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[NOTES/QM-17001] Angular Momentum in Quantum Mechanics --- Summary of results Node id: 4813page$\newcommand{\DD}[2][]{\frac{d^2 #1}{d^2 #2}}$ $\newcommand{\matrixelement}[3]{\langle#1|#2|#3\rangle}$ $\newcommand{\PP}[2][]{\frac{\partial^2 #1}{\partial #2^2}}$ $\newcommand{\dd}[2][]{\frac{d#1}{d#2}}$ $\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}$ $\newcommand{\average}[2]{\langle#1|#2|#1\rangle}$ qm-lec-17001 |
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[NOTES/QCQI-04002] Two Qubit GatesNode id: 5031page |
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[QUE/TH-13004] TH-PROBLEMNode id: 5175pageThe fundamental equation for a system is given by \begin{equation*} u = \Lambda \frac{s^{3/2}}{v^{1/2}} \end{equation*} where \(\Lambda\) is a constant. Prove the following equations \begin{eqnarray} T &=& \frac{5}{2} \frac{\Lambda S^{3/2}}{NV^{1/2}}\\ P V^{2/3} &=& N \frac{N \Lambda 2^{1/2}}{5*{3/2}} T^{5/3}\\ \mu &=& - \Big(\frac{2}{5}\Big)^{5/2} \frac{2}{\Lambda ^{2/3}} \Big(\frac{V}{N}\Big)^{1/3} T^{5/3}. \end{eqnarray} |
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22-01-13 17:01:57 |
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[2019CM/Final-Part-A]Node id: 5338page Classical Mechanics July 4, 2019
Instructions
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Answers for Part-A to be written on this sheet itself. |
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The answers for PART-B are to be written in a separate answer book. |
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Maximum Time for Part-A is 30 mins. |
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Attempt ALL questions from Part-A. |
- Define generalized coordinates. How many generalized coordinates are required for a bead moving on the surface of a sphere? What coordinates will you use?
- For a particle in three dimensional potential \(\lambda r^4\), write expression of effective potential and plot \(V_\text{eff}(r)\) against \(r\) for non-zero angular momentum.
- Define canonical transformation in two different ways.
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[2008EM/HMW-10]Node id: 5407page |
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22-05-10 20:05:39 |
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[2003SM/LNP-22] Lecture 22 -- Perfect Fermi GasNode id: 5521page |
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22-07-04 10:07:32 |
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[1998TH/LNP-17]-Node id: 5587page |
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22-07-17 18:07:32 |
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[NOTES/ME-02009]-Vectors as geometrical objectsNode id: 5669pageConvention about vectors is described. Different different symbols are used for vectors without reference to any axis, components of vector w.r.t. a system of coordinate axes, column vector notation for components |
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22-08-14 10:08:24 |
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[NOTES/EM-09004]-Self Inductance and Mutual InductanceNode id: 5725pageWe define the mutual inductance for two loops and self inductance for a loop. We obtain an expression for energy of a circuit with self inductance \(L\) |
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22-08-24 16:08:17 |
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[NOTES/QM-09006] Heisenberg Picture of quantum mechanicsNode id: 4709page
The time evolution of states a quantum system is given by the time dependent Schrodinger equation. Besides this framework, called the Schr\"{o}dinger picture, other scheme are possible. In the Heisenberg picture, defined here, the observable evolve according to the equation \[\dd[X]{t} =\frac{1}{i\hbar}[F, H] \]This equation corresponds to the classical equation of motion in the Poisson bracket formalism.
