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Title+Summary	Name	Date
[QUE/QM-07005] Node id: 2811page In the classical kinetic theory of gases, molecules are assumed to be point particles ( of rigid shapes) in motion. Assume the average kinetic energy of the molecules to be ${3\over 2} kT$. Using the uncertainty principle, estimate the minimum uncertainty in position of a molecule. Compare this with average molecular distance for a gas of density $\rho$ ( $\rho=N/V$ , $N=$ number of molecules and $V=$ volume ) When do the quantum mechanical effects become significant ? Estimate this temperature for (a) He gas at normal pressure (b) an electron gas at density of $2.5 \times 10^{22} $ electrons/cc.	kapoor	22-04-10 11:04:00	n
[QUE/CM-04004] Node id: 2835page $\newcommand{\PB}[1]{\{#1\}_\text{PB}}$ \(\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}\) Let $\Psi$ be a function of $f,g,h,..$ which are in turn functions of $q$ and $p$. Let $ \phi({{\bf q}, {\bf p}})$ be a function of \(\bf q,p\), Show that the Poisson bracket of $\phi$ with $\Psi$ is given by $$\PB{\phi,\Psi} =\pp[\Psi]{f}\PB{\phi,f} + \pp[\Psi]{g}\PB{\phi,g} + \pp[\Psi]{h}\PB{\phi,h} + \cdots $$	kapoor	22-04-10 11:04:34	n
[QUE/CM-04002] Node id: 2834page Let $\Psi$ be a function of $f,g,h,..$ which are in turn functions of $q$ and $p$. Let $X(q, p)$ be a function having zero Poisson brackets with $ f,g,h...$. Show that the Poisson bracket of $\Psi({f,g,h})$ with $X(q,p)$ also vanishes.	kapoor	22-04-10 11:04:53	n
[QUE/QM-16001] Node id: 2827page For each of the six potentials, shown in the \Figref{3dpot1} below, answer the following. \label{16001Q1} Write the radial Schrodinger equation in different regions of $r$ values. Write the most general solution as a linear combinations of spherical Bessel functions with appropriate arugments. Impose the necessary regularity property at the origin and at $\infty$. Impose the matching conditions at the boundary of two neighbouring regions. Considering appropriate ranges of energy separately, state if energy eigenvalues are continuous or discrete? \label{16001Q2} Whenever bound states exist for a potential, obtain the condition on the bound state energy eigenvalues. \FigBelow{0,0}{150}{220}{3dpot1}{Radial Wells for Q. \ref{16001Q1}-\ref{16001Q2}}	kapoor	22-04-10 11:04:31	n
[QUE/QM-16005] Node id: 2824page Classically orbit of a particle in a spherically symmetric potential lies in a plane. Show that the classical result is approximated by quantum solution infor \(L_z=\ell\) the limit of large angular momentum \(\ell\).	kapoor	22-04-10 11:04:20	n
[QUE/QM-16003] Node id: 2826page What qualitative differences do you expect in the spectra of the H- atom and (i) singly ionized Helium atom\\ (ii) the deuterium (iii) positronium	kapoor	22-04-10 11:04:56	n
[QUE/QM-16004] Node id: 2825page For a classical electron moving in an orbit around a nucleus with charge $Ze$ obtain a relation between the number of revolutions per second and the energy of the electron. (Use only classical mechanics). Using known quantum mechanical solution, show that in the limit of large principal quantum number \(n\), the frequency of the line emitted in a transition from $(n+1)^{\mbox{th}}$ level to the $n^{\mbox{th}}$ level is precisely the same as the frequuency obtained in part \((a)\).	kapoor	22-04-10 11:04:06	n
[QUE/QM-16006] Node id: 2823page Classically orbit of a particle in a spherically symmetric potential lies in a plane. Show that the classical result is approximated by quantum solution in for \(L_z=\ell\) the limit of large angular momentum \(\ell\).	kapoor	22-04-09 17:04:10	n
[QUE/QM-16007] Node id: 2822page the following statements correctly describes the degeneracy?\\ Eigen values are non-degenerate just as for the bound states in one dimension. Continuous eignevalues are doubly degenerate as in some one dimensional problems. Continuous eignevalues are non-degenerate as in some one dimensional problems. Infinitely degenerate as is the case for free particle in three dimension. \label{16007Q2} The degeneracy is $(2l+1)$ as expected for a general spherically symmetric potential in three dimensions. \FigBelow{0,0}{150}{200}{3dpot2}{Radial Wells for Q.	kapoor	22-04-09 17:04:30	n
