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[QUE/QM-09010]Node id: 2518page\(\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}\) \(\newcommand{\dd}[2][]{\frac{d #1}{d #2}}\) \(\newcommand{\ket}[1]{\vert#1\rangle}\) Consider the anharmonic oscillator problem with Hamiltonian \[H = \frac{p^2}{2m} + \frac{1}{2}k x^2 + \lambda x^4 \] Investigate if we can define a new picture such that the operators evolve according to the equation \begin{eqnarray} \dd[X_n]{t} &=&\pp[X_n]{t} + \frac{1}{i\hbar }\big[X_n,H_1]\\ H_1&=&\frac{p^2}{2m_1} + \frac{1}{2}k x^2 \end{eqnarray} and the states evolve according to the Schrodinger equation \begin{eqnarray} i\hbar \dd[\ket{\psi\,t}_n]{t} &=& H_2 \ket{\psi\,t}_n\\ H_2 &=& \frac{p^2}{2m_2} + \lambda x^4 \end{eqnarray} where the suffix \(n\) means states and operators in the new picture. If such a picture exists, what should be the relation between \(m_1\) and \(m_2\)? |
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22-04-13 21:04:01 |
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[QUE/CM-01023] --- Variable Mass ProblemNode id: 2435pageOne end of link chain is fixed to a support at \(z=0\). The free end of an, open, link-chain of length $L$, and weight $\mu$ per unit length, is released from rest at $x=0$. Determine the tension $T$ in the chain at its support at $A$ in terms of $x$. also determine the energy loss, $\Delta E$ during the entire motion $x=0$ to $x=2L$.
Source::IIT/K
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22-04-13 21:04:54 |
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[QUE/CM-01022]Node id: 2434page
The left end of an, open, link-chain of length $L$, and weight $\mu$ per unit length, is released from rest at $x=0$. Determine the tension $T$ in the chain at its support at $A$ in terms of $x$. also determine the energy loss, $\Delta E$ during the entire motion $x=0$ to $x=2L$.
Source:: IIT/K
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22-04-13 21:04:23 |
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[QUE/CM-01020] --- Variable Mass ProblemNode id: 2432pageA two stage rocket is fired as follows in deep space ( no gravity ). Stage 1: Mass of the frame+ engine = $M_1$ Mass of the fuel + oxidizer =$m_1$ gas velocity relative to the rocket = $v_r$ Stage 2: After burning $M_1$ the second stage is ignited with Mass of the frame+ engine = $M_2$ Mass of the fuel + oxidizer =$m_2$ gas velocity relative to the rocket = $v_r$ Show that the velocity achieved is $$v_f= - v_r \ln \frac{(M_1+M_2+m_1+m_2)(M_2+m_2)}{(M_1+m_2+m_2)M_2} $$
Source::HSM 70
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22-04-13 21:04:14 |
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[QUE/CM-01019] Sand falling in a moving car.Node id: 2431pageAn empty freight car, which is travelling along a horizontal track, passes under funnel filled with sand as shown in the diagram. The sand is released into the car at the rate of $\dot{m}$. Find the force needed to keep the car at constant speed $v_0$.
Thanks::HSM 67
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22-04-13 21:04:26 |
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[QUE/CM-01016]Node id: 2428page Water drop falling through saturated atmosphere. A spherical water droplet falls, without friction and under influence of gravity, through an atmosphere saturated with water vapor. Let its initial radius ($t=0$) be $c$, its initial velocity, $v_0$. As a result of condensation the water drop experiences continuous increase in mass proportional to its surface; Show that its radius increases then increases linearly with time. Integrate the differential equation of motion by introducing $r$ instead of $t$ as an independent variable. Show that for $c=0$, the velocity increases linearly with time.
