Angular Momentum I (Worksheet)

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Work in groups on these problems. You should try to answer the questions without referring to your textbook. If you get stuck, try asking another group for help.

Angular Momentum

We can express the quantum mechanical angular momentum using operators:

$$\hat L_x=\hat y\hat p_z - \hat z\hat p_y = -i\hbar \left(y\dfrac{\partial}{\partial z}-z\dfrac{\partial}{\partial y}\right)$$

$$\hat L_y=\hat z\hat p_x - \hat x\hat p_z = -i\hbar \left(z\dfrac{\partial}{\partial x}-x\dfrac{\partial}{\partial z}\right)$$

$$\hat L_z=\hat x\hat p_y - \hat y\hat p_x = -i\hbar \left(x\dfrac{\partial}{\partial y}-y\dfrac{\partial}{\partial x}\right)$$

$$\hat L^2=\hat L_x^2+\hat L_y^2+\hat L_z^2=-\hbar^2 \left(\dfrac{\partial ^2}{\partial x^2}+\dfrac{\partial ^2}{\partial y^2} +\dfrac{\partial ^2}{\partial z^2}\right)$$

Q1

Given these definitions, show that $\hat L^2$ and $\hat L_x$ commute.

Use definitions in Cartesian coordinates to show that $[\hat L_x,\hat L_y]=i\hbar\hat L_z$, $[\hat L_y,\hat L_z]=i\hbar\hat L_x$, $[\hat L_z,\hat L_x]=i\hbar\hat L_y$

Define two new Hermitians operators:

• $\hat L_+ = \hat L_x+i\hat L_y$ and
• $\hat L_-=\hat L_x-i\hat L_y$

Do $\hat L_+$ and $\hat L_-$ commute with $\hat L^2$? If not, what is the value of the commutator?

Do $\hat L_+$ and $\hat L_-$ commute with $\hat L_z$? If not, what is the value of the commutator?

Q2

We know that $\hat L^2Y_{lm}(\theta,\phi)=\hbar^2l(l+1)$. Thus, $Y_{lm}(\theta,\phi)$ must also be an eigenfunction of $\hat L_z$. Given that $\hat L_z=-i\hbar\dfrac{\partial}{\partial\phi}$, show that $Y_{lm}(\theta,\phi)$ is also an eigenfunction for $\hat L_z$ by applying it directly. What is the eigenvalue for $\hat L_zY_{lm}(\theta,\phi)$?

Spherical Harmonics

$Y_{0,0}(\theta,\phi)=Y_0^0(\theta,\phi)=\dfrac{1}{(4\pi )^{1/2}}$	$Y_{1,0}(\theta,\phi)=Y_1^0(\theta,\phi)=\dfrac{3}{(4\pi )^{1/2}}\cos{\theta}$
$Y_{1,1}(\theta,\phi)=Y_1^1(\theta,\phi)=\dfrac{3}{(8\pi )^{1/2}}\sin{\theta e^{i\phi}}$	$Y_{1,-1}(\theta,\phi)=Y_1^{-1}(\theta,\phi)=\dfrac{3}{(8\pi )^{1/2}}\sin{\theta e^{-i\phi}}$
$Y_{2,0}(\theta,\phi)=Y_2^0(\theta,\phi)=\dfrac{5}{(16\pi )^{1/2}}(\cos^2{\theta}-1)$
$Y_{2,1}(\theta,\phi)=Y_2^1(\theta,\phi)=\dfrac{15}{(8\pi )^{1/2}}\sin{\theta}\cos{\theta e^{i\phi}}$	$Y_{2,-1}(\theta,\phi)=Y_2^{-1}(\theta,\phi)=\dfrac{15}{(8\pi )^{1/2}}\sin{\theta}\cos{\theta e^{-i\phi}}$
$Y_{2,2}(\theta,\phi)=Y_2^2(\theta,\phi)=\dfrac{15}{(32\pi )^{1/2}}\sin^2{\theta e^{2i\phi}}$	$Y_{2,-2}(\theta,\phi)=Y_2^{-2}(\theta,\phi)=\dfrac{15}{(32\pi )^{1/2}}\sin^2{\theta e^{-2i\phi}}$
$Y_{3,0}(\theta,\phi)=Y_3^0(\theta,\phi)=\dfrac{15}{(32\pi )^{1/2}}\sin^2{\theta}$

What is the form of $\hat L_+$ in spherical polar coordinates?

What is the form of $\hat L_-$ in spherical polar coordinates?

If you apply $\hat L_+$ or $\hat L_-$ to $Y_{lm}(\theta,\phi)$, what happens?