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Cell[CellGroupData[{
Cell["Physics", "Title"],
Cell[CellGroupData[{
Cell["Introduction", "Section",
CellMargins->{{12, 12}, {5, 15}}],
Cell["\<\
Hello, how are you? Welcome to this book. I am assuming that you want to \
become a theoretical physicist, but\[LongDash]for one reason or another\
\[LongDash]you do not want to go to a university. I am okay with that, I \
never went to a university, except to assist professors and researchers in \
their teaching and in their research. What you want to do is a long and hard \
road\[LongDash]there is no way around that fact; if you want the book, \
\[OpenCurlyDoubleQuote]Theoretical physics in ten minutes,\
\[CloseCurlyDoubleQuote] you will never be able to find it. And we are \
starting at the very beginning. Along the way we will cover a lot of \
interesting things that you might not know. Even those of you who are experts \
might find some surprises. We will explore different questions and ideas. \
There are several goals the I want to achieve here:\
\>", "Text"],
Cell[CellGroupData[{
Cell["Rewire your brain to think like a physicist.", "Item1Numbered"],
Cell["Rewire your brain to think mathematically.", "Item1Numbered"],
Cell["Show you the underlying principles of physics.", "Item1Numbered"],
Cell["\<\
Show you how to use those principles to invent physics for yourself. I will \
help by inventing parts of it as examples of the techniques that I will \
present to you.\
\>", "Item1Numbered"],
Cell["Show you how to represent physics mathematically.", "Item1Numbered"],
Cell[TextData[{
"Show you how to represent physics by the use of the computer software \
system ",
StyleBox["Mathematica",
FontSlant->"Italic"],
". (See the Preface for some details of my involvement with ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" and how to get it for yourself). Note that these sections can be ignored \
if you do not have access to ",
StyleBox["Mathematica",
FontSlant->"Italic"],
"."
}], "Item1Numbered"],
Cell["\<\
Show you that you do not need to go to a university to study theoretical \
physics.\
\>", "Item1Numbered"],
Cell["\<\
Finally, and most importantly, show you that theoretical physics is one of \
the greatest cultural achievements of mankind. That it is your right and \
privilege to engage in it.\
\>", "Item1Numbered"]
}, Open ]],
Cell[TextData[{
"Every once in a while I will request that you stop reading and do \
something. I recommend that you do what is requested in every case. At first \
I will give you some complete answers, then I will limit it to clues and \
guidance, and then I will just tell you what to do and leave all of the \
details to you.\n\tI also suggest that you actively keep a notebook. How do \
you actively keep a notebook? Here is one way: Given a notebook, draw a \
straight line down the page you are currently working on, place the line so \
you still have three-quarters of the page as a working surface. Why draw such \
a line? You can make notes on your notes on the quarter remaining. As you \
finish each chapter, review all of your notes up to the current point in the \
book. As ideas occur to you, write them into the open area on the page. \
Another way is to use ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" and keep your notebook on the computer (though you will want to make \
backups, and maybe printed backups). I recommend that you use one of the \
report formats that are included with ",
StyleBox["Mathematica",
FontSlant->"Italic"],
". Play around with the styles until you find those that you like.\n\tJot \
down each idea as it is presented. Try to word the idea so you can understand \
it. If you do not understand the idea, note that lack of understanding and \
move on. It is impossible to write a book the way the brain thinks; we must \
write down one topic, then the next, and so on\[LongDash]as if the topics \
were rungs on a ladder. The brain does not work that way! You might \
completely understand one section and be completely lost in the next. Don't \
sweat it, that's normal. Write down in your notebook that you didn't \
understand the idea and move on. You will likely get it at some point in the \
future, possibly when reviewing your notes.\n\tI recommend that you initially \
write the answers to the Stop Reading and Do Something tasks on a note pad or \
in loose sheets of paper at first, when you have completed them and checked \
the answers then write them into your notebook, keep your working notes in a \
file folder or something similar. As you work through the book connections to \
previous topics will occur to you, write those down too\[LongDash]possibly in \