Source::Sommerfeld
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22-04-13 21:04:50 |
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[QUE/QM-09003]Node id: 2523page\(\newcommand{\dd}[2][]{\frac{d#1}{d#2}}\) Solve the differential equation \begin{equation}\label{EQ21} \dd{t} U(t,t_0) = \lambda \hat{F}(t) U(t,t_0) \end{equation} for \(U(t,t_0)\) subject to the condition \(U(t,t_0)=\hat{I}\) for the following cases. Here \(\hat{F}\) is an operator (or a matrix). [A] Assume the operator \(\hat{F}\) to be independent of time. Expand \(U(t,t_0)\) in a power series in \(\lambda\) and prove \[\begin{equation} U(t,t_0) = \exp(\lambda (t-t_0)\hat{F}).\end{equation}\] [B] When \(\hat{F}\) depends on time but is such that the commutator \(\big[\hat{F}(t), \hat{F}(t')\big]\) vanishes for all \(t,t'\) show that \begin{equation} U(t,t_0) = \exp\big(\lambda \int_{t_0}^t \hat{F}(\tau)\, d\tau\big).\end{equation}
NOTE: The case when \(\hat{F}(t)\) and \(\hat{F}(t^\prime)\) do not commute for \(t \ne t^\prime\), requires more work and requires use of Time Ordered Exponentials in the solution. |
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22-04-13 21:04:51 |
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[QUE/QM-09004]Node id: 2522pageWithout assuming that the Hamiltonian is independent of time, start from the requirement, \[ \begin{equation} _H{\langle \psi t \vert X_H(t)}_H\vert \psi \rangle_H = _{S\hspace{-1pt}}\langle\vert {\psi t}{X_S}\vert \psi\rangle_S \end{equation}\] and directly derive the equation of motion \[ \begin{equation} \boxed{\frac{d X_H}{d t} = \frac{\partial X_H }{\partial t}+ \frac{1}{i\hbar}\big[X_H, H\big]_-.} \end{equation}\] for the general case when the Hamiltonian depends on time. |
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22-04-13 21:04:52 |
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[QUE/QM-09006]Node id: 2521page Prove that \[ \begin{equation} \int_{0}^ t \, dt_1 \int_{0}^ {t_1} dt_2\, H^\prime_I(t_1) H^\prime_I(t_2) = \frac{1}{2} \int_{0}^ t \, dt_1 \int_{0}^ t \, dt_2 T\big( H^\prime_I(t_1) H^\prime_I(t_2)\big) \end{equation} \] |
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22-04-13 21:04:08 |
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[QUE/QM-09007]Node id: 2520page(a) For harmonic oscillator \[H=\frac{\hat{p}^2}{2m} + \frac{1}{2}m\omega^2 \hat{x}^2\] set up and solve up the equations of motion for position and momentum operators in the Heisenberg picture. Hence show that \[\begin{eqnarray} \hat{x}(t) = \tilde{x} \cos\omega t + \frac{1}{m\omega} \tilde{p} \sin\omega t \\ \hat{p}(t) = \tilde{p} \cos\omega t - m\omega \tilde{x} \sin\omega t \end{eqnarray}\] where \(\tilde{x},\tilde{p}\) in the right hand sides of the above equations are the Schr\"{o}dinger picture operators. (b) Calculate the commutators \[\begin{eqnarray*} [\hat{x}(t), \hat{x}(t^\prime))],\qquad [\hat{x}(t), \hat{p}(t^\prime))], \qquad [\hat{p}(t), \hat{p}(t^\prime))], \end{eqnarray*}\] for \(t\ne t^\prime\). How do these commutators at unequal times compare with equal time commutators? Are \(\hat{x}(t), \hat{x}(t^\prime)\) compatible observables? (c) Verify that at equal times the commutators reduce to canonical commutation rule \([\hat{x}, \hat{p}] =i\hbar.\) |
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22-04-13 21:04:07 |
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Thermodynamics --- Notes For Lectures and Problems [TH-MIXED-LOT]Node id: 4890collection |
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22-04-12 22:04:24 |
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[QUE/QM-09001]Node id: 2524pageA particle is described by the Hamiltonian $$ \hat{H} = \frac{\vec{p}\,^2}{2m} + V(\vec{x}) $$ Find the equations of motion of a particle in the Heisenberg picture. Solve the equations of motion for the position and momentum $\hat{x}(t)$ and $\hat{p}(t)$. Find $x(t), p(t)$, if initial ($t=0$) conditions are given. Suppose the particle is in initial state $\psi$, what is the equation satisfied by the expectation value of $x$ and $p$ at a later time $t$? What does one expect classically? |
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22-04-12 20:04:25 |
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[QUE/QM-09002]Node id: 2525page\(\newcommand{\ket}[1]{\vert#1\rangle}\) Consider the state \(\ket{\psi}\) given by \(\begin{equation*} \ket{\psi} = c_1 \ket{E_1} + c_2 \ket{E_2}. \end{equation*} \)Answer the following questions.
