MPH-008 Solved Assignment 2026

Description

MPH-008 Solved Assignment 2026 Available

Assignment Code: MPH-008/TMA/2026

PART A

1. a) Write the space translation operator in quantum mechanics hat T(a) for a finite

translation a along the x direction. Calculate the commutator [ hat x , hat T(a) ]. You may use the Baker-Campbell-Hausdorff formula: (5) bullet^ lambda hat B bullet^ – lambda = hat B +[ hat A , hat B ]+ 1/2 [ hat A ,[ hat A , hat B ]]+…

b) Consider an operator O for which dot H ^ dagger delta dot H =- dot O Show that the expectation value of O 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 10 simple harmonic oscillator.

b) Define the action of the permutation operator tilde P_{12} for a system of two particles 1 and 2 and two states | forall P rangle and | psi P’ rangle Show that hat A_{2} ^ 2 = hat I and determine the eigenvalues of tilde P_{12}

3. a) Write down the eigenkets | j ,m j rangle for j = l_{1} + l_{2} with j_{1} = 2; j_{2} = 1/2

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: (0 <= x <= L) H_{T}(x) = V_{0}*nin((pi*x)/L)

5. Consider the following one-dimension simple harmonic oscillator Hamiltonian operator

Use a trial wave function psi(x) = Naxp(- (x ^ 2)/(2a ^ 2)) with a variational parameter a to hat H =- hbar^ 2 2m * (d ^ 2)/(d * x ^ 2) + 1/2 * mo * s ^ 2 * x ^ 2 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:

V(x) = d|x| x >= 0; 2s|x| x < 0

7. Consider the two state problem in which the unperturbed Hamiltonian tilde H_{0} has just two eigenkets, | 1 rangle and | 2 rangle with: tilde H 0 |1 rangle=E 1 |1 rangle; tilde H 0 |2 rangle – E_{2} |2 rangle. and E_{2} > E_{1} . The system is subjected to a time-dependent perturbation: hat V(t) =V 0 cos(cot[|1 rangle langle2|+|2 rangle langle1|].

Calculate the probability for the system to be in the state 2) at time t, given that it is in the state | 1 rangle at t = 0

8. A charged particle of mass m and charge q, is confined to a one-dimensional box of side L with 0 <= x <= L At t > 0 an electric field vec E =E 0 theta ^ (- alpha * t) vec i acts on the particle where a 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: int 0 theta ^ (- alpha * x) * sin(beta*x) dx= beta alpha^ 2 + beta^ 2 for alpha > 0 You may use: V(r) = V_{0} * a/r * exp(- r/a)

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.