An Introduction to Quantum Mechanics

From Facts to Formalism

Tilak Sinha

• Publisher: CRC Press
• Year: 2021
• Print ISBN:
  9780367547073
• eBook ISBN:
  9781000415643
• Topic: Physical Sciences
• Subtopic: Linear Algebra

The core content of even the most intricate intellectual edifices is often a simple fact or idea. So is it with quantum mechanics; the entire mathematical fabric of the formal description of quantum mechanics stems essentially from the fact that quantum probabilities interfere (i.e., from the superposition principle). This book is dedicated to substantiating this claim. In the process, the book tries to demonstrate how the factual content of quantum mechanics can be transcribed in the formal language of vector spaces and linear transformations by disentangling the empirical content from the usual formal description. More importantly, it tries to bring out what this transcription achieves.

The book uses a pedagogic strategy which reverse engineers the postulates of quantum mechanics to device a schematic outline of the empirical content of quantum mechanics from which the postulates are then reconstructed step by step. This strategy is adopted to avoid the disconcerting details of actual experiments (however simplified) to spare the beginner of issues that lurk in the fragile foundations of the subject.

In the Copenhagen interpretation of quantum mechanics, the key idea is measurement. But "measurement" carries an entirely different meaning from the connotation that the term carries elsewhere in physics. This book strives to underline this as strongly as possible.

The book is intended as an undergraduate text for a first course in quantum mechanics. Since the book is self contained, it may also be used by enthusiastic outsiders interested to get a glimpse of the core content of the subject.

Features:

• Demonstrates why linear algebra is the appropriate mathematical language for quantum mechanics.

• Uses a reconstructive approach to motivate the postulates of quantum mechanics.

• Builds the vocabulary of quantum mechanics by showing how the entire body of its conceptual ingredients can be constructed from the single notion of quantum measurement.

Chapter 1

Theoretical Framework - A Working Definition

The principal objective of this book is to make quantum mechanics seem natural. We wish to demonstrate how quantum mechanics is just another theoretical framework that conforms to the standard definition of a theory in the natural sciences, and it is our purpose to expose the facts that underpin the essential fabric of the formal description of quantum mechanics. Obviously, before we embark on this task, we must be clear in our minds what we mean by a theoretical framework and its formal description. So, in this opening chapter, we begin our journey by looking into the meaning of these terms.

What Is a Theory?

In the natural sciences, a theory is defined as follows:

A theory is a logical framework comprising some assumption(s) from which one can make some logical prediction(s) that, in principle, should be falsifiable by experiment¹.

The assumptions of the theory are usually motivated by factual observations². They go by various names such as laws, principles, axioms, and postulates. The various names are used in slightly different contexts, but they essentially play the same role in the logical system. Here is a simple example of a theory. It makes the following two assumptions:

A1 Red apples are sweet.

A2 Sweet apples are expensive.

Since we want to discuss this toy theory for a while, we will give it a name. We will call it “a theory on the price of apples” or ATOPA in short. From the two assumptions, A1 and A2, we can immediately deduce an implication:

¹The experiment need not necessarily be performed by humans. It could well be some natural phenomenon that humans can observe.

²Quite often, however, the assumptions are educated guesses which do not lend themselves to a direct observation. It is only their implications that can be subjected to observation in such cases.

T1 Red apples are expensive.

This, of course, is certainly falsifiable by experiment. All we need to do is go about asking the price of red apples in every market that we come across.

It is important to note that the statement “If A1 and A2 is true, then T1 is true”, is always correct (true) irrespective of the truth of A1, A2 or T1. Whether or not A1, A2 or T1 is actually true in real life is a separate issue. If T1 turns out to be false, then we say that ATOPA has been disproved or falsified by observation. In this case at least one of the assumptions of ATOPA (i.e., either A1 or A2) must be false.

It should be emphasized that a theory can never be proved; it can only be disproved. Every time a theoretical prediction fits observed data, it merely increases our confidence in the theory. No matter how many times this agreement (between observed data and theoretical prediction) occurs, the theory is not proved. The possibility of a hitherto unperformed experiment which does not agree with the theoretical prediction can never be ruled out. Moreover, even if there are no disagreements with observed data, it is always possible to imagine the existence of a yet undiscovered theory which makes the same predictions so that it is impossible to claim that a given theory is the correct one. On the contrary, a single contradiction of theoretical prediction with observation immediately falsifies the theory in question.

