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Title+Summary	Name	Date
[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
[QUE/QM-07015] Node id: 2801page $\newcommand{\ket}[1]{\|#1\rangle}$ {Let \(\ket{E_1},\ket{E_2}\) denote normalized energy eigenstates with energies \(E_1\ne E_2\). Let \(\psi\) be the superposition \[ \ket{\psi} = a\ket{E_1} + b\ket{E_2},\] \(a,b\) are complex constants. Obtain an expression for the uncertainty in energy \((\Delta E)_\psi\)in the state \(\ket{\psi}\). Find all conditions so that \(\Delta E\) may be zero, and interpret the answers you get.}	kapoor	22-04-11 13:04:52	n
[QUE/QM-07014] Node id: 2802page	kapoor	22-04-11 13:04:13	n
[QUE/QM-07013] Node id: 2803page Show that if an operator commutes with to components of angular momentum, it commutes with the third component as well. {Daniel F. Styer}	kapoor	22-04-11 13:04:19	n
[QUE/QM-07012] Node id: 2804page Consider the space of square integrable functions on a plane. \[ \iint dx\,dy \|\psi(x,y)\|^2 < \infty.\] Define radial and angular momenta operators \(\hat{p}_r, \hat{P}_\theta\) on the subset of functions satisfying \[\psi(r,\theta+2\pi) = \psi(r,\theta), \qquad \psi(r, \theta)\|_{r=0} = \psi(r,\theta)\|_{r\to \infty} =0 .\] Show that \({P}_{\theta}= -i\hbar\frac{\partial}{\partial\theta}\) satisfies \[\Big(\phi(r,\theta), \hat{P}_\theta \psi(r,\theta)\Big) = \Big(\hat{P}_\theta\phi(r,\theta), \psi(r,\theta)\Big) \] and is, therefore, a hermitian operator. Find the hermitian conjugate of the operator \(\hat{p}_r\equiv-i\hbar\frac{\partial}{\partial r}\). Show that \(\hat{p}_r\) is not a hermitian operator. Find a hermitian operator \(\hat{P}_r\) that may represent radial momentum in two dimensions. Consider the classical free Hamiltonian \( H_{cl} = \frac{P_r^2}{2m} + \frac{P_\theta^2}{2mr^2}.\) Replace the classical momenta \(P_r, P_\theta\) by corresponding hermitian momentum operators \(\hat{P}_r, \hat{P}_\theta\). Compare your answer for the operator so obtained with the free particle Schr\"{o}dinger Hamiltonian \[\widehat{H}_0 = -\frac{\hbar^2}{2m}\Big(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y}\Big).\] and give your comments.	kapoor	22-04-11 12:04:28	n
[QUE/QM-07011] Node id: 2805page A physical system has dynamical variables $X,Y,Z$ represented by the $2\times2$ Pauli matrices $\sigma_x,\sigma_y,\sigma_z$. Compute the average value of $\ell X + m Y + n Z $ in any one state in which $a X + b Y + c Z $ has a definite value. Assume $\ell^2 + m^2 +n^2=1$ and $a^2 + b^2 + c^2=1$.	kapoor	22-04-11 12:04:57	n
[QUE/QM-07009] Node id: 2807page A physical system has dynamical variables $X,Y,Z$ represented by the $2\times2$ Pauli matrices $\sigma_x,\sigma_y,\sigma_z$. Compute the average values of the following operators in the specified states. The average of $X$ in the state represented by the vector $\begin{pmatrix}1 \\ 2\end{pmatrix}.$ The average value of $Y+Z$ in the state in which the variable $X$ has a definite value -1 for $X$.	kapoor	22-04-11 12:04:21	n
[QUE/QM-07008] Node id: 2808page A physical system has dynamical variables $X,Y,Z$ represented by the $2\times2$ Pauli matrices $\sigma_x,\sigma_y,\sigma_z$. In each of the following cases find out if the dynamical variables can be measured simultaneously or not. A physical system has dynamical variables $X,Y,Z$ represented by the $2\times2$ Pauli matrices $\sigma_x,\sigma_y,\sigma_z$. In each of the following cases find out if the dynamical variables can be measured simultaneously or not. $X $ and $Y$ $Y$ and $Z$ $Y^2$ and $Z$.	kapoor	22-04-11 12:04:19	n
[QUE/QM-07007] Node id: 2809page A physical system has dynamical variables $X,Y,Z$ represented by the $2\times2$ Pauli matrices $\sigma_x,\sigma_y,\sigma_z$. Compute the average values and uncertainties of $X,Y$ and $Z $ in a state represented by $$ \chi = \begin{pmatrix} 1 + 2i \\ 1-3i \end{pmatrix} $$	kapoor	22-04-11 12:04:06	n
[QUE/QM-07006] Node id: 2810page Use the uncertainty principle to estimate the ground state energy of harmonic oscillator.	kapoor	22-04-10 18:04:07	n
[QUE/QM-16009] Node id: 2820page Show that the average value of kinetic energy for a particle in one dimension having the wave function \(\psi(x)\) is \[ \langle \text{K.E.}\rangle = \frac{\hbar^2}{2m}\int_{-\infty} ^ \infty \|\psi(x)\|^2\, dx.\] Obtain a similar formula for the average of kinetic energy in three dimensions for a particle if the wave function \(\psi(r)\) is independent of polar coordinates \(\theta, \phi\). Can one write down a similar result for the general case in polar coordinates when the wave fucntion depends on all the three variables \(r,\theta,\phi\)?	kapoor	22-04-10 18:04:04	n
[QUE/QM-16008] Node id: 2821page Express the angular momentum operators \begin{eqnarray} \hat{L}_x &=& -i\hbar{\hat{y} \frac{\partial}{\partial z} - \hat{z} \frac{\partial}{\partial y} }\label{EQ01}\\ \hat{L}_y &=& -i\hbar{\hat{z} \frac{\partial}{\partial x} - \hat{x} \frac{\partial}{\partial z} }\label{EQ02}\\ \hat{L}_z &=& -i\hbar{\hat{x} \frac{\partial}{\partial y} - \hat{y} \frac{\partial}{\partial x} }\label{EQ03} \end{eqnarray} in polar coordinates and show \begin{eqnarray} \hat{L}_x &=& i\hbar \left(\sin\phi \frac{\partial}{\partial \theta} +\cot\theta\cos\phi\frac{\partial}{\partial \phi} \right)\label{EQ04}\\ \hat{L}_x &=& i\hbar \left(\sin\phi \frac{\partial}{\partial \theta} +\cot\theta\cos\phi\frac{\partial}{\partial \phi} \right)\label{EQ05}\\ \hat{L}_z &= & i\hbar \frac{\partial }{\partial \phi}\label{EQ06} \end{eqnarray} Use the result of the previous part and show that The operator $\vec{L}^2$ in spherical polar coordinates is given by \begin{equation} \vec{L}^2 = \hat{L}_x^2 +\hat{L}_y^2 + \hat{L}_z^2 \label{EQ07} \end{equation} takes the form \begin{equation}\label{EQ08} \vec{L}^2 = -\hbar^2\left[ \frac{1}{\sin\theta} \frac{\partial}{\partial\theta}\left(\sin\theta \frac{\partial}{\partial\theta} \right) + \frac{1}{\sin^2\theta}\frac{\partial^2}{\partial \phi^2} \right] \end{equation}	kapoor	22-04-10 18:04:26	n
[QUE/QM-16010] Node id: 2819page How much energy in eV will be required to ionize a H atom which is in the excited state \(n=3\)? Use calculator to get the answer upto significant number of places. Take the value of the Rydberg constant to be \(R = 109737.3177 \mbox{cm}^{-1}\)	kapoor	22-04-10 11:04:07	n
[QUE/QM-16011] Node id: 2818page The wave function of a particle has the form \begin{equation} \psi(r\theta,\phi) = \chi(r)( A\cos^7\theta + B \cos^3\theta + C \cos2\phi) \end{equation} where \(\chi(r)\) is radial part of the wave function dependent only on \(r\). A measurement of \(\vec{L}^2\) and \(L_z\) is made. What values do you expect for \(\vec{L}^2\)? for \(L_z\)?	kapoor	22-04-10 11:04:01	n

