IGNOU MSCPH MPH 8 Quantum Mechanics-II Solved Assignment ENGLISH

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MPH 008 (January 2026 - July 2026) - ENGLISH

Tutor Marked Assignment
QUANTUM MECHANICS-II

Course Code: MPH-008
Assignment Code: MPH-008/TMA/2026
Max. Marks: 100

Note: Attempt all questions. The marks for each question are indicated against it.


PART A

1. a) 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:  

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

2. a) 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. 

b) 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 . 

3. a) Write down the eigenkets for with .

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

4. 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: . 

5. 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. 

PART B

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

7. 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 .

8. 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.

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

10. a) 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?
b) Derive the current conservation equation from the Dirac equation.
 

MPH 8 2025 - English

Tutor Marked Assignment QUANTUM MECHANICS-II

Course Code: MPH-008

Assignment Code: MPH-008/TMA/2025

Max. Marks: 100

Note: Attempt all questions. The marks for each question are indicated against it.

PART A

1. a) Write the space translation operator in quantum mechanics ) ˆT for an infinitesimal translation (a). Hence evaluate the commutators: and

b) Show that for a Hamiltonian  to commute with the time reversal operator the potential function V(x) must be real. 

2. a) Construct the wave function  for a system of 3 identical fermions in the states  respectively of an one-dimensional box of size A 

b) Calculate the ground state energy for a system of five identical spin  particles placed In a one-dimensional simple harmonic oscillator potential of frequency

3. For a system of two particles each with angular momentum one

(i) Construct the normalized states of highest and second highest J_z for total angular momentum 2.

(i) Construct the normalized state of highest J₂ for total angular momentum 1. 

4. 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: 

 

5. 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:

PART B

6. 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.

7. 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.

8. 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)

9. For a Dirac particle moving in a central potential, show that the orbital angular momentum is not a constant of motion.

10. 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. 

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