Download PTU (I. K. Gujral Punjab Technical University) MSc -Chemistry 2nd Semester May 2019 71665 QUANTUM CHEMISTRY Question Paper.
Roll No. Total No. of Pages : 03
Total No. of Questions : 09
M.Sc.(Chemistry) (2015 to 2017) (Sem.?2)
QUANTUM CHEMISTRY
Subject Code : MSCH-204
M.Code : 71665
Time : 3 Hrs. Max. Marks : 100
INSTRUCTION TO CANDIDATES :
1. Atttempt FIVE questions in all selecting ONE question from each UNIT. All questions
carry equal marks.
2. Q. No. 1 is Compulsory.
1. Answer Briefly : (2?10=20)
a) Determine whether the following operator is linear or nonlinear :
^
2
Af(x)= x f(x)
b) Show that the functions ?, ? ? and 2i ? represent same state; ? being real.
c) Calculate the number of radial node and angular node of 4d orbital.
d) What is the complex conjugate of the wave function ( ? = 4 + 3i)?
e) Calculate the number of degenerate states for Hydrogen atom for n = 4.
f) Determine whether the given statement is true or false. Justify your choice.
The function exp [? ?x
2
] is an acceptable wave function.
g) Write down the Hamiltonian equation of He atom.
h) A particle in one dimensional box simple harmonic oscillator in x-direction is
perturbed by a potential ?x. What is the 1
st
order correction for ground state?
i) Calculate the magnitude of the angular momentum of an electron that occupies the
following atomic orbitals: 1s and 3d.
j) Calculate the number of radial node and angular node of 3p orbital.
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1 | M-71665 (S17)-1724
Roll No. Total No. of Pages : 03
Total No. of Questions : 09
M.Sc.(Chemistry) (2015 to 2017) (Sem.?2)
QUANTUM CHEMISTRY
Subject Code : MSCH-204
M.Code : 71665
Time : 3 Hrs. Max. Marks : 100
INSTRUCTION TO CANDIDATES :
1. Atttempt FIVE questions in all selecting ONE question from each UNIT. All questions
carry equal marks.
2. Q. No. 1 is Compulsory.
1. Answer Briefly : (2?10=20)
a) Determine whether the following operator is linear or nonlinear :
^
2
Af(x)= x f(x)
b) Show that the functions ?, ? ? and 2i ? represent same state; ? being real.
c) Calculate the number of radial node and angular node of 4d orbital.
d) What is the complex conjugate of the wave function ( ? = 4 + 3i)?
e) Calculate the number of degenerate states for Hydrogen atom for n = 4.
f) Determine whether the given statement is true or false. Justify your choice.
The function exp [? ?x
2
] is an acceptable wave function.
g) Write down the Hamiltonian equation of He atom.
h) A particle in one dimensional box simple harmonic oscillator in x-direction is
perturbed by a potential ?x. What is the 1
st
order correction for ground state?
i) Calculate the magnitude of the angular momentum of an electron that occupies the
following atomic orbitals: 1s and 3d.
j) Calculate the number of radial node and angular node of 3p orbital.
2 | M-71665 (S17)-1724
UNIT-I
2. a) If A is a linear operator and A ?
1
= a ?
1
and A ?
2
= a ?
2
then prove that any linear
combination of ?
1
and ?
2
say C
1
?
1
+ C
2
?
2
or C
1
?
1
? C
2
?
2
is an eigen function of ?A?
with the same eigen value ?a? where C
1
and C
2
are constants.
State Heisenberg?s uncertainty principle and using it show that electrons cannot
reside in nucleus.
b) Calculate the expectation value of x-component of momentum of a free particle in a
box of length 1,
2
= sin
n x
t l
? ? ?
?
? ?
? ?
. Show that e
ax
is an eigen function of the operator
d
n
/dx
n
. What is the eigen value? Prove that eigen values of Hermitian operator are
real. (10, 10)
3. a) Write down the quantum mechanical postulates with proper explanation.
b) For the ground state of a particle in 1-d box, calculate
> and <(p
x
)
2
>. Explain the
physical interpretations of your outcomes. (10, 10)
UNIT-II
4. a) Find out the probability of finding the 1s electron within the first Bohr orbit a
0
.
Tabulate all of the allowed microstates of p
2
electronic configuration.
b) Sketch ? and | ? |
2
for n = 1, n = 2 states of a particle in a one dimensional box of
length 1 and indicate the most likely locations of the particle in these states. (10, 10)
5. a) Plot the shapes (polar plots) of the atomic orbitals corresponding to 2p
x
, 2p
y
and 2p
z
for a hydrogen-like atom using the following equations :
2 2 2
Asin cos , A sin .sin A cos
x y z
p p p
and ? ? ? ? ? ? ? ? ? ? ?
Where,
5/2 /2
1
4 2
Zr
A Z re
?
?
?
. Denote the range of ? and ? used for the polar plots and
label the axes properly.
b) Find out the probability density of finding the 1s electron of hydrogen atom described
by the wave function
0
3/2
/2
0 0
1 1
2
r a
r
e
a a
?
? ? ? ?
?
? ? ? ?
