ABSTRACT
An equation describing the change of the state of a quantum system with respect to time is called a time evolution equation. One of the fundamental evolution equation of quantum mechanics is the Schrodinger equation; a dynamical system that has yielded closed form solution only in specialized and simplified cases. By observing that the time-independent Hamiltonian for the system is provided by a one parameter group of unitary operators, we cast the evolution equation within the frame work of this group of unitary operators, and thus obtained close form solutions for the time-dependent case, the equation dynamics specified a Hamiltonian that is the sum of two terms, a perturbed and the unperturbed part. The solution for the perturbed part was obtained in the form of an integral equation which can be iteratively constructed. The exact solution for the unperturbed part was obtained and probability that at time t the system can be found in a specific state from it’s initial state was computed.
TABLE
OF CONTENTS
Title Page i
Declaration page ii
Certification page iii
Dedication iv
Acknowledgement v
Table of content vi
Abstract vii
Acronyms
x
CHAPTER 1: INTRODUCTION
1.1
Background Of Study 1
1.2
Statement of Problem 4
1.3
Aim
& Objective 4
1.4
Motivation
4
1.5
Limitation And Scope of Study
5
1.6
Justification of Study
5
1.7
Significance of study
5
1.8
Definition of Terms
6
2.0
CHAPTER
2: LITERATURE
REVIEW 9
2.2 Theoretical Review 10
2.3 Empirical Review 11
2.4
Knowledge Gap 13
CHAPTER 3: RESEARCH METHOD
3.1 Introduction
14
3.2 Postulates
of quantum mechanics
14
3.3 Time
evolution in quantum mechanics
16
3.4 Stone’s
Theorem 21
CHAPTER 4: MAIN RESULTS AND DISCUSSION
4.1 Introduction 26
4.2 Harmonic
Oscillator
26
4.3 Time Independent
Shrodinger Equation 30
4.4 Time-Dependent Perturbation Theory
31
CHAPTER 5: SUMMARY, CONCLUSION AND RECOMMENDATIONS
5.0 Summary
40
5.1 Conclusion
40
5.2 Recommendation 40
REFERENCES 41
ACRONYMS
Probability
amplitude: this is a complex number used
in describing the behavour
of systems. The sequence of the modulus of this
quantity represents a probability
or probability density. This can also be defined as the relationship between wave function of a system and the
results of observations of that system.
wave number: this is
the number of the complete cycles of a wave over it’s wave length given by 
angular velocity:
refers to how fast an object rotates or revolves relative to another point, i.e how fast the angular position or orientation of an object changes
with time 
momentum: this is
the product of the mass and velocity of an object. It is a
vector quantity possessing magnitude and direction. 
particle
position
time
Plank’s
constant: this is a physical constant that is the quantum of electromagnetic
action, which relates the
energy carried by a photon to it’s frequency.
A photon’s energy is equal to it’s
frequency multiplied by the planck constant. 
Reduced Plank’s
constant: is a modified form of planck’s constant, in which h equals h divided
by pie, and is the quantization of angular momentum.
Wavelength: this can
be defined as the distance between two successive crests or troughs of a wave.
It is measured in the direction of the wave. 
m
Mass: this is measure
of it’s resistance to acceleration .
CHAPTER 1
INTRODUCTION
1.1 BACKGROUND OF STUDY
Quantum Mechanics is a branch of
mathematical physics that deals with atomic and subatomic systems and their
interaction with radiation in terms of observable quantities (Arnold et al, 2006).
It is an outgrowth of the concept that all forms of energy are released in
discrete units or bundles called quanta.
It
is one of the more sophisticated fields in physical science that has affected
our understanding of nano-meter length scale systems important for chemistry,
material sciences, optics, electronics, and quantum information. The existence
of orbitals and energy levels in atoms can only be explained by quantum
mechanics (Dams et al ,2003). Quantum
mechanics explains the radiation of a hot body or black body, and how it
changes in color with respect to temperature. The presence of holes and the transport of holes
and electrons in electronic device can also be well understood with the
understanding of Quantum mechanics. To soar high as a Scientist or an Engineer
in this present day one has to have a good knowledge of Quantum Mechanics
because emerging technologies require a good knowledge of quantum mechanics.
