11 - Angular Momentum

Angular Momentum Operators

Angular Momentum Operator

$$\mathbf{L}=(L_1,L_2,L_3)$$

has components $L_i$ for $i=1,2,3$ defined by

$$L_1\equiv X_2{P}_3-X_3{P}_2,L_2\equiv X_3{P}_1-X_1{P}_3,L_3\equiv X_1{P}_2-X_2{P}_1$$

Note that $L$ is known as the Orbital Angular Momentum

Alternate Form

$$L_i=\sum _{j,k=1}^3{ϵ}_{i,j,k}{X}_j{P}_k=(\mathbf{X}\wedge \mathbf{P}{)}_i$$

where

$$e_{ijk}=\{\begin{array}{ll}+1& \text{ if }ijk\text{ is an even permutation of }123\\ -1& \text{ if }ijk\text{ is an odd permutation of }123\\ 0& \text{ otherwise}\end{array}$$

Angular Momentum Commutator Relations

Angular Momentum Operator satisfies
1)

$$L_i,X_j=i\hslash \sum _{k=1}^3{ϵ}_{ijk}{X}_{ik}$$

$$[L_i,P_j]=i\hslash \sum _{k=1}^3{ϵ}_{ijk}{P}_k$$

$$[L_i,L_j]=i\hslash \sum _{k=1}^3{ϵ}_{ijk}{L}_k$$

Proof

$(1)$

$$\begin{array}{rl}[L_i,X_j]& =\sum _{m,n=1}^3{ϵ}_{imn}[X_m{P}_n,X_j]=\sum _{m,n=1}^3{ϵ}_{imn}(X_m[P_n,X_j]+[X_m,X_j]P_n)\\ & =\sum _{m,n=1}^3{ϵ}_{imn}(-i\hslash X_m{\delta }_{nj}+0)=-i\hslash \sum _{m=1}^3{ϵ}_{imj}{X}_m\\ & =i\hslash \sum _{m=1}^3{ϵ}_{ijm}{X}_m\end{array}$$

$(2)$
Similar to $(i)$

$(3)$

$$\begin{array}{rl}[L_1,L_2]& =[L_1,X_3{P}_1-X_1{P}_3]=X_3[L_1,P_1]+[L_1,X_3]P_1-X_1[L_1,P_3]-[L_1,X_1]P_3\\ & =0+i\hslash (-X_2)P_1-X_{1}i\hslash (-P_2)-0\\ & =i\hslash (X_1{P}_2-X_2{P}_1)\\ & =i\hslash L_3\end{array}$$

Angular Momentum Operators - General Form

Let $\mathbf{J}$ be any self-adjoint vector operator with components $J_i$ for $i=1,2,3$ with

$$[J_i,J_j]=i\hslash \sum _{k=1}^3{ϵ}_{ijk}{J}_k$$

then $\mathbf{J}$ is an angular momentum operator

Property of Angular Momentum Operators

Let $\mathbf{A}$ and $\mathbf{B}$ be operators with

$$[J_i,A_j]=i\hslash \sum _{k=1}^3{ϵ}_{ijk}{A}_k\text{ and }[J_i,B_j]=i\hslash \sum _{k=1}^3{ϵ}_{ijk}{B}_k$$

Then

$$[J_i,\mathbf{A}\cdot \mathbf{B}]=0$$

where $\mathbf{A}\cdot \mathbf{B}\equiv A_1{B}_1+A_2{B}_2+A_3{B}_3$

Proof

$$\begin{array}{rl}[J_i,\mathbf{A}\cdot \mathbf{B}]& =\sum _{j=1}^3[J_i,A_j{B}_j]=\sum _{j=1}^3(A_j[J_i,B_j]+[J_i,A_j]B_j)\\ & =i\hslash \sum _{j,k=1}^3{ϵ}_{ijk}(A_j{B}_k+A_k{B}_j)=0\end{array}$$

Commutative Operators of $L_i$ corollary

$L_i$ commutes with

$$\mathbf{X}\cdot \mathbf{X},\mathbf{P}\cdot \mathbf{P},\mathbf{X}\cdot \mathbf{P},\mathbf{L}\cdot \mathbf{L}$$

