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Prove the following commutation relation \begin{equation} [L_z, (L_+)^n] =  n \hbar (L_+)^n\end{equation}

Compute average of kinetic and potential energies     \[  \text{K.E.} = \frac{p^2}{2m},\qquad \text{P.E.}=  \frac{1}{2}m\omega^2x^2.\]   in the \(n^\text{th}\) excited state and show that the the average of total   energy is \(\frac{1}{2}\hbar \omega\).

Let $Y_{\ell m}(\theta,\phi)$ denote the  simultaneous normalized eigenfunctions of $L^2$ and $L_z$ operators. Use the properties of the ladder operators, $L_\pm$, and construct the expressions for $Y_{lm}(\theta,\phi)$ for $l=2$ and $m=2,1,0,-1,-2$.

1. Note that $Y_{\ell\ell}(\theta,\phi)$ satisfies\begin{eqnarray} L_z Y_{\ell\ell}(\theta,\phi) &=& \ell \hbar Y_{\ell\ell}(\theta, \phi)\nonumber\\ L_+ Y_{\ell\ell}(\theta,\phi) &=& 0. \nonumber\end{eqnarray}  Set up the above differential equations and solve them using separation of  variables and find (normalized) $Y_{2,2}$.
2. Next apply $L_-$ repeatedly on \(Y_{2,2}\) and use $$L_- Y_{\ell, m} = \sqrt{\ell(\ell + 1)-m(m - 1)}\,  \hbar\,Y_{\ell(m-1)}$$ to successively construct $Y_{2,m}$ for other values $m =  1, 0,-1,-2$.
3. Normalize your answers and compare them with known expressions of $Y_{2m}(\theta,\phi), m=-2,-1,0,1,2.$

Hint:
Use coordinate space representation  for angular mormentum operators.

The the exact zero point energy of a system  several uncoupled oscillators, all having frequency \(\omega_0\), is given to be \(314.5 \hbar \omega_0\). What is the energy of first excited state?

Compute average value \(\langle{n}|{q^4}|\rangle\) of \(q^4\) in the \(n^\text{th}\)   energy state of  harmonic oscillator.

The potential energy of two protons in hydrogen molecule ion in a model is given below \begin{eqnarray}
   V(x) &=& |E_1| f(x) \\
   f(x) &=& - 1 + \frac{2}{x}\left[ \frac{(1-(2/3)x^2) e^{-x} + (1+x) e^{-2x}}
   {1+(1+x+x^2/3) e^{-x} }\right], \qquad x=R/a
\end{eqnarray}
$E_1= 13.6 \text{ eV}$ is the ground state energy of H atom and $a$ is the Bohr  radius $\hbar^2/me^2$. The graph of this  function $f(x)$ is reproduced below. Find numerical values of the bond length in ${A^o}$, the zero point energy and spacing of vibrational spectrum, both energies in electron volts.

NoteThe expression for $V(x)$ is taken from an approximate variational calculation of energy of the H molecule ion in Born Oppenheimer approximation.

The spherical harmonics $Y_{lm}(\theta,\phi)$ are normalized simultaneous eigenfunctions of $L^2$ and $L_z$ operators. Use the co-ordinate space expressions

\begin{eqnarray*}
L_x &=& i\hbar \Big( \sin\phi {\partial\over \partial\theta } + \cot \theta
 \cos\phi{\partial\over \partial \phi} \Big)\\
L_y &=& i\hbar \Big(-\cos\phi {\partial\over \partial\theta } + \cot \theta
 \sin\phi{\partial\over \partial \phi} \Big)\\
L_z &=& -i\hbar {\partial\over\partial \phi}
\end{eqnarray*}

Note : For the orbital angular momentum operators and the properties of the ladder operators, $L^\pm$, and construct expressions for $Y_{lm}(\theta,\phi)$ for
$l=2$ and $m=2,1,0,-1,-2$.

     Let $V(r)$ be a function of $r$ alone and independent of $\theta$,  and   $\phi $. Show that the angular momentum operators $ L_x, L_y$ and  $L_z$ commute with $\hat{V}(r).$     

{[ Hint: Derive expressions for angular momentum operators in $r, \theta$             and $\phi$ variables ]}