?
? ? ? ?
at the nucleus and at a distance a
0
from the nucleus. Also find out the relative probability of finding the 1s electron in
Bohr?s first orbit and at a distance of 1 ? 10
?4
a
0
from the nucleus. (10,10)
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1 | M-71665 (S17)-1724
Roll No. Total No. of Pages : 03
Total No. of Questions : 09
M.Sc.(Chemistry) (2015 to 2017) (Sem.?2)
QUANTUM CHEMISTRY
Subject Code : MSCH-204
M.Code : 71665
Time : 3 Hrs. Max. Marks : 100
INSTRUCTION TO CANDIDATES :
1. Atttempt FIVE questions in all selecting ONE question from each UNIT. All questions
carry equal marks.
2. Q. No. 1 is Compulsory.
1. Answer Briefly : (2?10=20)
a) Determine whether the following operator is linear or nonlinear :
^
2
Af(x)= x f(x)
b) Show that the functions ?, ? ? and 2i ? represent same state; ? being real.
c) Calculate the number of radial node and angular node of 4d orbital.
d) What is the complex conjugate of the wave function ( ? = 4 + 3i)?
e) Calculate the number of degenerate states for Hydrogen atom for n = 4.
f) Determine whether the given statement is true or false. Justify your choice.
The function exp [? ?x
2
] is an acceptable wave function.
g) Write down the Hamiltonian equation of He atom.
h) A particle in one dimensional box simple harmonic oscillator in x-direction is
perturbed by a potential ?x. What is the 1
st
order correction for ground state?
i) Calculate the magnitude of the angular momentum of an electron that occupies the
following atomic orbitals: 1s and 3d.
j) Calculate the number of radial node and angular node of 3p orbital.
2 | M-71665 (S17)-1724
UNIT-I
2. a) If A is a linear operator and A ?
1
= a ?
1
and A ?
2
= a ?
2
then prove that any linear
combination of ?
1
and ?
2
say C
1
?
1
+ C
2
?
2
or C
1
?
1
? C
2
?
2
is an eigen function of ?A?
with the same eigen value ?a? where C
1
and C
2
are constants.
State Heisenberg?s uncertainty principle and using it show that electrons cannot
reside in nucleus.
b) Calculate the expectation value of x-component of momentum of a free particle in a
box of length 1,
2
= sin
n x
t l
? ? ?
?
? ?
? ?
. Show that e
ax
is an eigen function of the operator
d
n
/dx
n
. What is the eigen value? Prove that eigen values of Hermitian operator are
real. (10, 10)
3. a) Write down the quantum mechanical postulates with proper explanation.
b) For the ground state of a particle in 1-d box, calculate
> and <(p
x
)
2
>. Explain the
physical interpretations of your outcomes. (10, 10)
UNIT-II
4. a) Find out the probability of finding the 1s electron within the first Bohr orbit a
0
.
Tabulate all of the allowed microstates of p
2
electronic configuration.
b) Sketch ? and | ? |
2
for n = 1, n = 2 states of a particle in a one dimensional box of
length 1 and indicate the most likely locations of the particle in these states. (10, 10)
5. a) Plot the shapes (polar plots) of the atomic orbitals corresponding to 2p
x
, 2p
y
and 2p
z
for a hydrogen-like atom using the following equations :
2 2 2
Asin cos , A sin .sin A cos
x y z
p p p
and ? ? ? ? ? ? ? ? ? ? ?
Where,
5/2 /2
1
4 2
Zr
A Z re
?
?
?
. Denote the range of ? and ? used for the polar plots and
label the axes properly.
b) Find out the probability density of finding the 1s electron of hydrogen atom described
by the wave function
0
3/2
/2
0 0
1 1
2
r a
r
e
a a
?
? ? ? ?
?
? ? ? ?
?
? ? ? ?
at the nucleus and at a distance a
0
from the nucleus. Also find out the relative probability of finding the 1s electron in
Bohr?s first orbit and at a distance of 1 ? 10
?4
a
0
from the nucleus. (10,10)
3 | M-71665 (S17)-1724
UNIT-III
6. a) Calculate the energy value of H
2
molecule ion by using LCAO-MO wave function.
b) Write a short note on degenerate perturbation theory. (10,10)
7. a) Briefly describe the differences between perturbation method and variation method.
Calculate the bond order of the following molecules : (i) He
2
, (ii) H
2
, (iii) H
2
2+
,
(iv) He
2
2+
and (v) H
2
+
b) State and prove the variation theorem. (10,10)
UNIT-IV
8. a) Derive the Huckel MO theory for ethylene/ethane. Draw simple schematics of the
bonding and anti-bonding energy level diagrams.
b) Draw and explain the MO diagram of H
2
O. (10,10)
9. a) Derive the Huckel MO theory for 1,3-butadiene. Draw simple schematics of the
bonding and anti-bonding energy level diagrams.
b) Write a short note on Born-Openheimer approximation method. (10,10)
NOTE : Disclosure of Identity by writing Mobile No. or Making of passing request on any
page of Answer Sheet will lead to UMC against the Student.
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This post was last modified on 05 December 2019