- Give condition(s) on \(E_1, E_2\) so that \(\ket{\psi}\) may be a stationary state for arbitrary \(c_1, c_2\).Can the state represented by \(\ket{\psi}\) be a stationary state when \(E_1\ne E_2\)? If yes, give condition(s) on \(c_1,c_2\).
- Let \(E_1, E_2, c_1, c_2\) be such that \(\ket{\psi}\) is {\bf not} a stationary state. Compute the probability that a measurement of energy gives a value\(E_1\) at time \(t\). Does the probability vary time. WHY?
- Let \(E_1, E_2, c_1, c_2\) be as in part \ref{IT3}. Now under what conditions that a dynamical variable \(X\) must obey so that the average of \(X\) remains constant?
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22-04-12 19:04:43 |
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[QUE/QM-09009]Node id: 2526page |
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22-04-12 19:04:17 |
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[QUE/QM-09008]Node id: 2527page |
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22-04-12 19:04:21 |
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[QUE/CM-02002]Node id: 2533pageA wire is bent in the shape of a parabola \( x^2=4ay\). A bead slides on the wire which is kept in a vertical plane with \(y\)- pointing upwards. How many generalized coordinates are needed? Obtain the Lagrangian and derive the Euler Lagrange equations of motion. |
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22-04-12 19:04:19 |
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[QUE/CM-02003]Node id: 2535page
Set up the Lagrangian of a system of a heavy rod of length $L$ and a body of mass $M$. One end of the rod is fixed and the body is attached to the other end, the rod is free to rotate in the vertical plane. |
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22-04-12 19:04:52 |
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[QUE/CM-02013]Node id: 2540pageLet $C$ and $C^\prime$ be two neighbouring paths in configuration space $$ C: q_k=q_k(t) \qquad C^\prime : q_k = q_k(t) + \delta q_k(t) \qquad t_1 < t < t_2 $$ which do not have the same end points. Show that the difference in action along the two paths receives contribution only from the end points if and only if Euler Lagrange equations of motion are obeyed. In other words, show that the action integral depends on the differences $\delta q_1 $ and $\delta q_2$ at the end points $t_1$ and $t_2$ only and not on values of $\delta q(t)$ for other values of $t$ between $t_1$ and $t_2$.
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22-04-12 19:04:30 |
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[QUE/CM-03001] Action for harmonic oscillatorNode id: 2541page For a harmonic oscillator in one dimension the Lagrangian is $$L = {1\over 2}\,m\, \dot{x}^2 -{1\over 2}\,m\,\omega^2 x^2 $$ Write the solution of the equations of motion subject to the boundary conditions at times $t_1$ and $t_2$ given by $$ x(t_1) = x_1 \qquad x(t_2) = x_2 $$ Use the solution to compute the action integral. Show that the action integral is given by $$ S(x_2, t_2; x_1,t_1) = \frac{m\omega}{2\sin \omega(t_2-t_1) }\big[ (x_1^2 + x_2^2) \cos\omega(t_2-t_1) -2 x_1x_2\big] $$
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22-04-12 19:04:21 |
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[QUE/CM-03003]Node id: 2543pageLet $C$ and $C^\prime$ be two neighbouring paths in configuration space $$ C: q_k=q_k(t) \qquad C^\prime : q_k = q_k(t) + \delta q_k(t) \qquad t_1 < t < t_2 $$ which do not have the same end points. Show that the difference in action along the two paths receives contribution only from the end points if and only if Euler Lagrange equations of motion are obeyed.( Show that the action integral depends on the differences $\Delta q_1 $ and $\Delta q_2$ at the end points $t_1$ and $t_2$ only and not on values of $\delta q(t)$ for other values of $t$ between $t_1$ and $t_2$ |
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22-04-12 19:04:55 |
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