At the turn of the twentieth century, so
many experimental results showed that atomic particles are also wave-like in
nature. Therefore, it was considered ok to say that a wave equation can be use
to explain the behavior of atomic particles. The first person who wrote down
such a wave equation is Schrodinger. The eigevalue of the wave equation were
shown to be the same as the energy levels of the quantum mechanical system, the
equation proved credible when it was used in solving for the energy levels of
the hydrogen atom which were found to be in accord with the Rydberg's Law.
The Schrödinger equation is used
in describing the time evolution of a quantum state, having
the Hamiltonian which is the operator that corresponds to the total energy of the system that
generates the time evolution. The time evolution of wave function predicts what
the wave function will be at any time given the wave function at the initial
time.
The intensity of the wave function
must determine those probabilities. More precisely, we postulate that
It
is a probability that a particle turns up within the volume dV around
point r. The particle must be
somewhere in the interval
at any given time. That is, the probability for the particle
to be in this interval must be unity at any given time. So
The
Schrodinger equation for the wave function must be Differential, Linear and Reproduce
correct dispersion relations. We start with
Remark
1.
Schrödinger equation does not
determine the wave function uniquely: if 
is a solution then 
is also a solution.
2. If 
defines a probability density, it must obey certain
additional conditions. On the other
hand, the change of the initial wave function into another, later wave function
is not deterministic, it is unpredictable (i.e., random) during a measurement. Wave functions change
as time progresses. The Schrödinger equation
explains how wave functions change in time, playing a role similar to Newton's second law in classical mechanics.
The mathematics of
quantum theory precisely predicts and reproduces the results of physical
experiments. We are applying these principles in making delicate instruments
and building precise computers. Therefore, Quantum Mechanics is an actual science.
Solutions to Schrodinger’s
equation describe not only molecular, atomic and subatomic system, but also macroscopic systems ,
possibly even the whole universe (Laloe 2012).
1.2
STATEMENT
OF PROBLEM
Given
any dynamical system of the form: 
, where for each 
.This
problem has a unique solution for all
. It’s solution can be written as 
. We observe that
the sense
that
With its infinitesimal generator as
Thus,
the existence, uniqueness and continuous dependence property for all time of
the solution of the given initial value problem is intimately connected to the
family 
of
uniformly continuous unitary group of bounded linear operators whose generator
is
.
In
this research work, we employ the unitary group 
of operators as described above in the
solution of a quantum-evolution equation which takes the state of the system
from time
to the time
and satisfy the
operator form of the Schrödinger’s equation.
Where H is
the Hamiltonian of the particle. The solution is obtained by a time-dependent
perturbative technique.
1.3 MOTIVATION
We
are motivated to embark on this research work on finding solution of a quantum
evolution using a unitary group of operators, because unitary groups fall into
the category of elements of abstract algebra, and yet it is interesting to
discover that they have important applications in quantum theory. Secondly we
seek a way of getting a simplified version of the schrodinger equation that will
give us an accurate result.
1.4 AIM AND OBJECTIVES
Aim
The aim of this work is to solve
a quantum evolution equation.
Objectives
·
To determine the family
of quantum evolution operators 
which takes the state of the system from time to to time t
and implement the dynamics of the
quantum system.
·
To apply the family of
quantum evolution operators in solving the quantum evolution equation.
·
To compute an approximate
solution of our quantum evolution equation by perturbation method.
1.5 SIGNIFICANCE OF
STUDY
· Schrodinger's equation is an evolution equation which the physicist
and most of chemist use in tackling problems
of atomic structure of matter; it is a powerful
mathematical tool and forms a whole
basis of wave mechanic.
