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Title+Summary	Name	Date
[QUE/QM-10012] Node id: 2781page In the position representation a function of \(\vec{r}\), \(f(\vec{r})\), can be thought of as an operator \(\hat{f}\): \[ \hat{f} \psi(\vec{r}) = f(\vec{r}) \psi(\vec{r}). \] Use spherical harmonics, \(Y_{\ell m}(\theta, \phi)\) to define operators \(\widehat{Y}_{\ell m}\) in the above sense. Use properties of spherical harmonics and known expressions for angular momentum operators in polar coordinates to show \begin{eqnarray} [L_z, \widehat{Y}_{\ell m}] &=& m\hbar\, \widehat{Y}_{\ell m}, \nonumber\\{} [L_{\pm}, \widehat{Y}_{\ell m}] &=& \sqrt{\ell(\ell+1) - m(m\pm1)}\,\hbar\,\widehat{Y}_{\ell\pm 1 m}.\nonumber \end{eqnarray}	kapoor	22-04-15 20:04:41	n
[QUE/QM-10001] Node id: 2762page Find the momentum space wave function if the coordinate space wave function is $\psi(x) = C \exp(-\|x\|/L)$ $\psi(x) = C \exp(-x^2/2\alpha^2)$	kapoor	22-04-11 13:04:22	n
[QUE/QM-10002] Node id: 2763page 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 $m=2,1,0,-1,-2$.	kapoor	22-04-11 13:04:45	n
[QUE/QM-10013] Node id: 2780page What tells you that $\|\langle x\|\psi\rangle\|^2$ gives the probability for position? A particle has definite momentum $p_0$, what is the differential equation satisfied by its coordinate space wave function? WHY ? A particle has definite position $x_0$, what is the differential equation satisfied by its momentum space wave function? WHY ? Outline the sequence of steps showing that the coordinate and momentum wave functions should be Fourier transforms of each other.	kapoor	22-04-11 13:04:31	n
[QUE/QM-10011] Node id: 2782page Show that the parity operator $P$ defined by $$ P \psi(\vec{r}) = \psi(-\vec{r}) $$ obeys the commutation relations. $[ \hat{P}, \hat{x} ]_+ =0$ $[ \hat{P}, \hat{p}_x ] _+ =0$ $[ \hat{P}, \hat{x}^2 ] =0$ $[ \hat{P}, \hat{\vec{p}}^{2}] =0$	kapoor	22-04-11 13:04:53	n
[QUE/QM-10010] Node id: 2783page 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 ]}	kapoor	22-04-11 13:04:15	n
[QUE/QM-10008] Node id: 2784page Compute $$U(a)\ \vec{r} \ U^\dagger(a) $$ where $$U(a) = \exp(-i\vec{a}\cdot \hat{\vec{p}})$$ where $\vec{a}$ are numbers.	kapoor	22-04-11 13:04:58	n
[QUE/QM-10007] Node id: 2785page Show that $\displaystyle \left[ \hat{p}_x, F(\vec{r}) \right] = -i \hbar {\partial F\over \partial x} $ $\displaystyle \left[ \hat{x}, G(\vec{p}) \right] = -i \hbar {\partial G\over \partial p_x}$	kapoor	22-04-11 13:04:33	n
[QUE/QM-10006] Node id: 2786page Express the following operators in terms of $a$ and $a^\dagger$. $$ (i) \ \hat{x}\qquad (ii)\ \hat{p}\qquad (iii)\ \hat{x}^2 \qquad (iv)\ \hat{p}^2 $$ Using the properties of operators $a$ and $a^\dagger$ compute the $m n$ matrix elements of the four operators given in part (a) in the harmonic oscillator basis. What answer do you expect for the matrix elements of the Hamiltonian operator $$ \hat{H} = { \hat{p}^2\over 2m} +{1\over2}m \omega^2 \hat{x}^2 $$ Using the answers obtained in part (b) check if your guess is correct.	kapoor	22-04-11 13:04:06	n
[QUE/QM-10005] Node id: 2787page Find the matrices representing the operators $J_+, J_- \text{and} J_z$ in the basis $\|jm\rangle.$ Use your answers in part (a) to find the matrices for the operators $J_x$ and $J_y$. What answer do you expect for the matrix for $J^2$? Check if your guess is correct or not by computing the matrix for $L^2$ using the matrices found above.	kapoor	22-04-11 13:04:54	n
[QUE/QM-10004] Node id: 2788page Express the operators $a$ and $a^\dagger$ defined by $$ a = {( p -i m\omega x) \over \sqrt{2m\omega \hbar}}, \qquad a^\dagger = {( p +i m\omega x) \over \sqrt{2m\omega \hbar}} $$ in the co-ordinate representation and solve the equation $$ a \psi(x) = 0 $$ to determine the ground state wave function. Applying $a^\dagger$ on the ground state wave function, find the first two excited state eigen functions for the harmonic oscillator. Normalize the ground state and the two excited state eigen functions found above.	kapoor	22-04-11 13:04:38	n
[QUE/QM-10003] Node id: 2789page 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} \samepage{for the orbital angular momentum operators and the properties of thelladder\\operators, $L^\pm$, and construct expressions for $Y_{lm}(\theta,\phi)$ for $l=2$ and $m=2,1,0,-1,-2$.}	kapoor	22-04-11 13:04:36	n

