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Problem/QM-23014

A particle moves in a one dimensional box with rigid boundary walls at $x=0$ and $x=\pi$. A small perturbation $V_1$ given by $$ V_1 (x)=\epsilon \sin 3x  $$ is applied. Calculate the lowest order nonvanishing correction to the ground  state and the first excited state energies of the particle in a box.

kapoor kapoor's picture 28/12/18

Problem/QM-23016

The Hamiltonian for a harmonic oscillator is   \[ H=\frac{p^2}{2} + \frac{x^2}{2} \]   A perturbation \(\lambda x^3\) is applied. Compute the second order correction to the ground state energy.

kapoor kapoor's picture 28/12/18

Problem/QM-23015

Use the first order non-degenerate perturbation theory to compute the correction to the $\ell=1$ level of ( use units so that $\hbar=1$ ) $$ H = L^2 + \alpha L_z  + \beta L_x $$ Use the splitting of $H$ as $$ H_0=L^2 + \alpha L_z, \qquad \mbox{and} \qquad H^\prime =  \beta L_x $$ in terms of unperturbed Hamiltonian and perturbation Hamiltonian $H^\prime.$

  1. Find the corrections to the $\ell=1$ energy level using  perturbation theory to  the lowest non-vanishing order in $H^\prime$. Compare your answer with the exact answers.
  2. Obtain the eigenvectors of $H$ upto lowest order in $\beta$.
kapoor kapoor's picture 28/12/18

Problem/QM-23017

There is a degenerate two level system. A perturbation \(V\) is applied. The matrix element of \(V\) w.r.t.the unperturbed state are \(V_{11}, V_{12},V_{21},V_{22}\). How does the energy level split?

kapoor kapoor's picture 28/12/18

Problem/QM-23018

  1. For a relativistic harmonic oscillator, show that the Hamiltonian can be approximated by \[ H = \frac{p^2}{2m} +\frac{1}{2}m \omega^2 x^2 -\frac{p^4}{8c^2m^3}\]
  2. Show  that \[\langle{n}\vert{p^4}\vert{0}\rangle = \frac{m\hbar\omega^2}{2}\Big(3\delta_{n,0} -6\surd 2 \delta_{n,2} - 2\surd6 \delta_{m,4}\Big)\]
  3. Calculate the leading non-vanishing energy shift of the ground state due to this relativistic perturbation.
  4. Calculate the leading corrections to the ground state eigenvector \(\vert{0}\rangle\).


{Daniel F Styer*}

kapoor kapoor's picture 28/12/18

Problem/QM-23019

 Find corrections to the eigenvalues of the matrix \(H\) given below to lowest nonvanishing order in \(\epsilon\)
\begin{equation}
H=  \begin{pmatrix}
  1 & \epsilon & 0 \\
  \epsilon & 2 & \epsilon\\
  0 & \epsilon  & 2
  \end{pmatrix}
\end{equation}

kapoor kapoor's picture 28/12/18

Problem/QM-12004

Find the energy eigenvalues and eigenfunctions for a particle in a box, with  coordinates of the boundary given to be \(x=L\) and \(x=2L\). \[ V(x) =  \begin{cases} 0 , & L \le x \le 2L \\ \infty, & x < L \text{ or } x > 2L .\end{cases}\]

ANSWER: \[ E_n= \frac{\hbar^2k_n^2}{2m}, \quad u_n(x) = \sqrt{\tfrac{2}{L}} \sin k_n(x-L)\] where \( k_n = \frac{n\pi\hbar }{L}\)

 

kapoor kapoor's picture 28/12/18

Problem/QM-16002


Use the following data on the wavelengths of the first few lines of the Lyman series of the hydrogen atom to find best fit to the value of the Rydberg constant.

Wavelengths in  Angstrom units :: 1215.68, 1025.83, 972.54

Use your value of the Rydberg constant to predict the wavelengths of the lines
specified below.  

(i) next two lines in the Lyman series.
(ii) the first nine lines in the Balmer series.
(iii) the first three lines of the Ritz Paschen series.
(iv) the first two lines of the Brackett series.
(v) Compare your answers with the experimental results given at the
end and comment on the agreement of the theory with the experiments.

Remarks:
Experimental wavelengths are as follows
(i) Balmer series : 6562.79,
4861.33, 4340.17, 4104.74, 3970.07, 3889.05, 3835.39, 3797.90, 3770.63 , all in
Angstrom units.
(ii) Ritz Paschen Series : 18751.1, 12818.1, 10938 , all in Angstrom units.
(iii) Brackett Series : 2.63 $\mu$, 4.05 $\mu$
Latest value of the Rydberg constant,  R = 109737.3177 $\mbox{cm}^{-1}$




kapoor kapoor's picture 28/12/18

Problem/QM-16004

  1. For a classical electron moving in an orbit around a nucleus with charge $Ze$ obtain a relation between the number of revolutions per second and the energy of the electron. (Use only classical mechanics).
  2. Using known quantum mechanical solution, show that in the limit of large principal quantum number \(n\), the  frequency of the line emitted in a transition from $(n+1)^{\mbox{th}}$ level to the $n^{\mbox{th}}$ level is precisely the same as the frequuency obtained in part \((a)\).

 

kapoor kapoor's picture 28/12/18

Problem/QM-21001

Using a trial wave function of the form $$\psi(x) = C \exp(-\alpha^2 x^2/2) $$ estimate the ground state energy of  a particle in the $\delta-$ function potential $$ V(x) = - \gamma \delta(x) , \qquad \gamma > 0. $$

kapoor kapoor's picture 28/12/18

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