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

$\newcommand{\ket}[1]{|#1\rangle}$

{Let $\ket{E_1},\ket{E_2}$ denote normalized energy eigenstates with energies $E_1\ne E_2$. Let $\psi$ be the superposition \[ \ket{\psi} = a\ket{E_1} + b\ket{E_2},\] $a,b$ are complex constants. Obtain an expression for the uncertainty in energy $(\Delta E)_\psi$in the state $\ket{\psi}$. Find all conditions so that $\Delta E$ may be zero, and interpret the answers you get.}

Show that if an operator commutes with to components of angular momentum, it commutes with the third component as well.

{Daniel F. Styer}

Consider the space of square integrable functions on a plane. \[ \iint dx\,dy |\psi(x,y)|^2 < \infty.\] Define radial and angular momenta operators $\hat{p}_r, \hat{P}_\theta$ on the subset of functions satisfying \[\psi(r,\theta+2\pi) = \psi(r,\theta), \qquad \psi(r, \theta)|_{r=0} = \psi(r,\theta)|_{r\to \infty} =0 .\]

Show that ${P}_{\theta}= -i\hbar\frac{\partial}{\partial\theta}$ satisfies \[\Big(\phi(r,\theta), \hat{P}_\theta \psi(r,\theta)\Big) = \Big(\hat{P}_\theta\phi(r,\theta), \psi(r,\theta)\Big) \] and is, therefore, a hermitian operator.
Find the hermitian conjugate of the operator $\hat{p}_r\equiv-i\hbar\frac{\partial}{\partial r}$. Show that $\hat{p}_r$ is not a hermitian operator.
Find a hermitian operator $\hat{P}_r$ that may represent radial momentum in two dimensions.
Consider the classical free Hamiltonian $ H_{cl} = \frac{P_r^2}{2m} + \frac{P_\theta^2}{2mr^2}.$ Replace the classical momenta $P_r, P_\theta$ by corresponding hermitian momentum operators $\hat{P}_r, \hat{P}_\theta$. Compare your answer for the operator so obtained with the free particle Schr\"{o}dinger Hamiltonian \[\widehat{H}_0 = -\frac{\hbar^2}{2m}\Big(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y}\Big).\] and give your comments.

