Physics Problems & Solutions: Quantum Mechanics

Which is the following is the orbital angular momentum eigenfuction $Y_l^m (\theta,\phi )$ in a state for which the operators $L^2$ and $L_z$ have eigenvalues $6\hbar^2$ and $-\hbar$ respectively?

$\\$A.$ \hspace{5 mm} Y_2^1 (\theta,\phi) \\\\$B.$ \hspace{5 mm} Y_2^{-1} (\theta,\phi) \\\\$C.$ \hspace{5 mm} \frac{1}{\sqrt{2}}[Y_2^1 (\theta,\phi)+Y_2^{-1} (\theta,\phi)] \\\\$D.$ \hspace{5 mm} Y_2^3 (\theta,\phi) \\\\$E.$ \hspace{5 mm} Y_3^{-1} (\theta,\phi)$

(GR0177 #81)

Solution:

Orbital angular momentum: $L=\sqrt{l(l+1)} \hbar$

$\begin{aligned} L^2=l(l+1) \hbar^2&=6\hbar^2 \\l(l+1)=l^2+l&=6 \\l^2+l-6&=0 \\(l+3)(l-2)&=0 \\l&=-3;2 \end{aligned}$

Allowed values of l:

$l=\left\{\begin{matrix} $0 1 2 3 4 ...$ \\ $s p d f g ...$ \end{matrix}\right.$

l cannot be negative. So in this case, $l=2$

The z-component of L: $L_z=m_l \hbar=-\hbar\rightarrow m=-1$

Thus, $Y_l^m (\theta,\phi)=Y_2^{-1} (\theta,\phi )$

Answer: B

Physics Problems & Solutions

Quantum Mechanics - Orbital Angular Momentum

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