A First Course in Mathematical Physics (eBook)

Colm T. Whelan (Autor)

eBook Download: EPUB

2016
Wiley-VCH (Verlag)
978-3-527-68715-2 (ISBN)

Infos

The book assumes next to no prior knowledge of the topic. The first part introduces the core mathematics, always in conjunction with the physical context. In the second part of the book, a series of examples showcases some of the more conceptually advanced areas of physics, the presentation of which draws on the developments in the first part. A large number of problems helps students to hone their skills in using the presented mathematical methods. Solutions to the problems are available to instructors on an associated password-protected website for lecturers.

Colm T. Whelan is a Professor of Physics and an Eminent Scholar at Old Dominion University, USA. He received his PhD in Theoretical Atomic Physics from the University of Cambridge in 1985 and his ScD in 2001. He is a fellow of both the American Physical Society and the Institute of Physics (UK). He has over 25 years of experience in undergraduate teaching in both the UK and the US.

Inhalt

Preface page ix

Part I: Mathematics 1

1 Functions of one variable 2
1.1 Limits 2
1.2 Elementary Calculus 6
1.2.1 Chain Rule 7
1.2.2 Differentiation products and quotients 8
1.2.3 Inverse Functions 9
1.3 Integration 10
1.4 The Binomial Expansion 13
1.5 Taylor's series 15
1.6 Extrema 17
1.7 Power Series 18
1.8 Basic Functions 19
1.8.1 Exponential 19
1.8.2 Logarithm 23
1.9 1st order ordinary differential equations 24
1.10 Trigonometric Functions 26
1.10.1 L'Hopital's rule 28
1.11 Problems 30

2 Complex numbers 33
2.1 Exponential function of a complex variable 34
2.2 Argrand Diagrams and the Complex Plane 36
2.3 Hyperbolic functions 38
2.4 The simple harmonic oscillator 40
2.4.1 Mechanics in one dimension 42
2.5 Problems 45

3 Functions of Several Variables 48
3.1 Partial derivatives 48
3.1.1 Definition of the partial derivative 48
3.1.2 Total derivatives 51
3.1.3 Some relations 53
3.1.4 Change of variables 55
3.1.5 Mechanics again 56
3.1.6 Exact differentials and thermodynamics 58
3.2 Extrema under constraint 60
3.3 Multiple Integrals 62
3.3.1 Triple Integrals 66
3.3.2 Change of variables 67
3.4 Problems 69

4 Vectors in R3 72
4.1 Basic operation 72
4.1.1 scalar triple product 79
4.1.2 Vector equations of lines and planes 80
4.2 Kinematics in three dimensions 81
4.2.1 Differentiation 81
4.2.2 Motion in a uniform magnetic field 81
4.3 Coordinate systems 83
4.3.1 Polar coordinates 83
4.4 Central Forces 84
4.5 Rotating Frames 88
4.5.1 Larmor Effect 91
4.6 Problems 93

5 Vector fields and operators 96
5.1 The gradient operator 96
5.1.1 Coordinate Systems 97
5.2 Work and energy in vectorial mechanics 101
5.2.1 Line integrals 104
5.3 A little fluid dynamics 107
5.3.1 Rotational motion 111
5.3.2 Fields 114
5.3.3 Surface integrals 115
5.4 The divergence theorem 118
5.5 Stokes' Theorem 121
5.5.1 Conservative Forces 123
5.6 Problems 126

6 Generalized Functions. 130
6.1 The Dirac delta function 130
6.2 Problems 139

7 Vector Spaces 140
7.1 Formal Definition of a vector space 140
7.2 Fourier Series 145
7.3 Linear Operators 148
7.4 Change of basis 160
7.5 Problems 163

Part II: Physics 168

8 Maxwell's Equations: A very short Introduction 169
8.1 Electrostatic: Gauss's Law 169
8.2 Gauss's Law for a magnetic field 173
8.3 Ampere's Law 173
8.3.1 Gauge conditions 174
8.4 Problems 177