A physical system has dynamical variables $X,Y,Z$ represented by the $2\times2$ Pauli matrices $\sigma_x,\sigma_y,\sigma_z$.
Compute the average value of $\ell X + m Y + n Z $ in any one state in which $a X + b Y + c Z $ has a definite value. Assume $\ell^2 + m^2 +n^2=1$ and $a^2 + b^2 + c^2=1$.

     A physical system has dynamical variables $X,Y,Z$ represented by the
     $2\times2$ Pauli matrices $\sigma_x,\sigma_y,\sigma_z$.

      Compute the average values of the following operators in the specified
      states.

The average of $X$ in the state represented by the vector $\begin{pmatrix}1 \\ 2\end{pmatrix}.$
The average value of $Y+Z$ in the state in which the variable $X$ has a definite value -1 for $X$.

A physical system has dynamical variables $X,Y,Z$ represented by the $2\times2$ Pauli matrices $\sigma_x,\sigma_y,\sigma_z$.

In each of the following cases find out if the dynamical variables can be measured simultaneously or not.

A physical system has dynamical variables $X,Y,Z$ represented by the $2\times2$ Pauli matrices $\sigma_x,\sigma_y,\sigma_z$. In each of the following cases find out if the dynamical variables can be measured simultaneously or not.

$X $ and $Y$
$Y$ and $Z$
$Y^2$ and $Z$.

A physical system has dynamical variables $X,Y,Z$ represented by the $2\times2$ Pauli matrices $\sigma_x,\sigma_y,\sigma_z$.

Compute the average values and uncertainties of $X,Y$ and $Z $ in a state represented by
$$ \chi = \begin{pmatrix} 1 + 2i \\ 1-3i \end{pmatrix} $$

Use the uncertainty principle to estimate the ground state energy of harmonic oscillator.

Show that the average value of kinetic energy for a particle in one dimension having the wave function $\psi(x)$ is \[ \langle \text{K.E.}\rangle = \frac{\hbar^2}{2m}\int_{-\infty} ^ \infty |\psi(x)|^2\, dx.\]
Obtain a similar formula for the average of kinetic energy in three dimensions for a particle if the wave function $\psi(r)$ is independent of polar coordinates $\theta, \phi$.
Can one write down a similar result for the general case in polar coordinates when the wave fucntion depends on all the three variables $r,\theta,\phi$?

Express the angular momentum operators \begin{eqnarray} \hat{L}_x &=& -i\hbar{\hat{y} \frac{\partial}{\partial z} - \hat{z} \frac{\partial}{\partial y} }\label{EQ01}\\ \hat{L}_y &=& -i\hbar{\hat{z} \frac{\partial}{\partial x} - \hat{x} \frac{\partial}{\partial z} }\label{EQ02}\\ \hat{L}_z &=& -i\hbar{\hat{x} \frac{\partial}{\partial y} - \hat{y} \frac{\partial}{\partial x} }\label{EQ03} \end{eqnarray} in polar coordinates and show \begin{eqnarray} \hat{L}_x &=& i\hbar \left(\sin\phi \frac{\partial}{\partial \theta} +\cot\theta\cos\phi\frac{\partial}{\partial \phi} \right)\label{EQ04}\\ \hat{L}_x &=& i\hbar \left(\sin\phi \frac{\partial}{\partial \theta} +\cot\theta\cos\phi\frac{\partial}{\partial \phi} \right)\label{EQ05}\\ \hat{L}_z &= & i\hbar \frac{\partial }{\partial \phi}\label{EQ06} \end{eqnarray}
Use the result of the previous part and show that The operator $\vec{L}^2$ in spherical polar coordinates is given by \begin{equation} \vec{L}^2 = \hat{L}_x^2 +\hat{L}_y^2 + \hat{L}_z^2 \label{EQ07} \end{equation} takes the form \begin{equation}\label{EQ08} \vec{L}^2 = -\hbar^2\left[ \frac{1}{\sin\theta} \frac{\partial}{\partial\theta}\left(\sin\theta \frac{\partial}{\partial\theta} \right) + \frac{1}{\sin^2\theta}\frac{\partial^2}{\partial \phi^2} \right] \end{equation}

How much energy in eV will be required to ionize a H atom which is in the excited state $n=3$? Use calculator to get the answer upto significant number of places. Take the value of the Rydberg constant to be $R = 109737.3177 \mbox{cm}^{-1}$

The wave function of a particle has the form \begin{equation*} \psi(r\theta,\phi) = \chi(r)( A\cos^7\theta + B \cos^3\theta + C \cos2\phi) \end{equation*} where $\chi(r)$ is radial part of the wave function dependent only on $r$. A measurement of $\vec{L}^2$ and $L_z$ is made. What values do you expect for $\vec{L}^2$? for $L_z$?

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