SOLUTION OF A QUANTUM EVOLUTION EQUATION USING A UNITARY GROUP OF OPERATORS

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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 its 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 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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