9 Special Relativity:4-vector formalism 179
9.1 Lorentz transformation 179
9.1.1 Inertial frames 179
9.1.2 Properties and consequences of the Lorentz transformation
182
9.2 Minkowski space 183
9.2.1 Four vectors 183
9.2.2 Time Dilation 189
9.3 Four velocity 190
9.3.1 Four momentum 191
9.4 Electrodynamics 197
9.4.1 Maxwell's equations in 4-vector form 197
9.4.2 Field of a moving point charge 200
9.5 Problems 202

10 Quantum Theory 205
10.1 Formalism 205
10.1.1 Dirac notation 206
10.2 Probabilistic interpretation 207
10.2.1 Commutator relations 208
10.2.2 Functions of observables 210
10.3 The Stern-Gerlach experiment 210
10.3.1 Spin space 210
viii Contents
10.3.2 Explicit matrix representation 210
10.3.3 EPR paradox 210
10.3.4 Bell's Theorem 210
10.4 Quantization 210
10.4.1 Time evolution 210
10.4.2 The harmonic oscillator, coherent states 210
10.5 Problems 212

11 Atoms, molecules, solids, wave mechanics in one dimension
216
11.1 Atom 217
11.1.1 The square well 218
11.1.2 The delta function potential 219
11.2 Molecules 221
11.3 Solids 223
11.3.1 Block's Theorem 224
11.3.2 Band structure 226
12 An informal treatment of variational principles and their
history 229

Appendix 1 Conic Sections 230
Appendix 2 Vector Relations 232
Index 237

Textauszug

Chapter 1
Functions of One Variable

1.1 Limits

It is often said that most mathematical errors, which get published, follow the word “clearly” and involve the improper interchange of two limits. In simple terms, a “limit” is the number that a function or sequence “approaches” as the input or index approaches some value. For example, we will say that the sequence approaches the limit 0 as moves to infinity Or, in other words, we can make arbitrarily small by choosing big enough. We often write this as

We can also take the limit of a function, for example, if then

A sequence of numbers is said to converge to a limit if we can make the difference arbitrarily small by making big enough. If such a limit point does not exist, then we state that the sequence diverges. For example, the sequence of integers

is unbounded as , while the sequence

oscillates and never settles down to a limit. More formally, we state

Definition 1.1

Let f be a function defined on a real interval then the limit as exists if there exists a number such that given a number no matter how small, we can find a number , where for all satisfying

we have

Notice that we do not necessarily let ever reach but only get infinitesimally close to it. If in fact , then we state that the function is continuous. Intuitively, a function that is continuous on some interval will take on all values between and (Figure 1.1). For a more formal discussion see [1]. An intuitively obvious result is the intermediate value theorem.

Figure 1.1 If is a continuous function on if we pick any value, , that is between the value of and the value of and draw a line straight out from this point, the line will hit the graph in at least one point with an value between and .

Theorem 1.1

Let be a continuous function on a closed interval . If is a number such that , then there exists a number such that

Proof

For a formal proof see for example [1]

Consider the sequence of partial sums

1.1

if the sequence of partial sums converges to some limit as then we say that the infinite series is convergent.

Example 1.1

The geometric series

Let

then

hence

Clearly, therefore, if the series converges. Its value being given by

if , the series diverges.

1.2 Elementary Calculus

Assume that we are observing an object moving in one dimension. We measure its position to be at time and at time , thus its average speed is

1.2

Of course, this is only an average value the object could accelerate and decelerate during the time interval; if we need to know its speed at any given point, then we must shorten the time interval, and to know the “instantaneous” speed at a time , we need to let lead to zero, that is,

1.3

This motivates us to define the derivative of a function.

Definition 1.2

If is only a function of , then the first derivative of at is defined to be

1.4

If this limit exists, then the function is said to be differentiable. The function is said to be continuously differentiable if the derivative exists and is itself a continuous function.

Frequently, we use the notation as a shorthand, that is,

Example 1.2

If , then

If , where is a constant, then for all , consequently . A partial converse to this result is as follows. If on some interval , then , where is a constant. This is a consequence of the intermediate value theorem; see Problem 1.1. Clearly if

1.5

then

1.6

where is a constant.

1.2.1 Differentiation Products and Quotients

Assuming can be written as the product of two functions, for example,

then

1.7

We may rewrite the numerator in (1.7) as

in the limit as and

it immediately follows that

1.8

It is also possible to show, Problem 1.10, that

1.9

Lemma 1.1

Proof

If , clearly true since ; assume to be true for all , consider

then from (1.8)

1.10

Now, by assumption

and as we have already seen

hence

1.11

Hence, by principle of induction, true for all integers.