the margins you have created. Sometimes such connections can lead to \
interesting research ideas. When this happens, stop reading and explore the \
idea for a little while. Try to invent tests to see if you are correct. After \
a few minutes of this, note where you are leaving this idea and move on. You \
don't want to get so sidetracked that you stop working through the book!\n\tI \
also recommend that you perform at least one of the projects suggested \
throughout the book. There is nothing wrong with doing all of them. I \
recommend that you do more than one of them in the whole book. Since these \
tend to be long projects, do them outside of your reading time. Record the \
sessions in your notebook as you proceed. I recommend that when you have \
completed a project that you write up your results outside of your notebook \
as if you were presenting a research paper.\n\t",
StyleBox["How do you present a research paper?",
FontSlant->"Italic"],
" Give your paper a title followed by your name, then write a 100 word (or \
smaller) summary of the work\[LongDash]this is called an ",
StyleBox["abstract",
FontSlant->"Italic"],
". Then write an introduction where you explain why you were interested \
enough to try the project and major developments done by others in the field \
that you found productive (it is a good idea to find outside resources, many \
are free through the World Wide Web). Then explain in one or more sections \
how you did the work and what results you got. Finish the paper with a \
section listing all of the sources of information you used by the name of the \
author, the date of publication, the title, and the publisher (even if this \
is a web site).\n\tAfter you have worked through a few chapters in the book \
make a review of those chapters\[LongDash]a synthesis of your notes. Do this \
over four or five chapters. This will involve noting what you think are the \
most important points covered in your notes up to the current point. When you \
are done with the book write a final review where you do this for all of the \
reviews done throughout the book.\n\tA few months after you are done with the \
book go through your notes again. Make sure that you keep the notebook, you \
will return to it again as you progress on your path to learning to be a \
theoretical physicist."
}], "Text"],
Cell[TextData[{
StyleBox["Stop Reading and Do Something #1: Acquire a physical notebook or \
create a ",
FontSlant->"Plain"],
StyleBox["Mathematica",
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StyleBox[" file (also called a notebook\[LongDash]See Appendix 1: \
Introduction to ",
FontSlant->"Plain"],
StyleBox["Mathematica",
FontSlant->"Italic"],
StyleBox[" if you do this).",
FontSlant->"Plain"]
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Cell[CellGroupData[{
Cell["The Nature of Physics", "Section",
CellMargins->{{12, 12}, {5, 15}}],
Cell[TextData[{
"What is physics? At this point we have no real way to define physics \
without assuming a knowledge of physics. If I were to say that physics is the \
study of matter, energy, and their interactions I would be technically \
correct. Unfortunately, this definition of physics is meaningless without \
knowing what matter, energy, and interactions are. Another definition is that \
physics is what physicists study. This is sort of like saying that physics is \
physics, what we call a ",
StyleBox["circular definition",
FontSlant->"Italic"],
", where we use the word we are trying to define in the definition. So maybe \
we can\[CloseCurlyQuote]t define physics exactly, maybe we will just have to \
understand what we mean when we use the word physics. Such ideas are called ",
StyleBox["undefined terms",
FontSlant->"Italic"],
", or we could call them ",
StyleBox["technical terms",
FontSlant->"Italic"],
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}], "Text"],
Cell["\<\
Term 1: Undefined/Technical Term
A word or phrase used to describe an idea whose definition cannot be \
specified without resorting to a circular definition.\
\>", "Text",
CellFrame->True],
Cell["\<\
I suggest that you copy all such technical terms, along with your \
understanding of them, into your notebook.\
\>", "Text"],
Cell["\<\
Term 2: Circular Definition
An attempt to explain a word where the word is used in its definition. This \
use may be explicit\[LongDash]where the word is actually used in the \
definition, or it may be implicit\[LongDash]where a phrase amounting to the \
word is used in the definition.\
\>", "Text",
CellFrame->True],
Cell[TextData[{
"\tSo, let\[CloseCurlyQuote]s get back to the question at hand, what is \
physics? We can say that physics is a science. But what is a science? Science \
