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QPH702S - QUANTUM PHYSICS - 1ST OPP - NOVEMBER 2024 |
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nAml BIA UnlVERSITY
r
OF SCIEnCE TECHnOLOGY
FacultyofHealth,Natural
ResourceasndApplied
Sciences
Schoolof NaturalandApplied
Sciences
Department of Biology,
Chemistryand Physics
13JacksonKaujeuaStreet T: +264612072012
Private Bag13388
F: +264612079012
Windhoek
E: dbcp@nust.na
NAMIBIA
W: www.nust.na
QUALIFICATION: BACHELOR OF SCIENCE
QUALIFICATION CODE: 08BOSC
COURSE: QUANTUM PHYSICS
DATE: NOVEMBER 2024
DURATION: 3 HOURS
LEVEL:7
COURSECODE: QPH702S
SESSION: 1
MARKS: 100
EXAMINER:
MODERATOR:
FIRST OPPORTUNITY: QUESTION PAPER
Professor Dipti Ranjan Sahu
Professor Vijaya S. Vallabhapurapu
INSTRUCTIONS:
1. Answer all questions on the separate answer sheet.
2. Please write neatly and legibly.
3. Do not use the left side margin of the exam paper. This must be allowed for the
examiner.
4. No books, notes and other additional aids are allowed.
5. Mark all answers clearly with their respective question numbers.
PERMISSIBLE MATERIALS:
1. Non-Programmable Calculator
This question paper consists of 4 pages including this front page
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QUESTION 1:
[20 MARKS]
1.1 Consider a one-dimensional particle which is confined within the region O x a and whose
wave function is iv (x, t) = sin (n:x/a) exp (-iwt).
1.1.1 Find the potential V(x)
(5)
1.1.2 Calculate the probability offinding the particle in the interval a/4 x ~3a/4
(5)
1.2 List with reason, three properties of a valid wave of a bounded state.
(3)
1.3 Explain briefly why photoelectric effect is a quantum phenomenon?
(2)
1.4 Compare the energies and wavefunctions of 1-D infinite well and harmonic
(3)
oscillator
1.5 Why the de-Broglie wave associated with a moving car is not observable?
(2)
QUESTION 2:
2.1.
H The wave function of the particle is given by
q, (x) = sin(narrx), x E [O,a) , n=1, 2, 3....
[20 MARKS]
Find the average kinetic energy T (p) of the particle described by this wavefunction
(5)
2.2 How to describe a system in quantum mechanics?
(2)
2.3 Write the quantum mechanical operators of the classical mechanical expressions
K.E.= ½ rhv2 in three-dimensional space.
(3)
2.4 Consider a potential step of height Vas shown in the figure. A particle of energy
(10)
E> V propagates from -00 to +00
(1)
I, (2)
X
0
Given A1is the amplitude of the incident wave from wave vector K1and A2 be the amplitude of the
transmitted wave from wave vector Kz.Find the relation between A1and A2.
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QUESTION 3:
0
3.1
By applying L2 operator to the harmonic state Y (0, cp), determine the
2
eigenvalue of the state.
3.2
Evaluate the radial eigenfunctions R32 ofthe lithium atom
[20 MARKS]
(10)
(10)
QUESTION 4:
4.1
f Evaluate the spin matrices Szfor a particle with spins=
[20 MARKS]
(5)
4.2 Evaluate the commutation of L2 and band state the consequence of your results.
(5)
4.3. Evaluate the matrix of L2for/= 2. Why is the matrix not diagonal?
(10)
QUESTION 5:
[20 MARKS]
5.1 The perturbation H'= bx4, where bis a constant, is added to the one dimensional
harmonic oscillator potential V (x) = ! mw 2 x2• Determine the correction to the
2
ground state energy to first order in b. The normalized ground state wave
function of the one dimensional harmonic oscillator potential is
(10)
4'o=
mw
-mwx2
(-)1/4ezit
tin
5.2 A particle moves in a one-dimensional box with a small potential dip
(10)
V
0 ----
-b
0
2I
..•. X
'
V= oo for x< 0 and x>I,
V = -b for O < 2 < (1/2) E,
V = 0 far (1/2) E < x < I.
Treat the potential dip as a perturbation to a regular rigid box (V =oo for x < 0 and x >
I, V = 0 far O < x < I). Find the first order energy of the ground state. The ground state
ff energy and wavefunction is given by
r.{0}
c,
::= _r;J /12
t.rnF'
ij,(0 '(.c}=
-,·sin 1-:rX
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Useful Standard Integral
n even Ie~1,'de-y=I/(:J
O, · n odd
Spherical harmonics
I
Y/"(8,(f)) =[(2/ + l) (l-m)f]2 eimq,P/"(x)
4Jr (l+m)!
Associated Legendre polynomials:
P/" (x)
= (-1 )'
(l-x2)11112( d
,
-
)1+111
(l -
x 2 ) 1 , where
x = cos 0
2 /! dx
Radial eigenfunctions of hydrogen-like atoms:
-(2ZJ¾[ ]½(2JZ' -: R,,1(r) - -
a0 n
(n[-/-1)!
2n (n + /)!
]3
- r e 0:·,L,,2_1,+_(,1p ) , where
a0 n
[c /) L2,1,+_1,_(,p)--
n"-/'-"1'(-1
k=o
1]2 k
)k
• (n
-/-l.-k)
n
+
1.(·2/
p
+
I
+
k)
.1k
1. '
2Z and
p- -
-r
aon
END OF QUESTION PAPER
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