1.2.2 Chain Rule

Assume that

For example,

then for such a function:

Lemma 1.2

If is differentiable at the point and that is differentiable at the point , then

1.12

or in other words

Proof

1.13

We can now take the limit as and we have the result. In fact, to be more rigorous, we should worry about the possibility of passing through 0. For a treatment where this problem is explicitly dealt with see [2].

Example 1.3

Newton's second law can be written as

1.3

Thus, force can be defined as the rate of mass times acceleration or as the rate of change of the kinetic energy with distance

1.2.3 Inverse Functions

Consider the functions shown in Figure 1.2. Both are continuous but for the equation

does not have a unique solution for , while the equivalent equation for will have such a solution. The difference between the two functions is that is strictly increasing over the entire interval but is not.

Figure 1.2 Over the range shown, the function is invertible, is not.

Definition 1.3

If is a continuous strictly increasing function on , with , then from the intermediate value theorem, Theorem 1.1, we know that the set

forms the interval . We may define a function

It is clear that we could just as well have constructed an inverse for a strictly decreasing function. The only time we will have a problem is when but . Usually, we write as . Even for , we can define an inverse if we agree to only look at the intervals and separately. We can thus talk about a local inverse, that is, given a point if we can find an interval around it for which the function is strictly increasing or decreasing, then we can find an inverse valid in this region. We know that (see Problem 1.3) a function is strictly increasing or decreasing on an interval once its derivative does not change sign; so if the derivative is continuous and we are not at a point , where , then we can always find an interval, however, small so that the function is locally invertible. More formally, we may state the inverse function theorem.

Theorem 1.2

For functions of a single real variable, if is a continuously differentiable function with nonzero derivative at the point , then is invertible in a neighborhood of , the inverse is continuously differentiable and

where .

Proof

If has a nonzero derivative at , then it follows that there is a interval around where it is either increasing or decreasing then (Problem 1.4), is continuous. Let , then

since is continuous the result follows.

1.3 Integration

There are number of equivalent ways of looking at integrals. Perhaps the most intuitive is to consider

as the area in the plane bounded by the curves . Conventionally, we often describe this quantity as the area under the curve ; see Figure 1.3. As a first approximation, we could simply assume that the function could be approximated by its initial value over the entire range, and we would have

See Figure 1.3(a). Now, we clearly lose some area by this approximation. We can improve it by taking a point , with and approximating the integral by two rectangles of area and .

Figure 1.3 (a) Approximating the value of the integral as , (b) picking a point where and approximating integral as , and (c) picking another point where and approximating integral as .

We can continue this process by adding more and more subintervals. If

1.15

then we can approximate

1.16

where

If we make these intervals arbitrarily small, that is, let , then we should get anaccurate measure of the area under the curve. This prompts the following definition.

Definition 1.4

The integral from to of the function is given by

1.17

where

We can use the intermediate value theorem to establish the following theorem.

Theorem 1.3

Mean value theorem for integrals.Let f be continuous on , then there exists a s.t.

Proof

From Definition 1.2, if is the minimum value of on and its maximum, then

Hence, by the intermediate value theorem, there exists s.t.

We can define a...

Produkt-Details

• Erscheint lt. Verlag: 28.3.2016
• Sprache: englisch
• Themenwelt: Naturwissenschaften ► Physik / Astronomie
• Themenwelt: Technik
• Schlagworte: Angewandte Mathematik • Calculus • Computational / Numerical Methods • Differential • Einführung in die Physik (Analysis) • Einführung in die Physik (Analysis) • General • geometry</p> • Introductory Physics (Calculus-based) • <p>algebra • Maschinenbau • Mathematical & Computational Physics • Mathematische Physik • mechanical engineering • Physics • Physik • Rechnergestützte / Numerische Verfahren im Maschinenbau • Rechnergestützte / Numerische Verfahren im Maschinenbau • Relativity • Special
• ISBN-10: 3-527-68715-7 / 3527687157
• ISBN-13: 978-3-527-68715-2 / 9783527687152