is a word. We can think of it as being two words. The first word, science, is \
a noun and represents that facts and ideas developed by scientists. The \
second word, science, is a program of discovery that produces the facts and \
ideas of the noun. This program can be called ",
StyleBox["the scientific method",
FontSlant->"Italic"],
", but that is somewhat misleading\[LongDash]each scientist develops their \
own collection of methods. You begin by observing the real world. From such \
observations you develop an idea about a pattern in nature. You then assume \
the perceived pattern to be true and predict the consequences of the \
assumption in various situations. You then compare those predictions with the \
real world\[LongDash]either confirming or refuting the idea. Here is a \
program to apply this idea to physics:"
}], "Text"],
Cell[CellGroupData[{
Cell["\<\
Use of observation to gather enough information to find a pattern.\
\>", "Item1Numbered"],
Cell["Analyzing this pattern to develop an idea.", "Item1Numbered"],
Cell["\<\
Make measurements of something that you cannot derive mathematically or \
computationally.\
\>", "Item1Numbered"],
Cell["\<\
Use the concepts and methods of theoretical physics to see if the idea is \
consistent with what we already know about physics.\
\>", "Item1Numbered"],
Cell["\<\
Use the concepts and methods of mathematical physics to convert the idea into \
a problem in mathematics, solve that problem, and then return the result to \
its physics context.\
\>", "Item1Numbered"],
Cell["\<\
Use the concepts and methods of computational physics to convert the idea \
into a problem in computation, solve that problem, and then return the result \
to its physics context.\
\>", "Item1Numbered"],
Cell["A body of work established by steps 1-6 is called a theory.", \
"Item1Numbered"],
Cell["\<\
Use the concepts and methods of experimental physics to verify predictions \
made using the theory.\
\>", "Item1Numbered"]
}, Open ]],
Cell["\<\
Term 3: Theory
An established body of work in science.\
\>", "Text",
CellFrame->True],
Cell["\<\
At its most basic level physics has established some physical quantities that \
can be measured:\
\>", "Text"],
Cell["\<\
Term 4: Position
Where something is located.\
\>", "Text",
CellFrame->True],
Cell["\<\
Term 5: Size
An objects length, width, height, surface area, and volume.\
\>", "Text",
CellFrame->True],
Cell["\<\
Term 6: Weight
The result of weighing an object.\
\>", "Text",
CellFrame->True],
Cell["\<\
Term 7: Temperature
The result of taking a thermometer reading.\
\>", "Text",
CellFrame->True],
Cell["Physics concerns itself with several large problems:", "Text"],
Cell[CellGroupData[{
Cell["\<\
The problem of motion. Or trying to predict how an object will move given \
some initial data.\
\>", "Item1Numbered"],
Cell["\<\
The problem of gravity. Or trying to determine how one body attracts another.\
\>", "Item1Numbered"],
Cell["The problem of what matter is.", "Item1Numbered"],
Cell["The problem of what heat is.", "Item1Numbered"],
Cell["The problem of what electricity is.", "Item1Numbered"],
Cell["The problem of what magnetism is.", "Item1Numbered"],
Cell["The problem of what light is.", "Item1Numbered"]
}, Open ]],
Cell["We will explore each of these in due time.", "Text"],
Cell["\<\
Term 8: Physics
The science relating to motion, gravity, matter, heat, electricity, \
magnetism, and light.\
\>", "Text",
CellFrame->True]
}, Open ]],
Cell[CellGroupData[{
Cell["The Nature of Mathematics", "Section",
CellMargins->{{12, 12}, {5, 15}}],
Cell["\<\
Now that we have some vague idea of what physics is, we turn to mathematics. \
The first question is, what is mathematics?\
\>", "Text",
CellMargins->{{12, 10}, {3, 0}}],
Cell[TextData[StyleBox["Stop Reading and Do Something #2: Try to answer this \
question. Do not take more than five minutes to do this then go ahead and \
read on.",
FontSlant->"Plain"]], "Text",
CellFrame->True,
FontWeight->"Bold",
FontSlant->"Italic"],
Cell["\<\
Mathematics is the study of specific abstract objects, and the relationships \
between those objects. We call such objects abstract because there is no need \
to specify what they are. We need only list their general properties and how \
they relate to one another. In fact, the central question of all mathematics \
is: \"What are the properties of specific objects and their relationships \
with each other?\"
\tWe can use that idea to see that mathematics can also be viewed as a kind \
of language for expressing ideas in a pure form. The objects become the nouns \
and the relations adverbs. We can use symbols to represent the objects and \
their relationships without much of the ambiguity associated with most \
languages.
