Quantum Mechanics I : Final Examination

4th October, 1999

General Instructions

• Attempt all 20 problems.
• All carry the same mark.
• You can use calculators but order-of-magnitude answers are also acceptable.
• No lengthy calculation is required.

1. Prove that isotropy of space is equivalent to the condition $[H, L] = 0$.
2. Show that $(p_x, p_y, p_z, L_x, L_y, L_z)$ are closed with respect to commutation (i.e., commutation between two elements of the set produces another element of this same set). This set forms a Lie-group, known as the Euclidean group $E_3$ in 3-D.
3. Do the hydrogen atom energy levels get modified significantly at the surface of a Neutron Star?
4. A recently discovered class of Neutron Stars have a surface magnetic field of $\sim 10^{15}$ G. These are known as Magnetars. Find the Zeeman splitting of the ($n=1, l=1$) state of hydrogen atom near a Magnetar.
5. The ionised hydrogen regions of Interstellar medium are known as HII region. The HII region formed around a bright star is known as the Stromgren Sphere. Find the radius of the Stromgren Sphere centred around a star of radius $R$ and surface temperature $T$ where the density of interstellar medium is $N$.
6. $H = \frac{p^2}{2m} + V_1 + i V_2$, where $V_1, V_2$ are real. Show that the probability is not conserved but has a source/sink.
7. Quantise the system consisting of two neutral particles of masses equal to those of electron and proton held together by gravity. Obtain the expression for the energy eigenvalues.
8. Show that in the limit $\omega \rightarrow 0$ the harmonic oscillator wave-functions approach free-particle limit.
9. Two particles of equal mass are placed in the potential $V = \frac{1}{2}\omega^2 (x_1^2 + x_2^2 + 2 \lambda x_1 x_2)$. Find the energy eigenvalues.
10. Show that the Laplace-Runge-Lenz vector ${\bf A} = \frac{1}{2 \mu} ({\bf L} \times {\bf p} - {\bf p} \times {\bf L}) - \frac{e^2 {\bf r}}{r}$ is a constant motion for hydrogen atom.
11. Find the zero point of the following system - a mass of $1$  gm is connected to a fixed point by a spring which is stretched by $1$ cm when subjected to a force of $10^4$ dyne. The mass point is forced to move only in one direction.
12. For $[a, a^{\dagger}] = 1$ prove that $[a^{\dagger}a, a] = -a$ and $[a^{\dagger}a, a^{\dagger}] = a^{\dagger}$. Show that with $\{a, a^{\dagger}\} = 1$ the above relations would still be valid but the number operator $N = a^{\dagger}a$ now has the eigenvalues 0 and 1. As you might have guessed, the former case refers to the Bosons and the latter to the Fermions in the second-quantised formulation.
13. $J_{+/-} = J_x +/- i J_y, [J_+, V+] = 0, [J_z, V_+] = V_+$
    If $\vert j,m>$ are the simultaneous eigenstates of $J^2, J_z$ then show that $V_+\vert j,j> \propto \vert j+1,j+1>$.
14. $\vec a = \frac{1}{\sqrt 2} (\vec X + i \vec P), \vec a^{\dagger} = \frac{1}{\sqrt 2} (\vec X - i \vec P),$
    where, $\vec X = (\frac{m \omega}{\hbar})^{1/2} \vec x, \vec P = {m \omega \hbar}^{-1/2} \vec p$. Show that $b, b^{\dagger}$ change the angular momentum by one unit, whereas $a^2$ changes the energy by one unit, where $b = a_x + i a_y$ and $b^{\dagger} = a_x^{\dagger} + i a_y^{\dagger}$.
15. Consider a relativistic electron with $K.E. = \frac{p^2}{2m} - \frac{p^4}{8m^3c^2}$. Assuming the extra term to be a perturbation show that the modification to the Hydrogen atom energy levels is given by $\Delta E = \left[ \frac{3}{8n^4} - \frac{1}{(2l+1)n^3}\right] \frac{me^4}{\hbar^2} (\frac{e^2}{\hbar c})^2$
16. A mono-energetic beam of particles is incident on a set of N identical static potentials. Show that the differential cross-section in the lower Born-approximation is
    

    $$\sigma(\vec q) = \sigma_0 (\vec q) \left\vert \Sigma_{n=1}^N e^{i \vec q. \vec r_n} \right\vert^2$$ (1)

    
    
    where $\hbar \vec q$ is the momentum transfer on collision, $\sigma_0(\vec q)$ is the cross-section for scattering off one potential and $\vec r_n$ is the centre of the nth potential.
17. Consider the perturbation $H^{\prime} = \lambda x$ over a 1-D harmonic oscillator potential. Find the change in the ground state energy. What happens if $H^{\prime} = \lambda \vert x\vert$?
18. Consider a hydrogen atom fixed at the origin. Two charges $Q_1$ and $Q_2$ are placed on z-axis, at $z = +/- b$ where $b » a_0$ and $Q_1, Q_2 « e$. What can be said about the states when $Q_1 = Q_2$ and $Q_1$ not equal to $Q_2$?
19. Consider an alkali atom placed in an external magnetic field. Find the minimum value of the magnetic field when the spin-orbit coupling term and Zeeman term in the Hamiltonian becomes equal.
20. A half-square-well potential is smoothed at the edges symmetrically, as shown in fig.() . How would the energy $E_0$ of the bound state change under such modification of the potential?

Figure: Question No. 20