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Phys 221: Quantum Mechanics 1 Question Paper
Phys 221: Quantum Mechanics 1
Course:Bachelor Of Education Science
Institution: Chuka University question papers
Exam Year:2013
CHUKA
UNIVERSITY
UNIVERSITY EXAMINATIONS
SECOND YEAR EXAMINATION FOR THE AWARD OF DEGREE OF
BACHELOR OF EDUCATION (SCIENCE)
PHYS 221: QUANTUM MECHANICS 1
STREAMS: B.Ed. (SCI) Y2S2 TIME: 2 HOURS
DAY/DATE: WEDNESDAY 7/8/2013 2.30 P.M. – 4.30 P.M.
INSTRUCTIONS:
This paper consists of FIVE questions. You are required to answer Question ONE (30 Marks) which is compulsory and ANY other TWO questions (20 marks each).
The following constants and Identities may be useful:
Plank’s Constant h = 6.63 x 1034Js
Plank’s Constant (2) h = 1.055 x 10-34Js
Mass of an electron me = 9.11 x 10-31Kg
1eV = 1.6 x 10-19J
QUESTION ONE: COMPULSORY (30 MARKS)
Explain using an example, the significance of h in quantum mechanics. [2 marks]
Explain the concept of wave particle duality. [2 marks]
Determine the wavelength associated with an electron with energy 2000
eV. [4 marks]
An electron has a wavelength of 1.66 x 10-10m. Determine its:-
Velocity [3 marks]
Energy in electron volts [3 marks]
Write down the distinct states for the hydrogen atom for n = 2 [2 marks]
Explain Born’s interpretation of a wave function. [2 marks]
A quantum mechanical state is described by the wave function:-
Determine:-
(i) Its probability density. What can be inferred from the probability density of
this state? [3 marks]
(ii) The energy expectation value [2 marks]
The uncertainty in the energy, and hence interpret your result. [3 marks]
State three characteristics that a wave function describing a quantum mechanical
state should have. [3 marks]
State one postulate of quantum mechanics. [1 mark]
QUESTION TWO: (20 MARKS)
Using suitable illustrations, distinguish between the expected classical behavior of a particle trapped in an infinite potential well and its expected quantum mechanical behaviour. [5 marks]
An electron is bound in a one-dimensional infinite well of width 1 x 10-10m.
Find the energy value in the ground state. [4 Marks]
A particle of mass m moves in a one-dimensional box of length l, with boundaries
at x = 0 and x = 1. Thus, U_((x))=0, for 0=x=l, and U_((x))=8, elsewhere. The normalized eigenfunctions of the Hamiltonian for this system are given by
withE_n=(n^2 p^2 h^2)/?2ml?^2 , where the quantum number n = 1,2,3…….
Assuming that the particle is in an eigenstate calculate:-
(i) The probability density and hence
(ii) The probability that the particle is found somewhere in the region 0=x=?(1/4)
QUESTION THREE (20 MARKS)
Compare the spacing of the energy levels in the harmonic oscillator potential to that of an infinite potential well. [2 marks]
How does the harmonic potential compare to that of a diatomic molecule?[3 marks]
Obtain an expression for the energy of the first excited state of a 3-D harmonic oscillator. How many states have this energy and what is this phenomenon referred to as?
[5 marks]
Describe quantum tunneling. [2 marks]
Consider an electron in the potential well shown below. E1 is the first excited state energy.
Sketch the first excited wave function. Explain your reasoning. [3 marks]
(ii) Sketch the wave function corresponding to E2. Explain your reasoning. [3 marks]
(iii) In the case of E2, where will the electron be most likely found? [2 marks]
QUESTION FOUR (20 MARKS)
Show that the Schrodinger Equation for a particle in a region in which forces acting on it cause it to have a Potential Energy Ux can be given in one dimension by:-
(?^2 ?)/??x?^2 +(2pp/h)^2 ?=0 [5 marks]
Show that the complex exponential ?_(x,t)=Ae^(i(kx-?t)) is a solution to the Schrodinger equation for a free particle moving in one dimension and it obeys the dispersion relation
[6 marks]
A one dimensional wave function of a stationary state can be expressed as
?_((x))=?_0 e^(-ikx). Assuming that the particle is not under the influence of any potential field, determine the stationary state energy of the particle represented by this wave function.
[4 marks]
The voltage in an electron diffraction experiment is 100 V. Determine its de Broglie wavelength. [5 marks]
QUESTION FIVE (20 MARKS)
Why are Hermitian operators used in quantum mechanics? [2 marks]
Determine whether the operator Ly below is Hermitian. [4 marks]
Ly = [¦(0&-i/v2&0@i/v2&0&(-i)/v2@0&i/v2&0)]
Work out the following commutation relations and state the significance of the outcome on measurement of the quantities involved.
(i) [x,p] [6 marks]
(ii) [x,p2] [4 marks]
(iii) [x,H] [4 marks]
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