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For a simple harmonic oscillator with Lagrangian $$ L = {1\over 2} m \dot{q}^2 - {1\over 2} m \omega^2 q^2  $$ show that the transformation $$ q \rightarrow q^\prime = q+ \epsilon \sin \omega t $$ is a symmetry transformation. Find the conserved quantity associated with this symmetry transformation. Using equations of motion verify explicitly that this quantity is indeed conserved.

From symmetry considerations alone find a conserved quantity for the Lagrangian $$L={1\over2 } m \dot{\vec{r}} - V_0 x \sin\left({ 2\pi z\over R}\right) - V_0 y \cos\left( {2\pi z \over R} \right) $$ where $V_0$ and $R$ are constants. What is symmetry of $L$? Find generalized co-ordinates one of which is cyclic. Hint : It can be proved that for an unconstrained particle moving in a potential the only time independent transformations under which the Lagrangian  is invariant are combinations of rotations and translations.

 Write the Lagrangian for several charged particle having masses $m_k$ and charges $e_k$ interact with electromagnetic fields. Write the Lagrangian and and show that a gauge transformation on the potentials changes the Lagrangian by a total time derivative and hence the equations of motion do not change if a gauge transformation is applied on the potentials.

Show that the second order correction to the ground state energy is always negative.

QBANK-QM23

A particle moves in two dimensional circular oscillator potential $$  V(x) = {1\over 2} m \omega^2(x^2+y^2) $$ (a) What are the quantum numbers of the first excited state? Is it degenerate or not?
(b) If a small perturbation $H^\prime =  \lambda xy $ is applied compute the lowest order correction to the energy of the first excited state.     

Following the steps as in the theoretical treatment of perturbation theory, find eigenvectors and eigenvalues of the matrix
 $$\left[ \begin{array}{rrr}        1&\epsilon& 0 \\         \epsilon & 1&-\epsilon \\          0 & -\epsilon& 2    \end{array}\right] $$ upto the lowest non-vanishing order in $\epsilon$.

test1  test2     test5

Test 9

A spectral spectral line due to a transition from an electronic state \(p\) to an \(s\) state splits into three Zeemnan lines in the presence of a strong magnetic field. At intermediate field strengths find  the number of spectral lines that will be observed for this transition.

\(\newcommand{\ket}[1]{\vert#1\rangle}\)
\(\newcommand{\bra}[1]{\langle#1\vert}\)In an orthonormal basis consisting of three elements $\{\ket{1},\ket{2}, \ket{3} \}$, the Hamiltonian of a system is given by \begin{equation} H = \ket{1}\bra{1} + i\epsilon \ket{1}\bra{2} - i\epsilon \ket{2}\bra{1} + 2 \ket{2}\bra{2} + 2\epsilon \ket{2}\bra{3} + 2 \epsilon\ket{3}\bra{2} + \ket{3}\bra{3} \end{equation} Find the eigenvalues and eigenvectors of the total Hamiltonian \(H\) upto first order in \(\epsilon\). [Hint: First construct the matrix representation for $H$ in the given basis.]