IGNOU MPH 8 SOLVED ASSIGNMENT

Ques 1.

Write the space translation operator in quantum mechanics  for a finite translation a along the x direction. Calculate the commutator . You may use the Baker-Campbell-Hausdorff formula: 

Ques 2.

 Consider an operator  for which . Show that the expectation value of  in a parity eigenstate is zero. 

Ques 3.

Determine the wave function and energy of the ground state and first excited state for a system of two identical bosons in 1D simple harmonic oscillator. 

Ques 4.

Define the action of the permutation operator  for a system of two particles 1 and 2 and two states  and . Show that  and determine the eigenvalues of . 

Ques 5.

Write down the eigenkets  for  with .

Ques 6.

Calculate the matrix elements for J² for a system of two spin half particles. 

Ques 7.

Determine the first and second order perturbation correction to the ground state energy eigenvalue of the one-dimensional infinite potential well of width L () with the perturbation: . 

Ques 8.

Consider the following one-dimension simple harmonic oscillator Hamiltonian operator
$
Use a trial wave function  with a variational parameter  to estimate the upper bound to the ground state energy. 

Ques 9.

 Determine the WKB approximation for the bound state energy of a particle of mass m in the potential:

Ques 10.

Consider the two state problem in which the unperturbed Hamiltonian  has just two eigenkets,  and  with: ; , and E₂ > E₁. The system is subjected to a time-dependent perturbation: .
Calculate the probability for the system to be in the state  at time t, given that it is in the state  at .

Ques 11.

A charged particle of mass m and charge q, is confined to a one-dimensional box of side L with . At t > 0, an electric field  acts on the particle where  is a constant. If the particle is in the ground state when t < 0, calculate the probability that it will be in the first excited state for t > 0.

Ques 12.

Using the Born Approximation, calculate the differential cross-section for a beam of particles of mass m scattered by a potential: . You may use:
$

Ques 13.

Explain how the expression for the energy levels obtained by solving Klein Gordon equation for a Coulomb field differs from the results derived from Schrödinger equation. Why is this solution not able to explain the fine-structure splitting of the energy levels?

Ques 14.

Derive the current conservation equation from the Dirac equation.
 

Ques 15.

Write the space translation operator in quantum mechanics  for a finite translation a along the x direction. Calculate the commutator . You may use the Baker-Campbell-Hausdorff formula: 

Ques 16.

 Consider an operator  for which . Show that the expectation value of  in a parity eigenstate is zero. 

Ques 17.

Determine the wave function and energy of the ground state and first excited state for a system of two identical bosons in 1D simple harmonic oscillator. 

Ques 18.

Define the action of the permutation operator  for a system of two particles 1 and 2 and two states  and . Show that  and determine the eigenvalues of . 

Ques 19.

Write down the eigenkets  for  with .

Ques 20.

Calculate the matrix elements for J² for a system of two spin half particles. 

Ques 21.

Determine the first and second order perturbation correction to the ground state energy eigenvalue of the one-dimensional infinite potential well of width L () with the perturbation: . 

Ques 22.

Consider the following one-dimension simple harmonic oscillator Hamiltonian operator
$
Use a trial wave function  with a variational parameter  to estimate the upper bound to the ground state energy. 

Ques 23.

 Determine the WKB approximation for the bound state energy of a particle of mass m in the potential:

Ques 24.

Consider the two state problem in which the unperturbed Hamiltonian  has just two eigenkets,  and  with: ; , and E₂ > E₁. The system is subjected to a time-dependent perturbation: .
Calculate the probability for the system to be in the state  at time t, given that it is in the state  at .

Ques 25.

A charged particle of mass m and charge q, is confined to a one-dimensional box of side L with . At t > 0, an electric field  acts on the particle where  is a constant. If the particle is in the ground state when t < 0, calculate the probability that it will be in the first excited state for t > 0.

Ques 26.

Using the Born Approximation, calculate the differential cross-section for a beam of particles of mass m scattered by a potential: . You may use:
$

Ques 27.

Explain how the expression for the energy levels obtained by solving Klein Gordon equation for a Coulomb field differs from the results derived from Schrödinger equation. Why is this solution not able to explain the fine-structure splitting of the energy levels?

Ques 28.

Derive the current conservation equation from the Dirac equation.
 

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