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[QUE/CM-04014]Node id: 2839pageObtain 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.
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22-04-09 16:04:30 |
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[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.
- \(\vec{r}\cdot\vec{p}\)
- \(\vec{r}\cdot\vec{L}\)
- \(\vec{L}\cdot\vec{p}\)
- \( \vec{L}^{\,2}\)
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22-04-09 15:04:49 |
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[QUE/CM-04010]Node id: 2847pageStarting form the Lagrangian $$ L = {1\over 2}M\, \dot{\vec{r}}^2 + {e\over c}\,\vec{A}(\vec{r})\cdot \dot{\vec{r}} -e\phi $$ for a charged particle in external electric and magnetic fields, $\vec{E}$ and $\vec{B}$, show that the Hamiltonian is given by $$ H = {1\over 2M}\,(\vec{p} - e \vec{A})^2 + e \phi$$ |
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22-04-09 15:04:53 |
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[QUE/CM-04009]Node id: 2846pageThe Lagrangian for a symmetric top, in terms of Euler angles $\theta,\phi,\psi$ is given by $$ L = {1\over2} I_1 \left(\dot{\theta}^2 + \dot{\phi}^2 \sin^2\theta \right) +{1\over2} I_3 (\dot{\psi} + \dot{\phi} \cos\theta)^2 - mgL\cos\theta $$
- Compute the canonical momenta conjugate to $\theta,\phi,\psi.$
- Show that the Hamiltonian in terms of momenta $p_\theta, p_\phi$ and $p_\psi$ the Euler angles is given by $$H=\frac{p_\theta^2}{2I_1} + \frac{(p_\psi -p_\phi \cos\theta)^2}{2 I_1 \sin^2\theta} + \frac{p_\psi^2}{2I_3} + mgL\cos\theta$$
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22-04-09 15:04:12 |
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[QUE/CM-04008]Node id: 2845pageCompute the following Poisson brackets.
- $[\vec{a}\cdot\vec{L},\vec{b}\cdot\vec{L}]_\text{PB}$
- $[L_1, x_3]_\text{PB}$
- $[\vec{p}, r^n]_\text{PB}$
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22-04-09 14:04:27 |
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[QUE/CM-04020]Node id: 2851pageCompute the Poisson brackets of \(x\)- component of angular momentum \(L_x\) with the following quantities.
- \(\vec{r}\cdot \vec{p}\)
- \( p_y\)
- \(z\)
- \(\vec{p}^{\, 2}\)
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22-04-09 14:04:52 |
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[QUE/CM-04017]Node id: 2852pageThe Lagrangian for a system with one degree of freedom is given by \[ L=\frac{1}{2} m \dot{x}^2 - \omega^2 x^2 -\alpha x^4 + \beta x \dot{x}^2\]
- Obtain the momentum conjugate to \(x\) and the Hamiltonian
- Write the Hamiltonian equations of motion.
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22-04-09 14:04:16 |
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[QUE/CM-05017]Node id: 2861pageA body of mass $\mu$ is projected from a hypothetical planet, mass $16M$ and radius $2R$, towards another planet of mass $M$ and radius $R$. Plot the gravitational potential seen by the body as function of distance from the center of the first planet. The distance between the centers of the planet is given to be $10R.$
- Find the equilibrium point, if any, on the line joining the two planets.
- Discuss what happens when the particle is displaced a little towards one of the planets. Is the equilibrium stable or unstable?
- Find the minimum velocity with which the particle be projected from the surface of the first planet so that it may reach the second planet.
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22-04-09 14:04:15 |
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[QUE/CM-05016]Node id: 2862pageWhen a small correction $\delta U(r)$ is added to the potential energy $U=-{\alpha\over r}$, the paths of finite motion are no longer closed, and at each revolution the perihelion is dipslaced through a small angle $\delta \phi$. Show that $\delta \phi$ is approximately given by $$ \delta \phi = {\partial \over \partial M} \int_{r_1}^{r_2} { 2M\delta U dr\over \sqrt{ 2m(E +\alpha/r)-M^2/r^2 }} = {\partial \over \partial M}\left(\frac{2m}{M} \int_{0}^{\pi} r^2\delta U d\phi \right) $$ where $r_1, r_2$ are the minimum and the maximum values of $r$. Find $\delta \phi$ when $\delta U = {\beta \over r^2 }$. |
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22-04-09 14:04:46 |
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[QUE/CM-05015]Node id: 2863pageFor a particle moving in a spherically symmetric potential the three components of angular momentum are conserved. For a potential depending on $r$ as well as $\vec{n}\cdot\vec{r}$ ( $\vec{n} = $ a fixed vector) prove that only the component of angular momentum along $\vec{n}$ is conserved. |
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22-04-09 14:04:42 |
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[QUE/CM-05014]Node id: 2864pageUnder what conditions does a particle moving in a potential $ -{\alpha \over r^2}$ fall to the center? Find the time taken in falling the center from point $r=r_0$. |
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22-04-09 14:04:52 |
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[QUE/CM-05012]Node id: 2866pageFind the time dependence of coordinates $(x,y)$ of a particle with energy $E=0$ moving in a parabola in a potential $$V(r)= - {k\over r}.$$ |
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22-04-09 14:04:41 |
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[QUE/CM-05011]Node id: 2867page Solve the equations of motion, in polar coordinates, for an isotropic oscillator in two dimension with potential given by $V(r) ={1\over 2} m \omega^2 r^2$ |
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22-04-09 14:04:45 |
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[QUE/CM-05010]Node id: 2868pageA particle moves in a central potential $$ V(r) = -{\alpha \over r^2} $$ and has energy $E$, angular momentum $L$. Show that the orbit of the particle is given by $$ {1\over r} = \sqrt{2mE\over L^2 -2m\alpha } \cos ( \lambda (\phi + \delta))$$ where $\lambda ^2 =1- {2m\alpha \over L^2}$ |
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22-04-09 14:04:53 |
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[QUE/CM-05009]Node id: 2869page For a particle moving in Coulomb potential $V(r) = k/r$ 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 |
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22-04-09 14:04:00 |
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[QUE/CM-05008]Node id: 2870pageConsider the motion in a spherically symmetric potential $$ V(r) = - V_0 \left( {3 R\over r } + {R^3\over r^3} \right)$$ If orbital angular momentum of the particle is given by $l^2 = 10 m V_0 R^2$, plot the potential as a function of $r$ and answer the following questions.
- What is the minimum energy so that particle coming from infinity may fall to the center?
- What should be the energy of the particle so that it may move in a circular orbit? Find the radius of the orbit.
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22-04-09 14:04:41 |
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[NOTES/QCQI-01004] Positive Operators Node id: 5021page |
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22-04-08 13:04:09 |
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[ NOTES/QCQI-01003] Trace and Partial TraceNode id: 5020page |
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22-04-08 13:04:46 |
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[NOTES/QCQI-04001] Single Qubit Quantum GatesNode id: 5030page |
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22-04-08 13:04:54 |
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[NOTES/QCQI-04002] Two Qubit GatesNode id: 5031page |
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22-04-08 13:04:01 |
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