Total Angular Momentum

Total Angular Momentum is

$$J^2\equiv \mathbf{J}\cdot \mathbf{J}=J_1^2+J_2^2+J_3^2$$

with

$$[J^2,J_i]=0\text{ for }i=1,2,3$$

Raising and Lowering Operators for Angular Momentum

Raising and Lowering Operators for Angular Momentum

$$J_{\pm }\equiv J_1\pm i{J}_2$$

Relation of Raising and Lowering Operators for Angular Momentum Operators

$$[J^2,J_{\pm }]=0$$

$$J_{\pm }{J}_{\mp }=J^2-J_3^2\pm \hslash J_3$$

$$[J_{+},J_{-}]=2\hslash J_3$$

$$[J_3,J_{\pm }]=\pm \hslash J_{\pm }$$

Proof

$(1)$
Holds from linearity using $[J^2,J_i]=0$

$(2)$

$$J_{\pm }{J}_{\mp }=(J_1\pm i{J}_2)(J_1\mp i{J}_2)=J_1^2+J_2^2\mp i(J_1{J}_2-J_2{J}_1)=J^2-J_3^2\pm \hslash J_3$$

$(3)$

$$[J_{+},J_{-}]=J_{+}{J}_{-}-J_{-}{J}_{+}=2\hslash J_3$$

$(4)$

$$[J_3,J_{\pm }]=[J_3,J_1\pm i{J}_2]=i\hslash J_2\pm \hslash J_1=\pm \hslash J_{\pm s}$$

Eigenstates of Angular Momentum Operators

Let $\psi$ be a common eigenstate of $J^2$ and $J_3$ with

$$J^2\psi =\lambda {\hslash }^2\psi \text{ and }{J}_3\psi =m\hslash \psi$$

Then

$$J^2{J}_{\pm }\psi =\lambda {\hslash }^2{J}_{\pm }\psi$$

$$J_3{J}_{\pm }\psi =(m\pm 1)\hslash J_{\pm }\psi$$

Proof

$(1)$
By Relation of raising and lowering operators for angular momentum operators then
Applying $\psi$ to $J^2{J}_{\pm }=J_{\pm }{J}^2$ gives result

$(ii)$
By Relation of raising and lowering operators for angular momentum operators (4) then

$$J_3{J}_{\pm }\psi =(J_{\pm }{J}_3\pm \hslash J_{\pm })\psi =J_{\pm }(m\hslash \pm \hslash )\psi =(m\pm 1)\hslash J_{\pm }\psi$$

Constructed Finite-Dimensional Inner Product Space ${\mathcal{H}}_j$

For non-negative half integers $j$
Complex Vector Space ${\mathcal{H}}_j$ has dimension $2j+1\in \mathbb{N}$ with preferred basis

$$\{{\psi }_{-j},{\psi }_{-j+1},\cdots ,{\psi }_{j-1},{\psi }_j\}$$

where $J_3$ and $J_{\pm }\equiv J_1\pm i{J}_2$ on basis vector ${\psi }_m$ by Eigenstates of angular momentum operators

Trivial Representation ${\mathcal{H}}_0$

As $j=0$ then $\dim {\mathcal{H}}_0=2j+1=1$ so $H_0=\mathbb{C}$
${\mathcal{H}}_0$ is spanned by single state ${\psi }_0$ with

$$J_3{\psi }_0=0,J_{\pm }{\psi }_0=0,J^2{\psi }_0=0$$

Spin Representation ${\mathcal{H}}_{\frac{1}{2}}$

As $j=\frac{1}{2}$ then $\dim {\mathcal{H}}_{\frac{1}{2}}=2j+1=2$ so $m=\pm \frac{1}{2}$
Define ${\psi }_{\pm }={\psi }_{\pm \frac{1}{2}}$ hence

$$J_{+}{\psi }_{+}=0=J_{-}{\psi }_{-},J_{+}{\psi }_{-}=\hslash {\psi }_{+},J_{-}{\psi }_{+}=\hslash {\psi }_{-}$$

with orthonormal basis vectors ${\psi }_{\pm }$

$${\psi }_{+}=\left(\begin{array}{c}1\\ 0\end{array}\right){\psi }_{-}=\left(\begin{array}{c}0\\ 1\end{array}\right)$$

with corresponding angular momentum matrices

$$J_3=\frac{1}{2}\hslash \left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)J_{+}=\hslash \left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)J_{-}=\hslash \left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right)$$