·
The Schrödinger equation
determines the wavefunction and its dynamics.
·
Schrodinger’s equation
can be used to calculate the outcomes of
different types of experiments. However, it is limited to calculations
of the behaviour of electrons and other particles when they are traveling
slower than a significant fraction of the speed of light.
·
The electronic structure
of atoms and molecules can be well explained using Schrodinger’s equation. It
also helps to describe the shape of orbitals and their orientations.
1.6 JUSTIFICATION OF STUDY
Most often obtaining an exact solution to the Schrodinger
equation is difficult, except possibly for the case of the hydrogen atom which
is a single electron system. For more complicated atomic configurations,
it is necessary to adopt a numerical procedure for the solution of the
Shrodinger equation. However another analytic method involves applying the unitary group of operators to
the Hamiltonian which can result into a simplified version of the Shrodinger
equation which nonetheless has the same solution as the original. We also know
that the Shrodinger equation is very important in studying electronic structure
of atoms and molecules as well as wave functions and it dynamics; hence we are
justified in going into this research.
1.7 LIMITATION AND SCOPE
OF STUDY
Limitation of study
Our
study is limited to the domain of subatomic particles.
Scope of study
The
scope of this work is limited to applying unitary groups of operators in
quantum mechanics.
1.8 DEFINITION OF TERMS
·
Quantum
mechanics
Quantum mechanics is the
branch of mathematical physics that deals with atomic and subatomic systems and
their interaction with radiation in terms of observable quantities. It is an
outgrowth of the concept that all forms of energy are released in discrete
units or bundles called quanta
·
Operator
An
operator is a rule that
transforms a given function into another function'. The differentiation
operator
is an example it
transforms a differentiable function
into another function
·
Linear operator
An operator operator that satisfies the following two
condition
·
Hermitian
and adjoint operators
For
a linear operator
where U and V are vector spaces
If
, which is possible only if
, the operator is said to be self-adjoint or Hermitian. If
, again possible only if
, the operator is said to be unitary.
·
Unitary
operator
A linear operator whose inverse is
its adjoint is called unitary. These operators can be said to be a generalizations of complex numbers whose
absolute value is 1, i.e
.
·
Group
A
group is a non-empty set
on which there is defined a binary
operation
satisfying
the following property;
(i)
Closure: if
then
(ii)
Associativity: if
(iii)
Identity: if
then
for all
(iv)
Inverse; if
then
such that
We remark that a group is Abelian if
for any
·
Hamiltonian
operator
In
quantum mechanics, a Hamiltonian is an operator corresponding to the sum
of the kinetic energies and the potential energies of all the particles in the system (this
sum is the total energy
of the system in most of the cases under analysis). It is usually denoted by
H {\displaystyle H}H ˇ {\displaystyle {\check {H}}} H ^
{\displaystyle {\hat {H}}}It’s spectrum is the set of
possible outcomes when one measures the total energy of a system.
·
Eigenvalue
and Eigenvector
Let
A be a
matrix. A non-zero
vector
is an eigenvector of A if there exists a scalar
such that,
The scalar λ is called
the eigenvalue of the matrix A, corresponding
to the eigenvector
.
·
Wave
function
The wave function which is
denoted by the Greek letter psi,
is a variable
quantity that mathematically describes the wave characteristics of a particle.
The value of the wave function of a particle at a given point of space and time
is related to the likelihood of the particle’s being there at that time. Born
interpreted it to be “probability waves” in the sense that the amplitude
squared of the waves gives the probability ability of detecting the particle in
some region.
·
Perturbation
theory
Perturbation
theory is used to estimate the energies and wave functions for a quantum system
described by a potential slightly different than a potential with a known
solution or a theory that gives us a method for relating the problem that can
be solved exactly to the one that cannot. A critical feature of the technique
is a middle step that breaks the problem into “solvable” and “perturbation”
parts.
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