Browse & Filter

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.

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 $\beta$.

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*}

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 \( k_n = \frac{n\pi\hbar }{L}\)

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 \((a)\).

An arbitrary quantum mechanical system is initially in the state $\vert{0}\rangle$. At time $t=0$ a perturbation of the form $H^\prime =H_0 \exp(-t/T)$ is swithced on. Show that at large times the probability of the system being in the state \(\vert{1}\rangle\) is given by $$ \frac{|\langle0\vert H_0\vert1\rangle|^2}{(\hbar/T)^2 + (\Delta E)^2} $$ where $\Delta E$ is the energy difference in the states of $\vert{0}\rangle$ and $\vert{1}\rangle$.

Write the energy eigenvalues and eigenfunctions for a particle in a rigid box \(0\le x \le 2L\). What is the probability that a particle  initially in the ground state will be found in the first excited state after the size of box is suddenly halved by moving the wall at \(x=0\)  to \(x=L\).}

Consider a particle of charge \(q\) and mass \(m\), which is in simple harmonic motion along the \(x-\)axis with Hamiltonian is given by \[ H_0 = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + \frac{1}{2}m\omega^2 x^2.\] A homogeneous electric field \(\mathcal{E}(t)\) is applied along the \(x\)-axis \[\Eca(t)= \mathcal{E}_0 e^{-(t/\tau)^2}\] where \(\mathcal{E}_0\) and \(\tau\) are constants. If the oscillator is in the ground state at \(t=0\), find the probability it will be in the \(n^\text{th}\) excited state as \(t\to\infty\).

Source{Bransden and Jochain-9.2}

A particle of charge $e$ is confined to a two dimensional square box of sides $L$. An electric field $\vec{E}$ given below is applied to the system. $$  \vec{E} = \left\{ \begin{array}{cl} 0, & t< 0 \\ \vec{E_0} \exp(-\alpha t), & t > 0 \end{array} \right. $$ where $\alpha>0$. The vector $E_0$ is perpendicular to one of the sides of the box. To the lowest order in $E_0$ calculate the probability that the charged particle, in the ground state at time $t=0$, is excited to the first state at time $t=\infty$.

Source{Bransden and Jochain 9.1*}