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[NOTES/EM-04002] Poisson Equation in Cylindrical coordinatesNode id: 5972page$\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}$ $\newcommand{\PP}[2][]{\frac{\partial^2#1}{\partial #2^2}}$ Problems with cylindrical symmetry can be solved by separating the variables of the Poisson equation in cylindrical coordinates. The separation of variables for this class of problems and boundary conditions are explained. |
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23-10-25 06:10:18 |
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[NOTES/QM-25001] Electormagnetic WavesNode id: 4928page$\newcommand{\DD}[2][]{\frac{d^2 #1}{d^2 #2}}$ $\newcommand{\matrixelement}[3]{\langle#1|#2|#3\rangle}$ $\newcommand{\PP}[2][]{\frac{\partial^2 #1}{\partial #2^2}}$ $\newcommand{\dd}[2][]{\frac{d#1}{d#2}}$ $\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}$ $\newcommand{\average}[2]{\langle#1|#2|#1\rangle}$ $\newcommand{\ket}[1]{\langle #1\rangle}$ qm-lec-25001 |
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22-03-12 18:03:06 |
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[QUE/EM-02023] EM-PROBLEMNode id: 5126pageUse Gauss's law to find the electric field inside a uniformly charged solid sphere of radius \(R\) and carrying charged density $\rho$. State facts other than Gauss's law which you might have used in your answer. |
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22-01-09 21:01:52 |
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[QUE/TH-06011] TH-PROBLEMNode id: 5206page(a) Show that the work done on an ideal gas to compress it isothermally is greater than that necessary to compress it adiabatically if the pressure change is the same in two processes. (b)~Show that the isothermal work is less than the adiabatic work if the volume change is the some in two processes (c)~As numerical example, compare the work done from initial pressure and volume to be 10$^6$N/m$^2$ 0.5 m$^3$ kilomole$^{-}$ in the isothermal and adiabatic process when (i)~the pressure is doubled (ii)~ volume is halved. |
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22-01-20 15:01:42 |
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[2019EM/HMW-06]Node id: 5368pageElectrodynamics Oct 17, 2018 Tutorial-V (VI)
- A cylindrical resistor of length \(\ell\), radius \(a\) and resistivity $\varrho$ carrying a current \(i\).
- Show that the Poynting vector $\vec{S}$ at the surface of the resistor is everywhere directed to the normal to the surface, as shown,
- Show the rate $\rho$ at which energy flows into the resistor through its cylindrical surface, calculated by integrating the Poynting vector over this surface, is equal to the rate at which Joule heat is produced, i.e. \begin{equation*} \int\vec{S}\cdot\vec{dA}=i^2R \end{equation*}
- Figure 1 shows a parallel-plate capacitor being charged,
- Show that the Poynting vector \(S\) points everywhere radially into the cylindrical volume.
- Show that the rate \(P\) at which energy flows into this volume, calculated by integrating the Poynting vector over the cylindrical boundary of this volume, is equal to the rate at which the stored electrostatic energy increases; that is, that \[\int \vec{S}\cdot\vec{dA} = A d \frac{1}{2}\frac{d}{dt}(\epsilon \vec{E}^2)\],where \(Ad\) is the volume of the capacitor and \(\frac{1}{2} \epsilon_0\vec{E}^2\) is the energy density for all points within that volume. This analysis shows that, according to the Poynting vector point of view, the energy stored in a capacitor does not enter it through the wires but through the space around the wires and the plates. \{Hint: To find \(S\) we must first find \(B\), which is the magnetic field setup by the displacement current during the charging process; Ignore fringing of the lines of \(E\).)\}

- A coaxial cable (inner radius \(a\) and outer radius \(b\)) is used as a transmission line between a battery $\epsilon$ and a resistor \(R\),
- [(a)] Calculate $\vec{E}, \vec{B}$ for $a < r < b$
- [(b)] Calculate Poynting vector S for $a< r < b.$
- [(c)] By suitably integrating the Poynting vector, Show that the total power flowing across the annular cross section $a<r<b$ is="" $\epsilon^{2}="" r$.="" this="" reasonable="" ?="" <="" li="">
- [(d)] Show that the direction of $\vec{S}$ is always away from the battery to resistor, no matter which way is the battery connected.

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[QUE/EM-02001]Node id: 5439page[1] A thin glass rod is bent into a semicircle of radius $R$. A charge $+Q$ is uniformly distributed along the upper half and a charge $-Q$ is distributed uniformly along the lower half as shown in the figure. Find the electric field at P, the center of the semicircle. |
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[1998TH/LNP-09] Lecture -9 Equation of State-IINode id: 5551page |
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22-07-08 07:07:29 |
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[RCQ/CV-05002] Recalling Reasoning for Singular PointsNode id: 5627page |
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22-08-06 19:08:16 |
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[NOTES/ME-14005]-Vectors and TensorsNode id: 5699page |
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22-08-16 16:08:18 |
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[NOTES/EM-12001]-Lorentz transformations Node id: 5757page
The basic equations of Maxwell's theory are written down in relativistic notation. Using Lorentz transformations of the potentials, the expressions of the scalar and vector potentials of a point charge moving with a uniform velocity are obtained.
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