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Published: 12.04.2021  ## Mathematics Of Classical And Quantum Physics

By Frederick W. Byron and Robert W. In this chapter we shall review informally the properties of the vectors and vector fields that occur in classical physics. But we shall do so in a way, and in a notation, that leads to the more abstract discussion of vectors in later chapters. The aim here is to bridge the gap between classical three-dimensional vector analysis and the formulation of abstract vector spaces, which is the mathematical language of quantum physics.

Many of the ideas that will be developed more abstractly and thoroughly in later chapters will be anticipated in the familiar three-dimensional setting here. This should provide the subsequent treatment with more intuitive content. This chapter will also provide a brief recapitulation of classical physics, much of which can be elegantly stated in the language of vector analysis—which was, of course, devised expressly for this purpose.

Our purpose here is one of informal introduction and review; accordingly, the mathematical development will not be as rigorous as in subsequent chapters. In elementary physics courses the geometric aspect of vectors is emphasized. A vector, x , is first conceived as a directed line segment, or a quantity with both a magnitude and a direction, such as a velocity or a force. A vector is thus distinguished from a scalar, a quantity which has only magnitude such as temperature, entropy, or mass.

In the two-dimensional space depicted in Fig. They form an equivalence class which may be represented by V0, the unique vector whose initial point is at the origin. We shall gradually replace this elementary characterization of vectors and scalars with a more fundamental one.

But first we must develop another language with which to discuss vectors. An algebraic aspect of a vector is suggested by the one-to-one correspondence between the unique vectors issuing from the origin that represent equivalence classes of vectors, and the coordinates of their terminal points, the ordered pairs of real numbers x 1, x 2. Similarly, in three-dimensional space we associate a geometrical vector with an ordered triple of real numbers, x 1, x 2, x 3 , which are called the components of the vector.

We may write this vector more briefly as xi , where it is understood that i extends from 1 to 3. But even though we can no longer construct physical vectors for n greater than three, we retain the geometrical language for these n -dimensional generalizations.

A formal treatment of the properties of such abstract vectors, which are important in the theory of relativity and quantum mechanics, will be the subject of Chapters 3 and 4. In this chapter we shall restrict our attention to the three-dimensional case. There are then these two complementary aspects of a vector: the geometric, or physical, and the algebraic. These correspond to plane or solid geometry and analytic geometry.

The geometric aspect was discovered first and stood alone for centuries until Descartes discovered algebraic or analytic geometry. Anything that can be proved geometrically can be proved algebraically and vice-versa, but the proof of a given proposition may be far easier in one language than in the other.

Thus the algebraic language is more than a simple alternative to the geometric language. It allows us to formulate certain questions more easily than we could in the geometric language.

For example, the tangent to a curve at a point can be defined very simply in the algebraic language, thus facilitating further study of the whole range of problems surrounding this important concept. It is from just this formulation of the problem of tangents that the calculus arose. It is said of Niels Bohr that he never felt he understood philosophical ideas until he had discussed them with himself in German, French, and English as well as in his native Danish.

The same is true of the study of vectors. It is all too easy to rely on the algebraic language to carry one through vector analysis, skipping blithely over the physical, geometric interpretation of the differential operators.

We shall try to bring out the physical meanings of these operators as well as review their algebraic manipulation. The basic operators of vector analysis crop up everywhere in physics, so it pays to develop a physical picture of what these operators do—that is, what features they measure of the scalar or vector fields on which they operate. One of the most important aspects of the study of vectors is the resolution of vectors into components.

In fact, this will remain a central feature in Chapter 5, where we deal with Hilbert space, the infinite-dimensional generalization of a vector space. In three dimensions, any vector x can be expressed as a linear combination of any three noncoplanar vectors.

The three vectors V 1, V 2, and V 3 need not be perpendicular to each other—any three noncoplanar vectors form a base, or basis , in terms of which an arbitrary vector may be decomposed or expanded.

But it is often most convenient to choose the basis vectors perpendicular to each other. In this case the basis is called orthogonal; otherwise it is called oblique. We shall deal almost exclusively with sets of orthogonal basis vectors. A particularly useful set of basis vectors is the Cartesian basis , consisting of three mutually orthogonal vectors of unit length which have the same direction at all points in space.

We shall denote unit vectors by the letter e in this chapter; accordingly, the Cartesian basis is the set e 1, e 2, e 3 shown in Fig. Such a set of base vectors is called orthonormal , because the vectors are orthogonal to each other and are normalized have unit length. We shall not distinguish between a basis and a coordinate system in this treatment. The basis or coordinate system in Fig.

If any one of the basis vectors is reversed, we have a left-handed orthogonal basis. A mathematical definition of handedness will be given in Section 1. There are a great many other orthonormal bases, such as those of cylindrical and spherical and other curvilinear coordinate systems, which can greatly simplify the treatment of problems with special symmetry features.

We shall deal with these in Section 1. The scalar inner or dot product of two vectors x and y is the real number defined in geometrical language by the equation. This equation has a familiar and reasonable appearance, but that is only because we automatically interpret it algebraically, where we usually take distributivity for granted.

