Some Mathematics Used in Quantum Mechanics

In Chapter 2 we introduced trigonometric functions, the logarithm function and the exponential function. We now revisit the concept of a function. Mathematical functions are useful in thermodynamics because thermodynamic variables behave exactly like mathematical functions, with some variables acting as independent variables and others as dependent variables. They are useful in quantum mechanics because all information about the state of a system is contained in a mathematical function called a wave function or state function. 

The idea in perturbation methods is that the problem at hand only differs slightly from a problem that has already been solved (exactly or approximately).The solution to the given problem should therefore in some sense be close to the solution to the already known system. This is described mathematically by defining a Hamiltonian operator that consists of two parts, a reference (Ho) and a perturbation (H ). The premise of perturbation methods is that the H operator in some sense is small compared with Ho. Perturbation methods can be used in quantum mechanics for adding corrections to solutions that employ an independent-particle approximation, and the theoretical framework is then called Many-Body Perturbation Theory (MBPT). 

Spectrum Intensity of a signal due to a process such as optical absorption or emission displayed as a function of some varying characteristic such as wavelength, energy, or mass. Also used in quantum mechanics and applied mathematics to specify the pattern of eigenvalues of a linear operator and in electrodynamics to specify the range of frequencies of electromagnetic radiation. 

Before considering in more detail the approximation methods used in quantum mechanics, we review some of the mathematical tools available 4 One of the most powerful of these concerns the representation of a function as a mixture, with carefully chosen coefficients, of a number of more elementary functions ... 

Bom coined the term "Quantum mechanics and in 1925 devised a system called matrix mechanics, which accounted mathematically for the posidon and momentum of the electron in the atom. He devised a technique called the Born approximation in scattering theory for computing the behavior of subatomic particles which is used in high-energy physics. Also, interpretation of the wave function for Schrodinger s wave mechanics was solved by Born who suggested that the square of the wave function could be understood as the probability of finding a particle at some point in space, For this work in quantum mechanics. Max Bom received the Nobel Prize in Physics in 1954,... 

In the following we present the axioms or basic postulates of quantum mechanics and accompany them by their classical counterparts in the Hamiltonian formalism. We begin the presentation with a brief summary of some of the mathematical background essential for the developments in the following. It is by no means a comprehensive presentation, and the reader is supposed to have some basic knowledge about quantum mechanics that may be obtained from any of the many introductory textbooks in quantum mechanics. The focus here is on results of particular relevance to the subjects of this book. We consider, for example, a derivation of a formal expression for the flux density operator in quantum mechanics and its coordinate representation. A systematic way of generating any representation of any combination of operators is set up, and is of immediate usage for the time autocorrelation function of the flux operator used to determine the rate constants of a chemical process. 

Such comparisons apply in quantum mechanics, too. Recognizing the symmetry of an atomic or molecular system allows one to simplify the quantum mechanics, sometimes dramatically. We have already seen some aspects of symmetry odd and even functions, the spherical nature of the hydrogen atom s Is orbital, the cylindrical shape of H2 and H2. All these are applications of symmetry. In this chapter, we will develop a general understanding of symmetry using a mathematical tool called group theory. Then, we can see how symmetry applies to some aspects of quantum mechanics. 

The solution to this problem has an exceptionally important role in quantum mechanics and especially in quantum chemistry. Firstly, this is a problem that can be solved analytically (though some special mathematical functions must be used). Secondly, the solution is of great importance for chemistry where the electronic orbits arise from the solution moreover the theory of the chemical bond has been worked out using the results. Thirdly, an empirically modified hydrogen atom s orbits are widely used generally for heavier atoms because there are no other ways of achieving results. 

We shall treat coupling of modes of motion in some detail because there are fundamental mechanical and mathematical topics involved that will be useful to us in both MM and quantum mechanical calculations. In the tieatment of coupled haiinonic oscillators, matrix diagonalization and normal coordinates are encountered in a simple form. 

Statistical mechanics is the mathematical means to calculate the thermodynamic properties of bulk materials from a molecular description of the materials. Much of statistical mechanics is still at the paper-and-pencil stage of theory. Since quantum mechanicians cannot exactly solve the Schrodinger equation yet, statistical mechanicians do not really have even a starting point for a truly rigorous treatment. In spite of this limitation, some very useful results for bulk materials can be obtained. 

The interactions between electrons are inherently many-body forces. There are several methods in common use today which try to incorporate some, or all, of the many-body quantum mechanical effects. An important term is that of electronic exchange [57, 58]. Mathematically, when two particles in the many-body wavefunction are exchanged the wavefunction changes sign ... 

Throughout the book I have tried to constrain the wonders of imagination inspired by the subject by using simple calculations. Can all of the water on the Earth have been delivered by comets if so, how many comets How do I use molecular spectroscopy to work out what is happening in a giant molecular cloud Calculations form part of the big hard-sell for astrochemistry and they provide a powerful control against myth. I have aimed the book at second-year undergraduates who have had some exposure to quantum mechanics, kinetics, thermodynamics and mathematics but the book could easily be adapted as an introduction to all of these areas for a minor course in chemistry to stand alone. 

Physical chemistry and physics may be different fields but they have some important features in common they are abstract they both use mathematics they overlap in some content areas (such as thermodynamics and quantum mechanics). To a large extent, science and physics educators started research on basic physics concepts that also are used in physical chemistry. Consequently, physical chemistry education research owns much to the work that has been done in physics education and has much in common with it. For example, they share some of the research methodology and an interest in studying the relationship between the physical description of phenomena and its mathematics description in the learner s mind. 

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