\tIt is also true that mathematics is a way of getting the results from \
calculations. So you can use mathematics as a tool for getting specific \
results. In physics these results are compared to measurements made in \
experiments for verification of the theory.\
\>", "Text"],
Cell[TextData[{
"\tNow I want to address the almost mythical quality of the so-called ",
StyleBox["unreasonable effectiveness of mathematics.",
FontSlant->"Italic"],
" This quality describes the wonder that we experience when we make \
mathematical predictions that are extremely close to what we can measure in \
reality. The fact that abstract ideas can be applied to the world around us \
seems magical at first glance, but these abstract ideas came from concrete \
applications involving\[LongDash]at the beginning of \
mathematics\[LongDash]the measurement of land areas and the exchange of \
currency. It is not surprising that from these humble beginnings the vast \
array of modern mathematical applications are good at describing the world\
\[LongDash]that is where the abstractions that led to the creation of these \
applications came from in the first place. In this regard mathematics can be \
seen as a sequence of abstractions from specific ideas to general principles \
to new specific applications, and so on.\n\tAs we proceed we will examine the \
objects of different branches of mathematics, then their relationships. \
Sometimes we will examine how we can use these objects and relationships as \
language, and sometimes we will examine how to use them to make actual \
calculations."
}], "Text",
CellMargins->{{12, 10}, {3, 0}}],
Cell["\<\
Term 9: Mathematics
The study of specific abstract objects, and the relationships between those \
objects. This study allows us to construct a language using the objects as \
nouns and the relationships as adverbs. These ideas allow us a way of getting \
the results from calculations. These ideas evolve through cycles of \
application and generalization.\
\>", "Text",
CellFrame->True]
}, Open ]],
Cell[CellGroupData[{
Cell["The Methods of Theoretical Physics", "Section",
CellMargins->{{12, 12}, {5, 15}}],
Cell["\<\
The ancient Greek philosophers, such as Aristotle (384 BC\[LongDash]322 BC), \
had the mistaken idea that gravitation was a natural tendency for objects to \
be attracted to a mystical place in the world. This place was the Center of \
The Earth. The heavier an object was the more strongly attracted it would be \
to that center. In other words, their weight determined their proper place \
and they all settled into that place. Today scientists laugh at that idea, \
but what is it that makes this idea wrong? What is the right idea?\
\>", "Text"],
Cell["\<\
Term 10: Aristotle\[CloseCurlyQuote]s Gravitation
The natural tendency for objects to be attracted to the center of the Earth, \
based on their weight\[LongDash]the heavier an object is, the more strongly \
it will be drawn to the center.\
\>", "Text",
CellFrame->True],
Cell["\<\
\tThe fact that Aristotle's idea of gravity was wrong took a long time to be \
realized. It was the medievel scientist Galileo Galilei (1564\[LongDash]1642) \
that put the proverbial \"nail in the coffin\" of Aristotle's idea. His \
argument went something like this; note\[LongDash]I will enumerate the \
arguments so they are easier to follow (this will be a standard procedure for \
proofs and derivations):\
\>", "Text"],
Cell[CellGroupData[{
Cell["\<\
We will assume that an object that is heavy falls faster than a lighter \
object as they are each trying to get to their proper places in the world. \
This explains why it was possible to pick up light objects, but not buildings \
or mountains; the latter being in their proper places.\
\>", "Item1Numbered",
TextJustification->1.],
Cell[TextData[{
"What happens when we strap a light object to a heavy one? There are two \
possibilities; either the combined object acts like a single object, or it \
does not. This idea is an example of the ",
StyleBox["law of the excluded middle",
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Cell["\<\
If the combination forms a single object, that single object is heavier than \
either of the two components. By the assumption in step 1 the single heavy \
object must fall faster than the heavier of the two component objects. \
\>", "Item1Numbered",
TextJustification->1.],
Cell["\<\
If the combination does not form a single object, then, by the assumption \
made in step 1, the light object will fall slower than the heavy object. \
Since they are connected by the strap (which is likely even lighter than the \