The reader will find it instructive to prove this by geometrical construction. It may be that they are perpendicular. Note that the length of a vector x is given by. If we expand two arbitrary vectors, x and y , in terms of the Cartesian basis,. This equation provides an independent way of associating with any vector, a number called its length.

We see that the notion of length need not be taken as inherent in the notion of vector, but is rather a consequence of defining a scalar product in a space of abstract vectors. Thus in Chapter 3. Then, in Chapter 4 we shall add an inner or scalar product to this vector space and focus on the enriched structure that results from this addition. We shall now introduce a notational shorthand known as the Einstein summation convention.

Einstein, in working with vectors and tensors, noticed that whenever there was a summation over a given subscript or superscript , that subscript appeared twice in the summed expression, and vice versa.

Thus one could simply omit the redundant summation signs, interpreting an expression like xiyi to mean summation over the repeated subscript from 1 to, in our case, 3. If there are two distinct repeated subscripts, two summations are implied, and so on.

In terms of this convention we have, for example,. We shall now consider the relationship between the components of a vector expressed with respect to two different Cartesian bases with the same origin, as shown in Fig. For example, in K we have. The vector x can be expressed in terms of either basis.

The nine terms aij defined by Eq. These numbers may be written as the square array,. R is known as the rotation matrix in three dimensions, since it describes the consequences of a change from one basis to another rotated basis.

Almost all authors use the convention of Eq. In fact, in abstract vector space theory, matrices are usually defined by a convention consistent with Eq. However, by replacing aij by aji in Eq. In chapter 4 we shall reconsider these issues in a more general setting that permits an easy and complete systematization.

It is apparent that the elements of the rotation matrix are not independent. Since the basis vectors form an orthonormal set, it follows from Eq. Equation 1. It is left to the reader to show by expanding the unprimed vectors in terms of the primed basis and taking scalar products that we also have the relation. The expressions 1. In an n -dimensional space, the rotation matrix will have n conditions, as the reader can verify.

In a two-dimensional space this leaves one free parameter, which we may take as the angle of rotation. In a three-dimensional space there are three degrees of freedom, corresponding to the three so-called Euler angles used to describe the orientation of a rigid body. Let us first express the xj the components of x with respect to the basis e j , the components of x. Using Eq. Now, since the basis vectors are orthogonal, we may identify their coefficients in Eq.

To derive the inverse transformation, one could, of course, repeat the above procedure, substituting for the unprimed vectors. That is, instead of using Eq. However, using the orthogonality relations, we can derive this result directly from Eq. Multiplying it by akj , summing over j , and using Eq. It should be understood that these equations refer to the components of one vector, x , as expressed with respect to two different sets of basis vectors, e i.

Thus unprimed basis vectors can be expressed in terms of the other primed basis. Thus aij is the j , expressed with respect to the unprimed basis, and aji is the j th component of e i expressed with respect to the primed basis. Example 1. The two-dimensional rotation matrix. From Fig. The first subscript of aij labels the row and the second subscript labels the column of the element aij.

From Eq. ## Mathematics of Classical and Quantum Physics

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By Frederick W. Byron and Robert W. In this chapter we shall review informally the properties of the vectors and vector fields that occur in classical physics. But we shall do so in a way, and in a notation, that leads to the more abstract discussion of vectors in later chapters. The aim here is to bridge the gap between classical three-dimensional vector analysis and the formulation of abstract vector spaces, which is the mathematical language of quantum physics. The home page for Bernard de Wit's quantum field theory notes have changed; they are now available at The Spinoza Institute's page Stage 4 Mathematical Physics. Mathematical Physics optional : Advanced studies of modern physics rely heavily on mathematics.

### 183081177 Byron F W Fuller R W Mathematics of Classical and Quantum Physics PDF

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Isham Published Computer Science Vector spaces linear operators properties in classical physics the general formalism of quantum … This textbook is designed to complement graduate-level physics texts in classical mechanics, electricity, magnetism, and quantum mechanics. If you like this book and want to support the idea of online versions, please consider buying this book: Mathematics of Classical and Quantum Physics, by Frederick W. Byron and Robert W. Fuller, Addison-Wesley Mathematical Methods in Quantum Mechanics is intended for beginning graduate students in both mathematics and physics and provides a solid foundation for reading more advanced books and current research literature.

Byron F. Dover Publications, Well-organized text designed to complement graduate-level physics texts in classical mechanics, electricity, magnetism, and quantum mechanics. Topics include theory of vector spaces, analytic function theory, Green's function method of solving differential and partial differential equations, theory of groups, more. Many problems, suggestions for further reading. Brown L. Feynman's Thesis.

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Byron and Robert W. Press, Saul A. Teukolsky, William. I will.

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Mathematics of classical and quantum physics / Frederick W. Byron, Jr., Robert. W. Fuller. p. cm. "Unabridged, corrected republication of the work first published.

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Start reading Mathematics of Classical and Quantum Physics for free online and get access to an unlimited library Frederick W. Byron, Robert W. Fuller Chapters 3, 4 and 5 — the backbone of the book — cover the theory of vector spaces.

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Final exam and all homework sets have been graded.