Debye temperature for diamond is 2230 K. Calculate the frequency of highest possible lattice vibration in diamond and its molar heat capacity at 20 K
See Answer →If the potential energy function is expressed as calculate the inter-molecular distance at which the potential energy is a minimum. Show that in the stable configuration the energy of attraction is eight times the energy of repulsion.
At what angle will a diffracted beam emerge from the (110) planes of a cubic crystal of unit cell length 0.6 nm? Assume diffraction occurs in the first order and that the X-ray wavelength is 0.154 nm ?
See Answer →Show that the volume of the primitive cell in the reciprocal lattice space is inversely proportional to the volume of the primitive cell in the direct lattice.
See Answer →A metallic element has a density of 7.15 g cm-3, a lattice constant of 2.880 A and an atomic weight of 51.9961. Calculate the number of atoms per unit cell of this element and predict its lattice crystal structure.
See Answer →Consider the scattering of a particle of mass m by the following potential Assuming that the scattering is mainly due to s-waves (1-0), calculate the s-wave phase shift.
For a Dirac particle moving in a central potential, show that the orbital angular momentum is not a constant of motion.
For a static (time-independent) perturbation V suddenly switched on at time t=0:
show that the transition probability for a transition of the system from a state
up to first order in perturbation theory is just:
How does the transition probability change if the time of application is doubled in the limit t→0. (6+4)
A particle is in the ground state of the simple harmonic oscillator of frequency. A t =0 frequency of the oscillator changes from particle to remain in the ground state at Calculate the probability for the particle to remain in the ground state at t > 0
Consider the motion of a quantum particle in a potential V(x)-ax. Use the WKB approximation to determine how the energy of the bound state E, varies with n and a for large values of n.
See Answer →Determine the upper bound to the ground state energy for a particle in a one-dimensional box of length L. using the trial wave function:
Consider the following symmetric two-dimensional infinite potential well otherwise
Determine the first order perturbation correction to the energy eigenvalue of the two-fold degenerate first excited state, for the following perturbation:
For a system of two particles each with angular momentum one
(i) Construct the normalized states of highest and second highest Jz for total angular momentum 2.
(i) Construct the normalized state of highest J₂ for total angular momentum 1.
See Answer →Calculate the ground state energy for a system of five identical spin particles placed In a one-dimensional simple harmonic oscillator potential of frequency
Construct the wave function for a system of 3 identical fermions in the states
respectively of an one-dimensional box of size A
Show that for a Hamiltonian to commute with the time reversal operator the potential function V(x) must be real.
Write the space translation operator in quantum mechanics ) ˆT for an infinitesimal translation (a). Hence evaluate the commutators: and
A charged particle is moving with relativistic speed in a static, uniform electric field E-(E,0,0) along the x-axis. Assume that the particle is initially at rest so that the motion is effectively one-dimensional. Express the energy of the particle as a function of instantaneous position x(t) and solve the resulting equation of motion to obtain the value of x(t).
See Answer →Explain the phenomenon of time dilation and obtain expression relating the time intervals between two event observed by different inertial observers.
See Answer →Obtain the expressions for scalar and vector potentials due to an oscillating electric dipole at a point located at large distance from the dipole.
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