Calculate the zero-point energy of a system consisting of a mass of 10^{-3}\,\mathrm{kg} connected to a fixed point by a spring which is stretched by 10^{-2}\,\mathrm{m} by a force of 10^{-1}\,\mathrm{N}. The system is constrained to move only in one direction.
The ground state wave function for hydrogen atom is \psi(r)=\frac{1}{\sqrt{\pi a_0^3}}e^{-r/a_0} where a_0 is the Bohr radius. Sketch the wave function and the probability density as a function of the separation distance r. Calculate the probability that the electron in the ground state is found beyond the Bohr radius.
The wave function of a particle is given as \psi(x)=\frac{1}{\sqrt{a}}e^{-\lvert x\rvert/a}. Find the probability of locating the particle in the range -a\leq x\leq a.
Prove the following identities:
(i) [\hat{p}_x,\hat{L}_y]=i\hbar\hat{p}_z
(ii) e^{i(\hat{\sigma}\cdot\hat{n})\theta}=\cos\theta+i(\hat{\sigma}\cdot\hat{n})\sin\theta
The general wave function of harmonic oscillator (one-dimensional) are of the form u_n(x)=\sum_{k=0}^{\infty}a_k y^k e^{-y^2/2} With y=\sqrt{\frac{m\omega}{\hbar}}x, and coefficients a_k are determined by recurrence relations a_{k+2}=\frac{2(k-n)}{(k+1)(k+2)}a_k Corresponding energy levels are E_n=\left(n+\frac{1}{2}\right)\hbar\omega. Discuss the parity of these wave functions. What happens, if the potential for x\leq 0 is infinite (half harmonic oscillator)?
Calculate the radius of electron orbit for \mathrm{Li}^{++} in ground state.
Show that for free electron gas, the density of states in three dimensions (3D) varies as E^{1/2}, and this dependence changes to E^0 for 2D (quantum well), E^{-1/2} for 1D (quantum wire) and \delta function for 0D (quantum dot).
What is Zeeman effect? Discuss the factors on which Larmor frequency is dependent.
If the temperature variation of heat capacity is known, how do you calculate the change of entropy during an isochoric process? According to Debye's theory of specific heat of a solid, the molar heat capacity of diamond crystal at constant volume varies with temperature (T) as follows: c_v=\frac{12}{5}\pi^{4}R\left(\frac{T}{\Theta}\right)^{3} where R is the molar gas constant =8.315\ \mathrm{J/mol\ K} and \Theta=2230\ \mathrm{K} for diamond. Calculate the change in entropy of diamond of 0.36 g mass when it is heated at constant volume from 0 K to 300 K.
The pressure on 100 g of solid copper is increased quasi-statically and isothermally at 0^\circ\mathrm{C} from 0 to 0.5 \times 10^{8}\ \mathrm{Pa}. Assuming the density and isothermal compressibility to remain at constant values of 8.96\ \mathrm{g/cm^3} and 7.16 \times 10^{-12}\ \mathrm{Pa^{-1}}, respectively, calculate the work done. Comment on the sign and magnitude of work.
A system having two energy levels, -\frac{1}{2}\Delta and +\frac{1}{2}\Delta with \Delta=10\mathrm{meV} is populated by 1000 particles at a low temperature close to 100\ \mathrm{K}. Obtain the average energy per particle using classical distribution law.
At 4^{\circ}\mathrm{C} temperature, the density of water is found to be maximum. Prove that heat capacity at the constant pressure (c_p) is equal to the heat capacity at constant volume (c_v) for water at 4^{\circ}\mathrm{C}.
Schematically, show the variation of density of states, D(\varepsilon) and distribution function, f(\varepsilon,T), of particles in a non-relativistic Fermi gas at high temperatures. At a temperature T, an electron occupies a state with energy 100\mathrm{meV} above the Fermi energy (\varepsilon_{\mathrm F}) with the probability of 1\%. Find the temperature T.
One mole of a gas obeys the following equation of state: \left(P + \frac{a}{v^{2}}\right)(v-b)=RT, where v is the molar volume and, a and b are constants. Show that internal energy of the gas increases as the volume increases, with the temperature remaining constant.
Write down Maxwell's equations in integral form. Explain the significance of each of these equations.
A 0.5\,\mathrm{m} long cylindrical medium between two conducting plates has uniform charge density of 100\,\mathrm{nC/m^3}. The axis of the cylindrical medium is along z-axis. The left plate is at z=0 and has a potential of 10\,\mathrm{kV} and the right plate is grounded. Determine the electric field at axial distance z=0.2\,\mathrm{m}.
Two solenoids have 500 and 800 turns of wire and are placed co-axially close to each other. A current of 5.0 A in the first solenoid produces an average flux of 200\mu\mathrm{Wb} through its each turn and a flux of 100\mu\mathrm{Wb} through each turn of the second solenoid. Find the self-inductance of the first solenoid and the mutual inductance of the solenoids.
Define a plane electromagnetic wave. A plane polarized wave is incident on the interface between two dielectric media. Obtain expressions for the amplitudes of the reflected and transmitted waves when the incident wave is polarized with its electric field B vector perpendicular to the plane of incidence. Discuss the phase relationships of the reflected and transmitted waves with respect to the incident wave.
A parallel plate capacitor has plate area =4.0\ \mathrm{cm^2} and plate separation =2.0\ \mathrm{mm}. An a.c. voltage V=20\sin(5\times10^{3}t) volts is applied across the plates. If the dielectric constant of the medium between the plates is \varepsilon_r=2.0, calculate the displacement current.
A uniformly magnetized sphere of radius R has magnetization \vec{M}=M_0\hat{z}. If the scalar magnetic potentials inside and outside the sphere are given as under \phi_m=\frac{M_0}{3}z;\ r\leq R and \phi_m=\frac{M_0R^3}{3r^2}\cos\theta;\ r>R where, r,\theta are two spherical coordinates, find the magnetic field inside and outside the sphere.