Formal Description of a Theory

To put the discussion into context, let us start with an anecdote.

The deaf composer

In a town there lived a music composer who had conceived a theory on the musicality of music. He was extraordinarily gifted and could actually recognize the patterns that made some sequence of notes musical, some discordant, and some in between. The composer wanted to develop and propagate his theory. He was so ambitious in his objective that not only did he want to specify a rule that would tell one which pieces will be musical and which not, he also wanted to invent a formula that would quantify the musical quality of a piece. In fact, he even wanted to develop a recipe that would produce good music!

Unfortunately, the composer lived in an era when musical notation was not yet invented. So the only way he could communicate his ideas was through actual demonstration. His ideas, naturally, did not propagate very far. Firstly, they were limited to his students and friends. Moreover, how much of his ideas would be grasped in a demonstration was crucially dependent on the perceptibility and sensitivity of the audience. The communication of the ideas through his followers was even more vulnerable to distortion because it was also heavily dependent on the quality of demonstration, in addition to the sensitivity and perceptibility of the recipients. So each layer of prospective followers in the chain was receiving an increasingly deviated version of the original theory. As was to be expected, the distorted versions of the theory were more and more, less and less impressive. The future of the theory looked grim! Our composer became very depressed at the inevitable fate of the science that he had given birth to.

At this point a mathematician in the town, who had only a modest appreciation of music, developed a scheme by which music could be represented in a visual form as a script, much the same way that spoken words are scripted using alphabets and punctuation. This was the birth of the musical notation!

Our composer immediately realized that this could save his theory. So he took the plunge and invested all his time and energy into developing and writing up his ideas in the newly invented language. He wrote down all the facets and features of his musical theory, starting from the recipe of identifying a truly musical piece to the formula for creating one. The results were exactly as he had expected. In a few years, there was an immense following of his theory. With the distortions hugely minimized, people really began to appreciate the true content and merit of his work. It was all because the visual, scripted representation provided a much more objective character to his theory; it was now much less open to subjective, personal interpretations.

Our composer quickly became a famous man. He came to be recognized as a father figure of the science of music. However, it was only after many years that he received his highest accolade. It was at a concert where he was invited as the chief guest. The showstopper of the evening was a piece composed by a young man about whom our composer had not even heard of before. The performance was so overwhelming that it left the audience speechless for some time. Then the entire audience rose to give the young composer a standing ovation. As he was being applauded by everyone, the anchor walked up to the middle of the stage and announced: “If you are all astounded by what you heard just now, I have more astonishment in stock for you. Our composer can't hear a single clap that you are showering on him. He was born deaf. He cultivated the art and skill of making music based on the theory and method that was developed by our chief guest tonight!”.

This was a fictional anecdote designed to demonstrate the purpose of what is known as a formal description. A formal description is to its informal counterpart what the scripted version of the musical theory was to its original version (that was almost going to get obliterated). The whole purpose of a formal description is to make it so completely free of subjective human interpretations that even a machine, which is programmed to read the language, will be equipped to use it in spite of the fact that it has no understanding of the underlying reality. Let us then try to illustrate, more concretely, what we mean by a formal description of a theory in the natural sciences.

Formal description in the natural sciences

In the natural sciences, a theory is essentially a logical scheme to describe our experience. It establishes a connection between some set of our experience (usually called phenomena) and some logical system. The more mature the state of the science, the clearer the connection. In the initial stages of the formulation of a theory, the description (connection of the phenomena to the logical system) usually suffers from having ambiguities and, consequently, interpretations can be subjective. The ambiguities may be so subtle that they may not cause practical problems for a long time until some day one finds oneself confronting a situation that demands...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Contents
• Preface
• Prologue
• 1 Theoretical Framework - A Working Definition
• 2 The Empirical Basis of Quantum Mechanics
• 3 States as Vectors
• 4 Observables as Operators
• 5 Imprecise Measurements and Degeneracy
• 6 Time Evolution
• 7 Continuous Spectra
• 8 Three Archetypal Eigenvalue Problems
• 9 Composite Systems
• A Probability
• B Linear Algebra
• Bibliography
• Index