IGNOU MSCPH MPH 4 Quantum Mechanics-I Solved Assignment ENGLISH

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MPH 004 (January 2026 - July 2026) - ENGLISH

Tutor Marked Assignment
QUANTUM MECHANICS-I

Course Code: MPH-004
Assignment Code: MPH-004/TMA/2026
Max. Marks: 100

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


PART A

1. a) Electrons are accelerated through a potential 150 V and incident on a crystal with interatomic spacing . Calculate the de Broglie wavelength and the first-order Bragg diffraction angle. 

b) Estimate the uncertainty in the energy of a photon localized within a distance of . 

c) i) Normalize the wave function:
$
ii) Calculate the expectation value of x for a particle in this state. 

d) A quantum mechanical particle in one dimension has the wave function:
$
Use the Schrödinger equation to determine the corresponding potential V(x). 
2. a) A 1.5 mA beam of electrons enters a sharply defined boundary with a velocity , and then its velocity reduces to , due to the difference in potential. Calculate the transmitted and reflected currents. 
b) Show that for any stationary state of a symmetric potential well: .

c) Calculate the expectation value of the potential energy for the first excited state of a simple harmonic oscillator.

d) i) Write down the eigenfunctions for the and states of the hydrogen atom.
ii) Write the eigenvalues of and for the states and .
iii) Calculate the most probable value of r for the hydrogen atom in the state .

PART B

3. a) Consider the following state vectors: . Calculate (i) the norm of and (ii) the inner product . 

b) If an operator is Hermitian and an operator is unitary, show that the operator is Hermitian.

c) The spectral representation of an operator in a two-dimensional orthonormal basis is
$
Determine the matrix elements of .

d) and are the orthonormal basis states of a two-dimensional Hilbert space. A Hermitian operator in this basis is given by the spectral representation: . For a normalized state given by
$

e) Derive the Heisenberg equations of motion for and for the Hamiltonian .

4. a) For the simple harmonic oscillator
i) Show that .
ii) Calculate the matrix element

b) i) Write down the angular momentum states and calculate the matrix elements of and for .
ii) For the angular momentum state show that and .

c) Show that in the terms of the and basis vectors defined by
; , we can write:

MPH 4 2025 - English

Tutor Marked Assignment

QUANTUM MECHANICS-I

Course Code: MPH-004

Assignment Code: MPH-004/TMA/2025

Max. Marks: 100

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

PART A

1.

a) Estimate the kinetic energy of the neutrons in a neutron beam that can be used to probe lattice structures with an interatomic spacing of 0.3 nm. The mass of the neutron is 1.675 x 10-²⁷ kg.

b) Calculate the probability current density for the wavefunction: w(x) = f(x)exp[ig(x)].

c) The wavefunction of particle of mass m confined to move in one dimension is ψ(x) = Nx exp(-ax), 0 ≤ x ≤∞. Calculate the normalization constant N and the expectation value of 1/x.

d) Show that [x², px²] = 2ih(xpx +p _xx)

2.

a) Consider a one dimensional potential well with an infinite barrier at x = 0 and a finite potential V for x ≥ L. Solve the Schrödinger equation for a particle of mass m inside the well with an energy E <V. Using the boundary conditions, derive the equation for its eigen energies.

b) Calculate the probability that a simple harmonic oscillator in its ground state will be found beyond the classical turning points.

c) Consider an electron in the state

where , are the Hydrogen atom eigenfunctions. Determine the normalization constant N and the expectation values of the energy, L² and L^z.

PART B

3. a) Show that a linear combination of the degenerate eigenvectors of an operator belonging to a particular eigenvalue of the operator is also an eigenvector belonging to the same eigenvalue.

b) The orthonormal bases for a three-dimensional Hilbert space is described by . The action of an operator Ô in this space is given by:

Obtain the matrix representation of this operator.

c) A two state system has the orthonormal energy eigenkets IE₁) and IE2) with eigenvalues E_₁ and E₂. If the initial state of the system is given by, determine .

d) Derive the Heisenberg equations of motion for the simple harmonic oscillator.

4.

a) i) Show that the Hamiltonian for a simple harmonic oscillator can be written in terms of the raising and lowering operators as Ĥ = 

ii) Obtain the value of * (n|p^-2|n) for the eigenket | n) of t of the simple harmonic oscillator.

b) i) Write down the angular momentum states |j,mj) for j = 2 and the eigenvalue of J² and J_₂ for each of these states.

ii) Determine J+ 2,1) and J_|2,1).

c) The Hamiltonian for an electron at rest in a magnetic field Bo along the z-direction is , where . Given that the initial state of the system is  determine the state and  at al later time t.

 

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