinsic property of particles that is not associated with any spatial coordinates. It is a form of angular momentum that is not due to the motion of the particle. The spin quantum number $s$ is a half-integer or integer value that determines the magnitude of the spin angular momentum.

Def Spin Angular Momentum Operator The spin angular momentum operators are denoted by ${\hat{S}}_x$, ${\hat{S}}_y$, and ${\hat{S}}_z$,

Prop The spin angular momentum operators satisfy the same commutation relations as the orbital angular momentum operators: $\begin{array}{r}[][{\hat{S}}_x,{\hat{S}}_y]=i\hslash {\hat{S}}_z\\ [{\hat{S}}_y,{\hat{S}}_z]=i\hslash {\hat{S}}_x\\ [{\hat{S}}_z,{\hat{S}}_x]=i\hslash {\hat{S}}_y\end{array}$ Def Spin Ladder Operators The spin ladder operators are defined as: ${\hat{S}}_{+}={\hat{S}}_x+i{\hat{S}}_y,{\hat{S}}_{-}={\hat{S}}_x-i{\hat{S}}_y$

Prop The action of the operators on states of definite spin is given by: $\begin{array}{rl}& {\hat{S}}^2|s,m_s⟩=s(s+1){\hslash }^2|s,m_s⟩\\ & {\hat{S}}_z|s,m_s⟩=m_s\hslash |s,m_s⟩\\ & {\hat{S}}_{+}|s,m_s⟩=\hslash \sqrt{s(s+1)-m_s(m_s+1)}|s,m_s+1⟩\\ & {\hat{S}}_{-}|s,m_s⟩=\hslash \sqrt{s(s+1)-m_s(m_s-1)}|s,m_s-1⟩\end{array}$

Definition

Def Eigenspinors The eigenfunctions of the spin operator ${\hat{S}}_z$ are called eigenspinors. The eigenspinors of the spin angular momentum operators are denoted by ${\chi }_s^{m_s}$. e.g. In matrix representation, ${\chi }_{\frac{1}{2}}^{+\frac{1}{2}}=\left(\begin{array}{c}1\\ 0\end{array}\right)$ and ${\chi }_{\frac{1}{2}}^{-\frac{1}{2}}=\left(\begin{array}{c}0\\ 1\end{array}\right)$. Thus an arbitrary spin state for such a particle can then be represented by $\chi =a{\chi }_{\frac{1}{2}}^{+\frac{1}{2}}+b{\chi }_{\frac{1}{2}}^{-\frac{1}{2}}=\left(\begin{array}{c}a\\ b\end{array}\right)$.

Definition

Def Pauli Spin Matrices The Pauli spin matrices are defined as: ${\sigma }_x=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),{\sigma }_y=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right),{\sigma }_z=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$e.g If a particle has intrinsic spin $s=1/2$, then the spin operator can be represented by the Pauli matrices: ${\hat{S}}_x=\frac{\hslash }{2}{\sigma }_x,{\hat{S}}_y=\frac{\hslash }{2}{\sigma }_y,{\hat{S}}_z=\frac{\hslash }{2}{\sigma }_z$

The Hydrogen Atom

Proposition

Prop In spherical polar coordinates, the squared angular momentum operator can be expressed as: ${\hat{L}}^2=-{\hslash }^2\left[\frac{1}{\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial }{\partial \theta }\right)+\frac{1}{{\sin }^2\theta }\frac{{\partial }^2}{\partial {\varphi }^2}\right]$Proof Simply because ${\hat{L}}^2={\hat{L}}_x^2+{\hat{L}}_y^2+{\hat{L}}_z^2$.

Bohr Radius

In an atomic system, the Bohr radius is a natural length scale: $a_0=\frac{4\pi {ϵ}_0{\hslash }^2}{e^2{m}_e}=0.529\times 10^{-10}\mathrm{m}$

Rydberg Constant

The Rydberg constant is a scale for energy that defined as $E_{\mathrm{R}}=\frac{{\hslash }^2}{2m_e{a}_0^2}=13.6\mathrm{eV}$

Definition

Def Effective Potential The effective potential in a hydrogen system is defined as: $V_{\mathrm{eff}}=\frac{l(l+1){\hslash }^2}{2m{r}^2}-\frac{e^2}{4\pi {ϵ}_{0}r}$

Proposition

Prop The minimum of the effective potential is achieved at $r_{\min }=l(l+1)a_0$ where $a_0$ is the Bohr radius. And the minimum value of the effective potential is given by $V_{\mathrm{eff}}(r_{\min })=-\frac{E_{\mathrm{R}}}{l(l+1)}$

Theorem

Thrm Bohr Expression The eigenvalues of the hydrogen system is given by $E_{n_r,l}=-\frac{E_{\mathrm{R}}}{(n_r+l+1{)}^2}=-\frac{E_{\mathrm{R}}}{n^2}$where $n=n_r+l+1$ is defined as the principle quantum number.

Spectroscopic Notation

In spectroscopic notation, we use combination of a number and a letter to denote the principle quantum number $n$ and orbital quantum number $l$ respectively. If $l=0$, we use $s$; If $l=1$, we use $p$; If $l=2$, we use $d$. e.g.

• $2s$ stands for $n=n_r+l+1=2$, $l=0$;
• $2p$ stands for $n=n_r+l+1=2$, $l=1$;
• The atom Trusconium has $8$ protons, $9$ neutrons and $8$ electrons, then its shell structure is $1s^{2}2s^{2}2p^4$