[QUE/CM-04013] Node id: 2838page The Lagrangian for particle in three dimension is given by $$ L= {m\over 2} \left( \dot{\rho}^2 + \rho^2 \dot{\phi}^2 + {\dot{z}^2\over{1+(\rho/\lambda)^2 }}\right) -V(\rho) $$ Find the canonical momenta conjugate to the generalized co- ordinates $\rho,\phi,z$. Compute the Hamiltonian of the system	kapoor	22-04-09 17:04:09	n
[QUE/CM-04012] Node id: 2837page For a relativistic particle the Lagrangian is given by $$ L = - Mc^2 \sqrt{1- \dot{\vec{r}}^{\,2}/c^2} $$ find the relativistic momenta and the relativistic Hamiltonian.	kapoor	22-04-09 17:04:43	n
[QUE/CM-05013] Node id: 2865page For each of the potentials below check if a circular orbit is possible or not. $\displaystyle{ -V_0 {\exp(-\alpha r)\over r}}$ $\displaystyle -{ \alpha \over r^4}$ $\displaystyle -{\alpha \over r^2}$ If the orbit exists, relate the radius of the orbit to the angular momentum and energy.	kapoor	22-04-09 17:04:09	n
[QUE/CM-04019] Node id: 2850page \(\newcommand{\PB}[1]{[#1]_\text{PB}}\) Compute the Poisson brackets of components of angular momentum \(\vec{L}\) and show that \( [L_j, x_k]_\text{PB} =\epsilon_{jk\ell}\, x_\ell \) \( [L_j, p_k]_\text{PB} = \epsilon_{jk\ell}\, p_\ell \) \([L_j, \vec{p}^{\,2}]_\text{PB} =0 \) \([L_j, f(r,p)]_\text{PB}=0\) Here \(f\) is an arbitrary function of \(r=\|\vec{x}\|\) and ,\(p=\|\vec{p}\|\).	kapoor	22-04-09 17:04:36	n
[QUE/CM-04007] Node id: 2844page If for a system $F(q,p)$ and $G(q,p)$ are constants of motion, show that their Poisson bracket $[F,G]_\text{PB}$ is also a constant of motion.	kapoor	22-04-09 17:04:09	n
[QUE/CM-04006] Node id: 2843page For a particle moving in Coulomb potential $V(r) = k/r$, using Poisson brackets, show that the Runge Lenz vector $$ \vec{N} = \vec{v}\times\vec{L} - k \vec{r}/r $$ is a constant of motion where $\vec{v}$ is the velocity, and $\vec{L}$ is the angular momentum of the particle.	kapoor	22-04-09 17:04:49	n
[QUE/CM-04005] Node id: 2842page \( \newcommand{\PB}[1]{ [#1]_\text{PB} } \) Without using the definition of the Poisson bracket, using only the properties given a single particle show that the components of angular momentum satisfy the Poisson bracket relations $$\PB{L_1,L_2} = L_3, \qquad \PB{L_2,L_3}=L_1, \qquad \PB{L_3,L_1}=L_2$$ and $$ \PB{\vec{L}^2, L_k}=0, \qquad k=1,\cdots, n $$	kapoor	22-04-09 17:04:10	n
[QUE/CM-04016] Node id: 2841page The ends \(A\) and \(B\) of a thin uniform rod of mass \(m\) and length \(2a\) can slide freely, \(A\) along a smooth horizontal wire \(OX\) and \(B\) along a smooth vertical wire \(OZ\), with \(OZ\) pointing upwards. The wire frame \(OXZ\) is made to rotate with constant angular velocity \(\Omega\) about \(OZ\). Show that if \(B\) is above \(O\) and the angle \(OBA\) is \(\theta\), then \[H= \frac{2}{3} ma^2(\dot{\theta}^2 -\Omega^2 \sin^2\theta) +mga \cos\theta\] and that \(H\) is a constant of motion. Is \(H\) equal to total energy? What is the Hamiltonian if, instead of being rotated about \(OZ\), the frame \(OXZ\) is made to rotate with constant angular velocity \(\omega\) about the horizontal axis \(OX\)? Is it conserved?The ends \(A\) and \(B\) of a thin uniform rod of mass \(m\) and length \(2a\) can slide freely, \(A\) along a smooth horizontal wire \(OX\) and \(B\) along a smooth vertical wire \(OZ\), with \(OZ\) pointing upwards. The wire frame \(OXZ\) is made to rotate with constant angular velocity \(\Omega\) about \(OZ\).	kapoor	22-04-09 17:04:00	n
[QUE/CM-04015] Node id: 2840page For relativistic particle of mass \(m\) in three dimensions the Hamiltonian is given by \( H= \sqrt{\vec{p}^{\,2} c^2 +m^2c^4}\). Obtain the Hamiltonian equations of motion. Derive the Lagrangian and write Euler Lagrange equations of motion.	kapoor	22-04-09 16:04:53	n
[QUE/CM-04014] Node id: 2839page Obtain the canonical momenta and the Hamiltonian for a free particle in each of the following three coordinate systems. Spherical polar coordinates in three dimensions; Parabolic coordinates \((\xi,\eta,\zeta)\); Cylindrical coordinates in three dimensions.	kapoor	22-04-09 16:04:30	n
[QUE/CM-04018] Node id: 2849page \(\newcommand{\PB}[1]{[#1]_\text{PB}}\) Show that the Poisson brackets of components of angular momentum \(\vec{L}\) with the following quantities vanishes. \(\vec{r}\cdot\vec{p}\) \(\vec{r}\cdot\vec{L}\) \(\vec{L}\cdot\vec{p}\) \( \vec{L}^{\,2}\)	kapoor	22-04-09 15:04:49	n

