M.Sc. (Semester II) (CBCS) Examination Mar/Apr-2018
Physics (Materials Science)
QUANTUM MECHANICS
Time: 2½ Hours
Max. Marks: 70
Instructions: Q.1 and Q.2 are compulsory. Attempt any three questions from Q. 3 to 7. Use of Non programmable calculator is allowed. All questions carry equal marks.
Q.1

Choose the correct alternative:
06
Heisenberg's uncertainty principle states
A particles position can be measured exactly
A particle's energy can be measured exactly
The more precise a particle's momentum can be measured, the less precise its position can be measured
The more precise a particle's momentum can be measured, the less precise its energy can be measured
The wave function for a particle must be normalizable because.
The particle's charge must be conserved
The particle's momentum must be conserved
The particle must be somewhere
The particle's angular momentum must be conserved
A particle has a total energy that is less than that of a potential barrier. When the particle penetrates the barrier, its wave function is
Exponentially decreasing
Exponentially increasing
A positive constant
Oscillatory
According to Schrödinger, a particle is equivalent to a
Single wave
Wave packet
Light wave
Cannot behave as wave
The energies of a particle in a box are given by
Continuous energy spectrum
n2 ħ22mL2
ħ22mL2n2

The wave function in the ground state of hydrogen atom is given as where r measures distance from nucleus and a is constant. The value of A is
1
1 πa3
1 π.a
1 πa5
Page 2 of 2
SLR-UN-475

State True or False
08
Bound states must vanish at infinity.
The time development of a wave function is Ψ.
The eigenfunctions belonging to different eigenvalues of a unitary operator are mutually orthogonal.
In a non-linear molecule where electronic degeneracy occurs there always exists a vibrational mode which can remove the degeneracy
The combined space and spin function of an electron is called a spin-orbital.
The wave functions for different states of a harmonic oscillator are mutually orthonormal.
For many electron atoms, the electron repulsion terms must be excluded in the potential energy term of the wave equation.
The Born-Oppenheimer approximation is not valid as long as the various energy levels in a molecule are widely separated from each other.
Q.2
Write a short note on:
Shape of atomic orbitals
05
Characteristics of the wave functions
04
Show that the operators commustes with
05
Q.3
V x =0→∞ x
Consider a symmetric "1-D rigid box" of length 2a,
Obtain the energy eigenvalues and eigenfunctions.
08
Normalize the energy eigen functions for a particle in a symmetric 1-D finite box (only odd parity)
06
Q.4
θ,∅ θ ∅±m ∅
Obtain the total wave function of a rigid rotator in the form
08
Calculate the spherical harmonics: a γ0,0 θ,∅ b θ,∅
06
Q.5
Show that how the Hartree and Hartree Fock self-consistent field methods are powerful for obtaining the ground state energy and wave functions of many-electron atoms.
10
Write down the 1s orbital of the hydrogen atom and obtain the probability density Ψ1s 2
04
Q.6
What is the Born-Oppenheimer approximation? Write and interpret each term of the wave equation for it.
08
How the linear combination of atomic orbitals (LCAO) is the basis for the calculation of approximate energies and molecular orbitals in molecules? Explain.
06
Q.7
Explain the fourth postulate of quantum mechanics.
08
Prove that, if two operators A and B commute then they have the same set of eigenfunctions.
06