Team Quanta presents all possible short questions of BS Physics book Quantum Mechanics-I’s Chapter#5: Angular Momentum.
Q.5.1 what is importance of angular momentum in quantum physics?
Answer: Angular momentum is just as important in quantum physics as in classical physics. It is very useful for studying the dynamical of systems that move under influence of spherically symmetric or central potentials, as in classical physics the orbits angular momentum of these systems is conserved. Angular momentum plays critical role in description of molecular rotations, the motion of electrons in atoms and motion of nucleons in nuclei.
Q.5.2 what is expression of orbital angular momentum in spherical polar coordinates?
Answer: Angular momentum is,
$$\vec{L}=-i\hbar r\hat{r}\times\Delta$$
$$\Rightarrow\vec{L}=-i\ \hbar r\hat{r}\times\left(\hat{r}\frac{\partial}{\partial r}+\frac{\hat{\theta}}{r}\ \frac{\partial}{\partial\theta}+\frac{\hat{\varphi}}{rsin\theta}\ \frac{\partial}{\partial\varphi}\right)$$
$$\Rightarrow\vec{L}=-i\hbar\left(\hat{\varphi}\frac{\partial}{\partial\theta}-\frac{\hat{\theta}}{sin\theta}\ \frac{\partial}{\partial\varphi}\right)$$
Q.5.3 Write Eigen values and simultaneous Eigen functions of L2 & Lz ( do not derive ).
Answer: Eigen value equations of L2 & Lz are
$$L^2Y_{lm}\left(\theta,\ \varphi\right)=l\left(l+1\right) \hbar^2 Y_{lm}\left(\theta,\ \varphi\right)$$ & $$L_z\ Y_{lm}\left(\theta,\ \varphi\right)=m \hbar Y_{lm}\left(\theta,\ \varphi\right)$$
Where $Y_{lm}\left(\theta,\ \varphi\right)$ are simultaneous Eigen functions of L2 & Lz .
Q.5.4 Verify that spherical harmonics $Y_{lm}\left(\theta,\ \varphi\right)$ are Eigen states of parity operator with Eigen value $\left(-1\right)^l$
Answer: Here,
$$\hat{P}Y_{lm}\left(\theta,\ \varphi\right)=Y_{lm}\left(\pi-\theta,\ \varphi+\pi\ \ \right)$$
$$\Rightarrow\hat{P}Y_{lm}\left(\theta,\ \varphi\right)=\left(-1\right)^m\sqrt{\frac{2l+1}{4\pi}\ \frac{\left(l-m\right)!}{\left(l+m\right)!}}\ P_l^m\left(cos\left(\pi-\theta\right)\right)e^{im\left(\pi+\theta\right)}$$
$$\Rightarrow\hat{P}Y_{lm}\left(\theta,\ \varphi\right)=\left(-1\right)^lY_{lm}\left(\theta,\ \varphi\right)$$
Q.5.5 what are possible values of m that can be measured for j =3/2.
Answer: Since j is half-odd integral, so m = 0 is not allowed. So
$$m=-\frac{3}{2},\ -\frac{1}{2},\ \frac{3}{2},\ \frac{1}{2}$$
Q.5.6 Prove that components of orbital angular momentum are hermitian.
Hint: We have to prove that,
$${\hat{L}}_x^\dag=\ {\hat{L}}_x,\ \ \ \ \ \ \ {\hat{L}}_y^\dag={\hat{L}}_y,\ \ \ \ \ \ {\hat{L}}_z^\dag={\hat{L}}_z$$