In the classical kinetic theory of gases, molecules are assumed to be point particles ( of rigid shapes) in motion. Assume the average kinetic energy of the molecules to be ${3\over 2} kT$. Using the uncertainty principle, estimate the minimum uncertainty in position of a molecule. Compare this with average molecular distance for a gas of density $\rho$ ( $\rho=N/V$ , $N=$ number of molecules and $V=$ volume ) When do the quantum mechanical effects become significant ? Estimate this temperature for (a) He gas at normal pressure (b) an electron gas at density of $2.5 \times 10^{22} $ electrons/cc.

$\newcommand{\PB}[1]{\{#1\}_\text{PB}}$ $\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}$
Let $\Psi$ be a function of $f,g,h,..$ which are in turn functions of $q$ and $p$. Let $ \phi({{\bf q}, {\bf p}})$ be a function of $\bf q,p$, Show that the Poisson bracket of $\phi$ with $\Psi$ is given by $$\PB{\phi,\Psi} =\pp[\Psi]{f}\PB{\phi,f} + \pp[\Psi]{g}\PB{\phi,g} + \pp[\Psi]{h}\PB{\phi,h} + \cdots $$

Let $\Psi$ be a function of $f,g,h,..$ which are in turn functions of $q$ and $p$. Let $X(q, p)$ be a function having zero Poisson brackets with $ f,g,h...$. Show that the Poisson bracket of $\Psi({f,g,h})$ with $X(q,p)$ also vanishes.

For each of the six potentials, shown in the \Figref{3dpot1} below, answer the
following.

\label{16001Q1} Write the radial Schrodinger equation in different regions of $r$ values.
Write the most general solution as a linear combinations of spherical Bessel functions with appropriate arugments.
Impose the necessary regularity property at the origin and at $\infty$.
Impose the matching conditions at the boundary of two neighbouring regions.
Considering appropriate ranges of energy separately, state if energy eigenvalues are continuous or discrete?
\label{16001Q2} Whenever bound states exist for a potential, obtain the condition on the bound state energy eigenvalues.

\FigBelow{0,0}{150}{220}{3dpot1}{Radial Wells for Q.
\ref{16001Q1}-\ref{16001Q2}}

Classically orbit of a particle in a spherically symmetric potential lies in a plane. Show that the classical result is approximated by quantum solution infor $L_z=\ell$ the limit of large angular momentum $\ell$.

What qualitative differences do you expect in the spectra of the
H- atom and
(i) singly ionized Helium atom\\
(ii) the deuterium
(iii) positronium

For a classical electron moving in an orbit around a nucleus with charge $Ze$ obtain a relation between the number of revolutions per second and the energy of the electron. (Use only classical mechanics).
Using known quantum mechanical solution, show that in the limit of large principal quantum number $n$, the frequency of the line emitted in a transition from $(n+1)^{\mbox{th}}$ level to the $n^{\mbox{th}}$ level is precisely the same as the frequuency obtained in part $(a)$.

Classically orbit of a particle in a spherically symmetric potential lies in a plane. Show that the classical result is approximated by quantum solution in for $L_z=\ell$ the limit of large angular momentum $\ell$.

the following statements correctly describes the degeneracy?\\

Eigen values are non-degenerate just as for the bound states in one dimension.
Continuous eignevalues are doubly degenerate as in some one dimensional problems.
Continuous eignevalues are non-degenerate as in some one dimensional problems.
Infinitely degenerate as is the case for free particle in three dimension.
\label{16007Q2} The degeneracy is $(2l+1)$ as expected for a general spherically symmetric potential in three dimensions.

