🧠 Remember: from previous courses

• Angular momentum vector operator: $\hat {\bold L} = \hat {\bold r} \times \hat {\bold p}$ with commutations as follows

  $$ [{\rm \hat L}_x, {\rm \hat L}_y]=i\hbar {\rm \hat L}_z \qquad [{\rm \hat L}_y,{\rm \hat L}_z]=i\hbar {\rm \hat L}_x \qquad [{\rm \hat L}_z,{\rm \hat L}_x]=i\hbar {\rm \hat L}_y \qquad [ \hat {\bold L}^2, \hat {\mathrm L}_i]=0 $$

  where $\hat{\bold L}^2=\hat {\mathrm L}^2_x+ \hat {\mathrm L}^2_y+\hat {\mathrm L}^2_z$

⚙️ Properties:

• $\hat {\bold L}^2$ commutes with the Hamiltonian if the potential $V(r)$ is independent of angle
• Eigenfunctions of $\hat {\bold L}^2$ and $\rm \hat L_z$ are the spherical harmonic $Y_l^m(\theta ,\phi)$
• Eigenvalues for $\hat {\bold L}^2$ they are $\hbar ^2 l(l+1)$
  • Where $l\in \N$
• Eigenvalues for $\rm \hat L_z$ they are $\hbar m$
  • Where $m=-l,-l+1,\ldots ,l$

General properties of angular momentum

🗒️ Note: Here we will take a more general approach (ie coordinate independent) thus we will call the angular momentum vector operator $\hat {\bold J }$ with the same units as $\hbar$

💼 Case: consider the Hermitian operators $\rm \hat J_1, \hat J_2, \hat J_3$ components of $\hat {\bold J}$ where we know that there commutation relations are

$$ [{\rm \hat J}_1, {\rm \hat J}_2]=i\hbar {\rm \hat J}_3 \qquad [{\rm \hat J}_2,{\rm \hat J}_3]=i\hbar {\rm \hat J}_1 \qquad [{\rm \hat J}_3,{\rm \hat J}_1]=i\hbar {\rm \hat J}_2 $$

which we can write as $[{\rm \hat J}_i,{\rm \hat J}j]=i\hbar \sum_k \epsilon{ijk}{\rm \hat J}_k$

🗒️ Note: assumptions end and math begins here (ie the rest can be derived from the above)

• Using the operator square $\hat{\bold J}^2=\hat {\mathrm J}^2_1+ \hat {\mathrm J}^2_2+\hat {\mathrm J}^2_3$ and the commutation relations we can show

  $$ [ \hat {\bold J}^2, \hat {\mathrm J}_i]=0 $$

  🗒️ Note: for convention we chose $i=3$ as the commuting operator

🥦 Eigenvector

• We define the normalised states $|\lambda , \mu \rang$ such that

  • The eigenvalues of $\hat {\bold J} ^2$ are $\hbar^2 \lambda$ where $\lambda$ is non-negative
  • The eigenvalues of $\rm \hat J_3$ are $\hbar \mu$ where $\mu$ is real

  🗒️ Note: $\hbar^2$ and $\hbar$ are there so that $\mu$ and $\lambda$ are unitless

• Using

  • the properties of Hermitian operators: $\lang \alpha | \rm \hat A^2 |\alpha \rang = \lang \hat A \alpha | \hat A \alpha \rang \ge 0$
  • $\hbar ^2 (\lambda - \mu^2)=\lang \lambda, \mu | ({\hat { \bold J}^2} -{\rm \hat J^2_3}|\lambda, \mu \rang =\lang \lambda , \mu |( {\rm \hat J_1^2 + J_2 ^2 }) | \lambda , \mu \rang \ge 0$

  we get the relation

  $$ \boxed{|\mu| \le \sqrt{\lambda}} $$

🫁 Raising and lowering operators $\rm \hat J _\pm$

• We define them as

  $$ {\rm \hat J_+}\equiv {\rm \hat J}_1+i{\rm \hat J}2 \qquad {\rm \hat J}- \equiv {\rm \hat J}_1 -i {\rm \hat J}_2 $$

  🗒️ Note: these are not Hermitian but ${\rm \hat J}-={\rm \hat J}+^\dag$

• We can show that these follow the following commutations

  $$ [{\rm \hat J}+,{\rm \hat J}-]=2\hbar {\rm \hat J}3 \qquad [{\rm \hat J}3,{\rm \hat J}+]=\hbar {\rm \hat J}+ \qquad [{\rm \hat J}3,{\rm \hat J}-]=-\hbar {\rm \hat J}- \qquad [\hat {\bold J}^2, {\rm \hat J}\pm ]=0 $$

  🗒️ Note: since $\rm \hat J_\pm$ and $\hat {\bold J}^2$ commute, measuring one doesn't affect the result of the other

🥟 Action of raising and lowering operators on eigenstates

• To achieve this we will need to do 2 calculations

  • Defining define $\hat{\bold J}^2$ in terms of ${\rm \hat J}_\pm$

    $$ \hat{\bold J}^2=\frac 12 ({\rm \hat J}+ {\rm \hat J}- + {\rm \hat J}- {\rm \hat J}+ )+ {\rm \hat J^2_3}={\rm \hat J}+{\rm \hat J}-+{\rm \hat J_3^2}-\hbar {\rm \hat J}3={\rm \hat J}- {\rm \hat J}_+ + {\rm \hat J^2_3}+\hbar {\rm \hat J}_3 $$

  • Considering the state $\rm \hat J_+|\lambda , \mu \rang$

    $$ {\rm \hat J_3} ({\rm \hat J_+}|\lambda , \mu \rang ) = {\rm \hat J_+}{\rm \hat J}3|\lambda, \mu \rang + \hbar {\rm \hat J}+ |\lambda , \mu \rang = \hbar (\mu+1)({\rm \hat J_+}|\lambda , \mu \rang ) $$

    • where we used

    🗒️ Note: either ${\rm \hat J_+} |\lambda,\mu \rang$ is an eigenstate of ${\rm \hat J}3$ with eigenvalue $\hbar (\mu+1)$ or =$| 0\rang$ and similarly for ${\rm \hat J-} |\lambda,\mu \rang$ with eigenvalue $\hbar (\mu -1)$