light object), the light object will slow the rate of fall of the heavy \
object, so the combination will not fall as fast as the single heavy object.\
\>", "Item1Numbered",
TextJustification->1.],
Cell[TextData[{
"These arguments lead to the prediction that the same combination of objects \
fall both faster and slower than the heavier of the two component objects. In \
a sense both answers are wrong since they oppose each other. A situation \
where every argument leads to a false outcome is called a ",
StyleBox["contradiction",
FontSlant->"Italic"],
". No assertion that leads to a contradiction can be true. This method of \
proof is ",
StyleBox["proof by contradiction",
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", or ",
StyleBox["reductio ad absurdum",
FontSlant->"Italic"],
". Generally, let us say that you are trying to prove an assertion. The \
first step in a proof by contradiction is to assume your assertion to be \
false. You then show that this falsehood leads to a contradiction. Since no \
assertion leading to a contradiction can be true, the falsehood is then \
itself false. This proves that your original assertion cannot be false. By \
the law of the excluded middle, if it cannot be false it must be true. This \
completes a proof by contradiction."
}], "Item1Numbered",
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Cell[TextData[{
"In this case we have proved that Aristotle's assertion that objects fall at \
a rate according to their weight is false; this is the same as proving that \
objects fall in a way that is independent of their weight. In fact, this \
principle is called ",
StyleBox["the law of falling bodies",
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Cell["\<\
Having made the prediction that objects fall independently of their weights, \
experiments were performed that confirmed this result.\
\>", "Item1Numbered",
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Cell[TextData[{
"This is a fantastic example of the method of theoretical physics that we \
call a ",
StyleBox["thought experiment",
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produced results that were contradictory, thus formulated a new hypothesis \
and confirmed it by both logical reasoning and physical experiment."
}], "Text"],
Cell["\<\
Term 10: Gravitation
The natural tendency for objects to be attracted to The Center of The Earth, \
independently of their weight\[LongDash]all objects are drawn towards The \
Center equally.\
\>", "Text",
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Cell[CellGroupData[{
Cell["The Nature of Mathematical Physics", "Section",
CellMargins->{{12, 12}, {5, 15}}],
Cell["\<\
While theoretical physics is rooted deeply in physical intuition, \
mathematical physics turns a question about physics into a question about \
mathematics. We solve that mathematics problem, and then turn that \
mathematical solution into a physics solution. How do we do this? We choose a \
suitable mathematical object to represent our physical situation. Then we \
study the properties of that structure in the light of the physical object. \
We check to make sure the physical situation obeys each of the properties of \
the mathematical object, this gives us confidence in that representation of \
the physical situation. Then we begin making predictions that can be tested. \
This is the process of making a mathematical model. You must guard yourself \
from getting too attached to such a model, no model is ever more than a \
shadow of reality.\
\>", "Text"],
Cell["\<\
Term 11: Mathematical Model
The direct application of mathematics to a problem in order to produce \
predictions that can be compared to reality.\
\>", "Text",
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"As one might expect, computational physics turns a question about physics \
into question about computation. Computation has been around at least as long \
as numbers. What makes it a separate area from mathematics is the advent and \
insurgence of computers into everything in the modern world. A modern smart \
phone is more powerful than a room-sized supercomputer of twenty five years \
ago. Instead of converting a physical phenomena into a mathematical object, \
we choose a computational object. Then we apply a method of solving a \
computational problem, what we call an ",
StyleBox["algorithm",
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called ",
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Term 12: Algorithm
A method of solving computational problems. \
\>", "Text",
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