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QPH702S - QUANTUM PHYSICS - 2ND OPP - JANUARY 2025 |
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nAml BIA UnlVERSITY
OF SCIEnCE AnDTECHnOLOGY
FacultyofHealth,Natural
ResourceasndApplied
Sciences
Schoolof NaturalandApplied
Sciences
Departmentof 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
SECOND OPPORTUNITY/SUPPLEMENTARY: QUESTION PAPER
EXAMINER:
MODERATOR:
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 paper consists of 3 pages including this front page
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QUESTION 1:
1.1 A particle has the wave function
¢ (r} = Ne-ar
where N is a normalization factor and a is a known real parameter.
1.1.1 Calculate the factor N.
[20 MARKS]
(3)
1.1.2 Calculate the expectation values <r>
(2)
1.2 What is physical significance of wave function?
(5)
1. 3 Consider the Hamiltonian
( 10)
H
=
Px 2
+P 2
Y
+V
(x,
y)
-
V (x, y) = { 0
I x I < 2L ' O < Y < l
zm
oo else
Determine the eigenvalue and normalized eigenfunction of the system
QUESTION 2:
[20 MARKS]
A potential barrier is defined by:
!1.2 eV - 00 < x < -2
V(x) = 0 - 2 < x < 2
1.2 eV 2 < x < oo
2.1 Sketch the graph of V(x)
(2)
A particle of mass m and kinetic energy 1.0 eV is incident on this barrier from -oo.
2.1.1 Evaluate the acceptable wave function of the particle.
(8)
2.1.2 Evaluate the conditions applicable at the boundaries
(5)
2.2 Calculate the energy of the first excited quantum state of a particle in the two-
dimensional potential V (x, y) = mw 2 (x2 + 4y2)
2
QUESTION 3
(5)
[20 MARKS
3.1 A particle is represented by the normalized wave function
lLjJ(x) =
0
v'15(a2-x2l
4al
, otherwise
for -a < x < a,
Determine, the uncertainty ~P in its momentum
(10)
3.2 The wave function for the ground state of a harmonic oscillator is
~Jo(.x) = Coe-a2x2;2
find a and the energy corresponding to this state.
(10)
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QUESTION 4:
[20 MARKS]
4.1 Evaluate the [x, Px] commutators and state the consequences of the results.
(5)
4.2 What are Pauli spin matrices and what value of spin they correspond?
(5)
4.3 Evaluate the matrix of Lxfor/= 1
QUESTION 5:
10)
[20 MARKS]
5.1 Show that in the usual stationary state perturbation theory, if the Hamiltonian (10)
can be written H =Ho + H' with Ho(/)o=Eo(/)o,then the correction ~Eois
~Eo::: < <DoIH' I<Do>
5.2 Evaluate the eigenfunction and the energy of the state n = 1 for a quantum system
with the
Potential energy V (x) =
{ -0.5x
0.5x
_j,_<x<O
2
O<x<½
using first order perturbation theory and the infinite potential well as the
unperturbed
system. Given
'¥
0
1
=
'1G-C:cos(L.l!..x)
Useful Standard Integral
_,,,
"'sy 11 e-y 2 dy=-;
-00
n
n even
1OC>e~,'e-P'dy=( : J½e"p'2
O' · n odd
Spherical harmonics
I
Y/"(0' tp) =[(2/ + 1) (I - m)! ]2 eimq,P/"(x)
4n (l + m)!
Associated Legendre polynomials:
P/" (x)
=
(-1
)1(I
-
,
x,-
)111(1,d
--
)1+111
(1- x 2 ) 1 ,
where
x = cos 0
2 /! dx
Radial eigenfunctions of hydrogen-like atoms:
2ZJ%[ ]¾(2Jz' R,,,()r =( -
a0 n
(n[-/-1)!
2n (n + /)!
]3
- r e-:O;·,L112_,_11+(p1) , where
a0 n
'° L21+1( )- n-/-1 (-l)k.
[cn+ /)'].2 p k
d --r2Z
n-t-i p - Lk=., o
(n -/-1-k)'(2/ . . + 1+ k) .1k .1' an p- aon
END OF QUESTION PAPER
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