\FigBelow{0,0}{150}{200}{3dpot2}{Radial Wells for Q.

The Lagrangian for particle in three dimension is given by $$ L= {m\over 2} \left( \dot{\rho}^2 + \rho^2 \dot{\phi}^2 + {\dot{z}^2\over{1+(\rho/\lambda)^2 }}\right) -V(\rho) $$

Find the canonical momenta conjugate to the generalized co- ordinates $\rho,\phi,z$.
Compute the Hamiltonian of the system

For a relativistic particle the Lagrangian is given by $$ L = - Mc^2 \sqrt{1- \dot{\vec{r}}^{\,2}/c^2} $$ find the relativistic momenta and the relativistic Hamiltonian.

For each of the potentials below check if a circular orbit is possible or not.

$\displaystyle{ -V_0 {\exp(-\alpha r)\over r}}$
$\displaystyle -{ \alpha \over r^4}$
$\displaystyle -{\alpha \over r^2}$

If the orbit exists, relate the radius of the orbit to the angular momentum and energy.

$\newcommand{\PB}[1]{[#1]_\text{PB}}$
Compute the Poisson brackets of components of angular momentum $\vec{L}$ and show that

$ [L_j, x_k]_\text{PB} =\epsilon_{jk\ell}\, x_\ell $
$ [L_j, p_k]_\text{PB} = \epsilon_{jk\ell}\, p_\ell $
$[L_j, \vec{p}^{\,2}]_\text{PB} =0 $
$[L_j, f(r,p)]_\text{PB}=0$

Here $f$ is an arbitrary function of $r=|\vec{x}|$ and ,$p=|\vec{p}|$.

If for a system $F(q,p)$ and $G(q,p)$ are constants of motion, show that their Poisson bracket $[F,G]_\text{PB}$ is also a constant of motion.

For a particle moving in Coulomb potential $V(r) = k/r$, using Poisson brackets, show that the Runge Lenz vector $$ \vec{N} = \vec{v}\times\vec{L} - k \vec{r}/r $$ is a constant of motion where $\vec{v}$ is the velocity, and $\vec{L}$ is the angular momentum of the particle.

$ \newcommand{\PB}[1]{ [#1]_\text{PB} } $

Without using the definition of the Poisson bracket, using only the properties given a single particle show that the components of angular momentum satisfy the Poisson bracket relations $$\PB{L_1,L_2} = L_3, \qquad \PB{L_2,L_3}=L_1, \qquad \PB{L_3,L_1}=L_2$$ and $$ \PB{\vec{L}^2, L_k}=0, \qquad k=1,\cdots, n $$

The ends $A$ and $B$ of a thin uniform rod of mass $m$ and length $2a$ can slide freely, $A$ along a smooth horizontal wire $OX$ and $B$ along a smooth vertical wire $OZ$, with $OZ$ pointing upwards. The wire frame $OXZ$ is made to rotate with constant angular velocity $\Omega$ about $OZ$.
Show that if $B$ is above $O$ and the angle $OBA$ is $\theta$, then \[H= \frac{2}{3} ma^2(\dot{\theta}^2 -\Omega^2 \sin^2\theta) +mga \cos\theta\] and that $H$ is a constant of motion. Is $H$ equal to total energy? What is the Hamiltonian if, instead of being rotated about $OZ$, the frame $OXZ$ is made to rotate with constant angular velocity $\omega$ about the horizontal axis $OX$? Is it conserved?The ends $A$ and $B$ of a thin uniform rod of mass $m$ and length $2a$ can slide freely, $A$ along a smooth horizontal wire $OX$ and $B$ along a smooth vertical wire $OZ$, with $OZ$ pointing upwards. The wire frame $OXZ$ is made to rotate with constant angular velocity $\Omega$ about $OZ$.

For relativistic particle of mass $m$ in three dimensions the Hamiltonian is given by $ H= \sqrt{\vec{p}^{\,2} c^2 +m^2c^4}$.

Obtain the Hamiltonian equations of motion.
Derive the Lagrangian and write Euler Lagrange equations of motion.

Obtain the canonical momenta and the Hamiltonian for a free particle in each of the following three coordinate systems.

Spherical polar coordinates in three dimensions;
Parabolic coordinates $(\xi,\eta,\zeta)$;
Cylindrical coordinates in three dimensions.

$\newcommand{\PB}[1]{[#1]_\text{PB}}$ Show that the Poisson brackets of components of angular momentum $\vec{L}$
with the following quantities vanishes.

$\vec{r}\cdot\vec{p}$
$\vec{r}\cdot\vec{L}$
$\vec{L}\cdot\vec{p}$
$ \vec{L}^{\,2}$

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