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QPH702S - QUANTUM PHYSICS - 1ST OPP - NOV 2022 |
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n Am I 8 I A Un IVE RSITY
OF SCIEnCE Ano TECHnOLOGY
FACULTYOF HEALTH,NATURALRESOURCEASNDAPPLIEDSCIENCES
DEPARTMENTOF NATURALAND APPLIEDSCIENCES
QUALIFICATION: BACHELOROF SCIENCE
QUALIFICATION CODE: 07BOSC
COURSE CODE: QPH 702S
SESSION: NOVEMBER 2022
DURATION: 3 HOURS
LEVEL: 7
COURSE NAME: QUANTUM PHYSICS
PAPER: THEORY
MARKS: 100
EXAMINER(S)
MODERATOR:
FIRST OPPORTUNITY EXAMINATION QUESTION PAPER
Prof Dipti R. Sahu
Prof Vijaya S. Vallabhapurapu
INSTRUCTIONS
1. Answer any Five questions.
2. Write clearly and neatly.
3. Number the answers clearly.
PERMISSIBLE MATERIALS
Non-programmable Calculators
THIS QUESTION PAPER CONSISTS OF 4 PAGES (Including this front page)
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Question 1
[20]
1.1 List with reason, three properties of a valid wave of a bounded state.
(3)
1.2 Replace the following classical mechanical expressions with their corresponding
(6)
quantum mechanical operators.
a. K.E.= ½ mv2 in three-dimensional space.
b. p = mv, a three-dimensional cartesian vector.
c. y-component of angular momentum: Lv= ZPx - xp,.
1.3. How to describe a system in quantum mechanics?
(4)
1.4 For a particle moving freely along the x-axis, show that the Heisenberg uncertainty
(5)
principle can be written in the alternative form: !:::.'fA:::.x 'A2./ 4n:where f:::.xis the
uncertainty in position of the particle and !:::.'iAs the simultaneous uncertainty in the
de Broglie wavelength.
1.5 What is the significance of wave packet
(2)
Question 2
[20]
2.1 Consider a one-dimensional particle which is confined within the region 0 :=x:; :=a:; and
whose wave function is ljJ (x, t) = sin (n:x/a) exp (-iwt).
(a) Find the potential V(x).
(5)
(b) Calculate the probability offinding the particle in the interval a/4 :=x:; :=:;3a/4.
(5)
2.2 Consider the one-dimensional wave function
(10)
LV(x)=A(x/xo)n e-xlxo
where A, n and xo are constants. Using Schrodinger's equation, find the potential
V(x) and energy f for which this wave function is an eigenfunction. (Assume that
as X
V(x) 0).
Question 3
[20]
3.1 The wavefunction of a particle moving in the x-dimension is
1/-(1X)
=
{
Nx(L-x)
0
0 <x < L
elsewhere
3.1.1 Normalize the wavefunction
(4)
3.1.2 Determine the expectation value of x
(4)
3.2 Evaluate the probability current density of the wavefunction,
(2)
'-I'(x) = 5exp(-3ix)
3.3
The potential function V(x) of the problem is given by
(10)
V(x)={Vo x>O
0 X< 0
where Vo is constant potential energy.
Find the wave function for E < Vo where Eis the incident particle energy and interpret
the results.
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Question 4
[20]
4.1
½ Obtain the spin matrix 52for spins= particle using the eigenstates of 52 as the basis (10)
4.2 Evaluate the commutation of L2,L3.
(5)
4.3 Consider a system which is initially in the state
(5)
'3 4J (8,
cp)=
1
{s
Y1,-1((8,cp)+
(8,
cp)+
1
{s
Y1,1(8,
cp),
Find < lj.JI L+I 4J>
Question 5
[20]
5.1 Consider an infinite well for which the bottom is not flat, as sketched here. If the
(5)
slope is small, the potential V= 8 lxl / a may be considered as a perturbation
on the square-well potential over-a/2 ::;x ::;a/2.
00
00
V(x)
-a/2
a/2
X
Calculate the ground-state energy, correct to first order in perturbation theory. Given
rr Ground state of box size a: 4Jo=
(2/a) Cos-1,aTX
Ground
state
energy
Eo=
-h- 2
2ma
2
2
5.2 The wave function of the ground state of hydrogen has the form.
(5)
-r
l!J100
=-
1
-
ero
jrrr5
Find the probability of finding the electron in a volume dV around a given point.
5.3 Evaluate the constant Bin the hydrogen-like wave function
(10)
4J (r,
8,
cp)= B r2sin28
e2
.
up
exp(-
-3Z-r)
3 ao
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Question 6
6.1 The wavefunction of a state of harmonic oscillator is given by:
CD(x) =
( mw
1
}4
(
4
mw
x2
-
2)
exp (-mwx 2/2li}
64nh
h
Obtain the corresponding energy of the state.
; -00 < x< oo
6.2 A particle moves in a one-dimensional box with a small potential dip
[20]
(10)
(10)
V
0
•
- b _______
11-
1
0
fl
V = oo for x<O and x > l
V = -b for O < l < (l/2) l
V = 0 for (l/2) L< x < l
Treat the potential dip as a perturbation to a regular rigid box (V = oofor x < 0 and x > L V = 0 for O<
x < D-Find the first order energy of the ground state. The ground state energy and wavefunction is
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given by E 0 = n h
1jJ0 (x) = {"is.in nx
2ml 2 '
l
.............................................................................. END...................................................................................... .
Useful Standard Integral
f00
e_Y2dy=
-00
OyOnJe-y2dy =-;
n
-00
n even
O; n odd
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