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### Problem/QM-23004

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.

kapoor 28/12/18

### Problem/QM-23005

Use the first order perturbation theory to compute the correction to the $l=1$ level of $$H = L^2 + \alpha L_x + \beta L_y$$ taking  $$H_0=L^2, \qquad \mbox{and} \qquad H^\prime = \alpha L_x + \beta L_y.$$ Find the corrections to the $l=1$ level using first order perturbation theory.

kapoor 28/12/18

### Problem/QM-23006

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 $\epsilon$.

kapoor 28/12/18

### Problem/QM-23007

Calculate  $\ell=1$ eigenvalues and eigenvectors to the lowest non vanishing order in perturbation theory taking  $$H_0= \vec{L}^2 + \gamma L_z$$ as the unperturbed Hamiltonian and    $$H^\prime = \epsilon (L_xL_y+L_y L_x)$$ as perturbation.

kapoor 28/12/18

### Problem/QM-23008

The unperturbed part and the perturbation part of  Hamiltonian for a system can be written as
H_0= \begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 2
\end{pmatrix} \quad H^\prime = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1
& 0 \end{pmatrix} Find the lowest order nonvanishing corrections to the eigenvalues.

kapoor 28/12/18

### Problem/QM-23009

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.

kapoor 28/12/18

### Problem/QM-23010

Summary (or abstract) not available for this node.
kapoor 28/12/18

### Problem/QM-23011 Time Independent Perturbation Theory

$$\newcommand{\ket}[1]{\vert#1\rangle}$$
$$\newcommand{\bra}[1]{\langle#1\vert}$$In an orthonormal basis consisting of three elements $\{\ket{1},\ket{2}, \ket{3} \}$, the Hamiltonian of a system is given by $$H = \ket{1}\bra{1} + i\epsilon \ket{1}\bra{2} - i\epsilon \ket{2}\bra{1} + 2 \ket{2}\bra{2} + 2\epsilon \ket{2}\bra{3} + 2 \epsilon\ket{3}\bra{2} + \ket{3}\bra{3}$$ 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.]

kapoor 28/12/18

### Problem/QM-23012

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

kapoor 28/12/18

### Problem/QM-23013

A particle, having charge $q$, moves in harmonic oscillator potential $V(x)={1\over 2} m \omega^2x^2$.  Find the shift in the energy level of  the $\mbox{n}^{\mbox{th}}$ excited state when uniform electric field $E$ is applied. Compute the corrections upto lowest order giving a nonzero value.

kapoor 28/12/18