Quantum Chemistry (CHMS4207)
BSc, Chemistry - CHMS
Semester: Second Semester
Level: 400
Year: 2018
THE UNIVERSITY OF BAMENDA FACULTY OF SCIENCE
DEPARTMENT OF CHEMISTRY
CHMS4207: QUANTUM CHEMISTRY TEST 31/05/2018
TIME ALOWED: 3 HOURS
ANSWER ALL THE QUESTIONS
m
e
= 9.1X10
-31
kg; m
p
= 1.67xl0
-27
kg a
o
= 5.30nm
1. The wavefunction of a particle of mass m at a time t = 0 is:
=
exp
where a =1.0000 nm.
(a) Show that the wavefunction is normalized. (b) Estimate the probability that the particle lies between x
= 1.5000 nm and 1.5001 nm. (c) Calculate the kinetic energy of the particle in electronvolts. The mass
of the particle is 9.1 x 10
-28
g.
2. (a) A particle of mass m is in a 2 dimensional well of sides a in the x-direction, and b in the y-direction.
The quantum numbers in the x and y directions are n
x
and n
y
respectively.
(i) Write out the eigen function for particle.
(ii) Write the expression for the eigenvalue of the particle.
(iii) Determine the zero-point energy of the particle.
(iv) Deduce whether the wavefunction is an eigen function of the momentum operator.
(b) Consider that the particle is in a one dimensional box with both sides having finite potential energy U.
The expression of the energy of the particle is
tan
=
. if f1 = tan
and f2 =
, and
L=2.5 nm; m = 9.1 x 10
-28
g; U = 1 eV,
(i) calculate f
2
for E = 0.4 eV. (ii) Describe briefly how you will obtain the quantized energies.
3. (a) The eigenfunctions of a harmonic oscillator of mass m, and force constant k is given by the
equation:
(y) = A
n
H
n
(y)exp
Where, A
n
=
!
and H
n
(y)=(-1)
n
exp(y
2
)
"
"
exp(-y
2
), and x=
(i) What names are given to A
n
and H
n
(y)? (ii) Show that H
2
(y) =4y
2
-2.
(ii) Write out the expression for the wavefunction,
(y)
(b). The energy of the oscillator is E
n
=h v(n +
#
$
!
).
(i) Determine the zero-point energy, and the also E
2
(ii) Determine the turning points, for the state n = 2. (iii)
Using the state n = 2, determine the y values for which the probability density Is maximum.
Approximate Mark Allocation: QI (20); Q2 (20); Q3 (20)
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