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QPH702S - QUANTUM PHYSICS - 2ND OPP - JAN 2023 |
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r,
(
nAm I BI A u n IVE RSITY
OF SCIEnCE TECHnOLOGY
FACULTYOF HEALTHN, ATURALRESOURCEASNDAPPLIEDSCIENCES
DEPARTMENTOF NATURALANDAPPLIEDSCIENCES
QUALIFICATIONB: ACHELOROF SCIENCE
QUALIFICATIONCODE:07BOSC
COURSECODE:QPH702S
SESSION:JANUARY2023
DURATION: 3 HOURS
LEVEL:7
COURSENAME: QUANTUM PHYSICS
PAPERT: HEORY
MARKS: 100
SUPPLEMEMTAY/SECONODPPORTUNITYEXAMINATIONQUESTIONPAPER
EXAMINER(S) Prof Dipti R Sahu
MODERATOR: Prof Vijaya S. Vallabhapurapu
INSTRUCTIONS
1. Answer any Fivethe questions.
2. Write clearly and neatly.
3. Number the answers clearly.
PERMISSIBLME ATERIALS
Non-programmable Calculators
THISQUESTIONPAPERCONSISTSOF 3 PAGES(Including this front page)
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Question 1
[20]
1.1 Consider a one-dimensional bound particle. Show that if the particle is in a
(10)
stationary state at a given time, then it will always remain in a stationary state.
1.2 Consider a one-dimensional bound particle, Show
(10)
ddx fc_ o 1/J(*x, t)-i/J(x, t)dt = 0, lJJneed not be a statio. nary state.
00
Question 2
[20]
2.1
A one-dimensional harmonic oscillator wave function is Axe-bx 2
2.1.1 Find b
(5)
2.1.2 The total energy E
(5)
2.2
The wavefunction of a particle confined to the x axis is Lj.=, e-x for x > 0 and Lj.=, e +x
(10)
for x < 0. Normalize this wavefunction and calculate the probability offinding the
particle between x = -1 and x = 1.
Question 3
[20]
3.1 An electron is trapped in a 1-D infinite well of width 5nm. Evaluate the
(5)
wavelength of radiation emitted when the electron makes a transition from
third to first excited states.
3.2 Compare the energies and wavefunctions of 1-D infinite well and harmonic
(3)
oscillator.
3.3. Explain quantum tunnelling and list three applications of it.
(5)
3.4 What does it mean to say that certain operators commute? Give examples of operators (4)
that commute and of operators that do not commute.
3.5 Why the de-Broglie wave associated with a moving car is not observable?
(3)
Question 4
[20]
4.1 A potential barrier is defined by:
1.2 eV - oo< x < -2
V(x) = 0 - 2 < x < 2
{ 1.2 eV 2 < x < oo
A particle of mass m and kinetic energy 1.0 eV is incident on this barrier from -co.
Evaluate the acceptable wave function of the particle.
(10)
4.2
Calculate the expectation value (r)21for the hydrogen atom and compare it with
(6)
the valuer at which the radial probability density reaches its maximum for the
j state n = 2, / = l. Given R21 (r) = re -r hao / 24a5
4.3
Show explicitly that S2 = n2s(s+l) I
(4)
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Question 5
[20]
5.1 What can be said about the Hamiltonian operator if L, is a constant in time?
(2)
5.2 What you conclude that two operators commuting?
(2)
5.3 Evaluate the matrix of Lxfor/= 1?
(6)
5.4 What are the kinetic, potential, and Hamiltonian operators for the hydrogen atom?
(5)
Write the Schrodinger equation for the H-atom.
5.5 Show explicitly in Cartesian (x, y, z) coordinates that the V2 and L, operators commute,
(5)
i.e., [V2, L,] = 0
Question 6
[20)
6.1 Show that in the usual stationary state perturbation theory, if the Hamiltonian
(10)
can be written H =Ho+ H' with Ho(})o= Eo(})ot,hen the correction l'.Eois
l'.Eo:::<: c:DIoH' Ic:Do>
6.2 A hydrogen atom is in a superposition state given by:
'Y= r1;;--;;[3\\Jf100-4\\Jl2120-\\Jl321]
-v29
Evaluate
6.2.1 The probability that the atom is found in each of the sub-states.
(3)
6.2.2 The expectation value of the energy in this superposition state. Given
(7)
E = = - 13.6 e V
n n2
n2
a:,yJ" e-yl dy =-;
-a:,
n
n even
-co
O' · n odd
3