Using definition of $J_1$ and $J_2$ in terms of $J_{\pm }$ then

$$J_1=\frac{1}{2}\hslash \left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),J_2=\frac{1}{2}\hslash \left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right)$$

Representation of Angular Momentum

Consider Vector Space $V$ with linear action of Angular Momentum Operators then

If vector space $V$ satisfies Angular momentum operators - General Form then
$V$ is a representation of angular momentum

$J^2$ in form $\{{\psi }_m\mid m=-j,-j+1,\cdots ,j-1,j\}$ satisfies

$$\begin{array}{rl}{J}_3{\psi }_m& =m\hslash {\psi }_m\\ J_{\pm }{\psi }_m& =\sqrt{(j\mp m)(j\pm m+1)}\hslash {\psi }_{m\pm 1}\end{array}$$

Spin Representation of Angular Momentum

Spin Representation of Angular Momentum is Matrix Representation of $J_i$

Pauli Matrices

Pauli Matrices are defined as

$${\sigma }_1\equiv \left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right){\sigma }_2\equiv \left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right){\sigma }_3\equiv \left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$$

where each matrix is traceless and Hermitian with

$$J_i=\frac{1}{2}\hslash {\sigma }_i\text{ for }i=1,2,3$$

and

$$[{\sigma }_i,{\sigma }_j]=2i\sum _{k=1}^3{ϵ}_{ijk}{\sigma }_k$$

Orbital Angular Momentum Spectrum

For Orbital Angular Momentum for Spectrum of Angular Momentum Operators then

$$j,m\in \mathbb{Z}$$

Generally label $j=l\in {\mathbb{Z}}^{\ge 0}$

Raising and Lowering Operators in Spherical Polar Coordinates

Raising and Lowering Operators in Spherical Polar Coordinates

$$\begin{array}{rl}{L}_{+}& =L_1+i{L}_2=\hslash e^{i\varphi }\left(\frac{\partial }{\partial \theta }+i\cot \theta \frac{\partial }{\partial \varphi }\right)\\ L_{-}& =L_1-i{L}_2=-\hslash e^{i\varphi }\left(\frac{\partial }{\partial \theta }-i\cot \theta \frac{\partial }{\partial \varphi }\right)\end{array}$$

with total angular momentum operator

$$L^2=L_1^2+L_2^2+L_3^2=-{\hslash }^2\left(\frac{{\partial }^2}{\partial {\theta }^2}+\cot \theta \frac{\partial }{\partial \theta }+\frac{1}{{\sin }^2\theta }\frac{{\partial }^2}{\partial {\varphi }^2}\right)$$

and $L_3$ is

$$L_3=-\hslash \frac{\partial }{\partial \varphi }$$

Spherical Harmonics

Function $Y_{l,m}(\theta ,\varphi )$ is called spherical harmonics where

$$Y_{l,m}(\theta ,\varphi )=P_{l,m}(\theta )e^{im\varphi }\text{ and }{P}_{l,l}(\theta )=a_l(\sin \theta {)}^l$$

and

$$Y_{l,m-1}(\theta ,\varphi )\equiv \frac{1}{\hslash \sqrt{(l+m)(l-m+1)}}{L}_{-}{Y}_{l,m}(\theta ,\varphi )$$

Normalising $a_l$ then $Y_{l,m}$ satisfy orthonormality property

$${\int }_{\theta =0}^{\theta =\pi }{\int }_{\varphi =0}^{\varphi =2\pi }\overline{Y_{l,m}(\theta ,\varphi )}{Y}_{l^{\prime },m^{\prime }}(\theta ,\varphi )\sin \theta d\theta d\varphi ={\delta }_{l,l^{\prime }}{\delta }_{m,m^{\prime }}$$

where ${\int }_{\theta =0}^{\pi }{\int }_{\varphi =0}^{2\pi }|a_l{|}^2(\sin \theta {)}^{2l}d\theta d\varphi =1$