NATIONAL OPEN UNIVERSITY OF NIGERIA
14/16 AHMADU BELLO WAY, VICTORIA ISLAND, LAGOS
SCHOOL OF SCIENCE AND TECHNOLOGY
MARCH/APRIL 2014 EXAMINATION
COURSE CODE: PHY 309
COURSE TITLE: QUANTUM MECHANICS
TIME ALLOWED: 2 ½ HRS
INSTRUCTION: ANSWER QUESTION ANY FIVE QUESTIONS
QUESTION ONE
A particle is confined within a one-dimensional region 0 x L. At time t = 0, its wave function is given as
i. Normalise the wave function.
ii. Find the average energy of the system at time t = 0 and at an arbitrary time t0 .
iii. Find the average energy of the particle.
iv. Write the expression for the probability that the particle is found within 0 x L/ 2?
QUESTION TWO
a. What are the allowable eigenfunctions and energy eigenvalues of the infinite potential well?
b. Check whether the following vectors are linearly independent.
2i 3 j k , i j 3k and 3i 2 j k
c. Find the inner product of the following vectors: ix2+2 and 2x-3i for
QUESTION THREE
The state of a free particle is described by the following wave function:
a. Find A using the normalization condition.
b. What is the probability of finding the particle within the interval [0, b]?
c. Calculate for this state.
d. Calculate the momentum probability density.
QUESTION FOUR
a. Assume that a photon is scattered by an electron initially at rest. Which photon scattering angle corresponds to the largest Compton shift and why? At what minimum photon energy can half of the photon energy be transferred onto the electron?
b. Write the function as a sum of odd and even functions h(x) = e2xsinx as sum of odd and even functions.
QUESTION FIVE
a. You are given the set
i. Are the linearly independent?
ii. Are they orthogonal?
iii. Are they normalized? If not, normalize them
b. Find the eigen values and the corresponding eigen functions of the matrix.
c. If this matrix represents a physically observable attribute of a particle, what is the
expectation value of the attribute in each of the possible states. Comment on your results.
QUESTION SIX
a. A quantum-mechanical oscillator of mass m moves in one dimension such that its energy eigenstate x) = ( y2/ exp(- y2x2/2) with energy E = ħ 2y2/2m
i. Find the mean position of the particle.
ii. Find the mean momentum of the particle
b. Normalise the eigen functions x) = Aexp Hence, find the probability that the particle subjected to harmonic oscillation lies in the range .
QUESTION SEVEN
a. Given the basis {(2, 3), (1, 4)}, write the expression for a transformation to {(0, 2), (-1, 5)}
b. What would the potential function be if is an eigenfunctions of the Schrödinger equation? Assume that when x→ ∞, V(X) → 0
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