In the position representation a function of $\vec{r}$, $f(\vec{r})$, can be thought of as an operator $\hat{f}$: \[ \hat{f} \psi(\vec{r}) = f(\vec{r}) \psi(\vec{r}). \] Use spherical harmonics, $Y_{\ell m}(\theta, \phi)$ to define operators $\widehat{Y}_{\ell m}$ in the above sense. Use properties of spherical harmonics and known expressions for angular momentum operators in polar coordinates to show \begin{eqnarray} [L_z, \widehat{Y}_{\ell m}] &=& m\hbar\, \widehat{Y}_{\ell m}, \nonumber\\{} [L_{\pm}, \widehat{Y}_{\ell m}] &=& \sqrt{\ell(\ell+1) - m(m\pm1)}\,\hbar\,\widehat{Y}_{\ell\pm 1 m}.\nonumber \end{eqnarray}

Find the momentum space wave function if the coordinate space wave function is

$\psi(x) = C \exp(-|x|/L)$
$\psi(x) = C \exp(-x^2/2\alpha^2)$

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 $m=2,1,0,-1,-2$.

What tells you that $|\langle x|\psi\rangle|^2$ gives the probability for position?
A particle has definite momentum $p_0$, what is the differential equation satisfied by its coordinate space wave function? WHY ?
A particle has definite position $x_0$, what is the differential equation satisfied by its momentum space wave function? WHY ?
Outline the sequence of steps showing that the coordinate and momentum wave functions should be Fourier transforms of each other.

Show that the parity operator $P$ defined by $$ P \psi(\vec{r}) = \psi(-\vec{r}) $$ obeys the commutation relations.

$[ \hat{P}, \hat{x} ]_+ =0$
$[ \hat{P}, \hat{p}_x ] _+ =0$
$[ \hat{P}, \hat{x}^2 ] =0$
$[ \hat{P}, \hat{\vec{p}}^{2}] =0$

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 ]}

Compute $$U(a)\ \vec{r} \ U^\dagger(a) $$ where $$U(a) = \exp(-i\vec{a}\cdot \hat{\vec{p}})$$ where $\vec{a}$ are numbers.

Show that

$\displaystyle \left[ \hat{p}_x, F(\vec{r}) \right] = -i \hbar {\partial F\over \partial x} $
$\displaystyle \left[ \hat{x}, G(\vec{p}) \right] = -i \hbar {\partial G\over \partial p_x}$

Express the following operators in terms of $a$ and $a^\dagger$. $$ (i) \ \hat{x}\qquad (ii)\ \hat{p}\qquad (iii)\ \hat{x}^2 \qquad (iv)\ \hat{p}^2 $$
Using the properties of operators $a$ and $a^\dagger$ compute the $m n$ matrix elements of the four operators given in part (a) in the harmonic oscillator basis.
What answer do you expect for the matrix elements of the Hamiltonian operator $$ \hat{H} = { \hat{p}^2\over 2m} +{1\over2}m \omega^2 \hat{x}^2 $$ Using the answers obtained in part (b) check if your guess is correct.

Find the matrices representing the operators $J_+, J_- \text{and} J_z$ in the basis $|jm\rangle.$
Use your answers in part (a) to find the matrices for the operators $J_x$ and $J_y$.
What answer do you expect for the matrix for $J^2$? Check if your guess is correct or not by computing the matrix for $L^2$ using the matrices found above.

Express the operators $a$ and $a^\dagger$ defined by $$ a = {( p -i m\omega x) \over \sqrt{2m\omega \hbar}}, \qquad a^\dagger = {( p +i m\omega x) \over \sqrt{2m\omega \hbar}} $$ in the co-ordinate representation and solve the equation $$ a \psi(x) = 0 $$ to determine the ground state wave function.
Applying $a^\dagger$ on the ground state wave function, find the first two excited state eigen functions for the harmonic oscillator.
Normalize the ground state and the two excited state eigen functions found above.

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

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