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Using Mathematica for Quantum Mechanics: A Student's Manual by Roman Schmied
English | PDF | 2019 | 163 Pages | ISBN : N/A | 8.90 MB
This book is an attempt to help students transform all of the concepts of quantum mechanics into concrete computer representations, which can be constructed, evaluated, analyzed, and hopefully understood at a deeper level than what is possible with more abstract representations.
Learning quantum mechanics is difficult and counter-intuitive. The first lectures I heard were filled with strange concepts that had no relationship with the mechanics I knew, and it took me years of solving research problems until I acquired even a semblance of understanding and intuition. This process is much like learning a new language, in which a solid mastery of the concepts and rules is required before new ideas and relationships can be expressed fluently.
The major difficulty in bridging the chasm between introductory quantum lectures, on the one hand, and advanced research topics, on the other, was for me the lack of such a language, or of a technical framework in which quantum ideas could be expressed and manipulated. On the one hand, I had the hand tools of algebraic notation, which are understandable but only serve to express very small systems and ideas; on the other hand I had diagrams, circuits, and quasi-phenomenological formulae that describe interesting research problems, but which are difficult to grasp with the mathematical detail I was looking for.
This book is an attempt to help students transform all of the concepts of quantum mechanics into concrete computer representations, which can be constructed, evaluated, analyzed, and hopefully understood at a deeper level than what is possible with more abstract representations. It was written for a Master’s and PhD lecture given yearly at the University of Basel, Switzerland. The goal is to give a language to the student in which to speak about quantum physics in more detail, and to start the student on a path of fluency in this language. We will revisit most of the problems encountered in introductory quantum mechanics, focusing on computer implementations for finding analytical as well as numerical solutions and their visualization.
On our journey we approach questions such as: You already know how to calculate the energy eigenstates of a single particle in a simple one-dimensional potential. How can such calculations be generalized to non-trivial potentials, higher dimensions, and interacting particles?
You have heard that quantum mechanics describes our everyday world just as well as classical mechanics does, but have you ever seen an example where such behavior is calculated in detail and where the transition from classical to quantum physics is evident?
How can we describe the internal spin structure of particles? How does this internal structure couple to the particles’ motion?
What are qubits and quantum circuits, and how can they be assembled to simulate a future quantum computer?
Most of the calculations necessary to study and visualize such problems are too complicated to be done by hand. Even relatively simple problems, such as two interacting particles in a one-dimensional trap, do not have analytic solutions and require the use of computers for their solution and visualization. More complex problems scale exponentially with the number of degrees of freedom, and make the use of large computer simulations unavoidable.
The methods presented in this book do not pretend to solve large-scale quantum-mechanical problems in an efficient way; the focus here is more on developing a descriptive language. Once this language is established, it will provide the reader with the tools for understanding efficient large-scale calculations better.

Why Mathematica? This book is written in the Wolfram language of Mathematica (version 11); however, any other language such as Matlab or Python may be used with suitable translation, as the core ideas presented here are not specific to the Wolfram language.
There are several reasons why Mathematica was chosen over other computer-algebra systems:
Mathematica is a very high-level programming environment, which allows the user to focus on what s?he wants to do instead of how it is done. The Wolfram language is extremely expressive and can perform deep calculations with very short and unencumbered programs.
Mathematica supports a wide range of programming paradigms, which means that you can keep programming in your favorite style. See section 1.9 for a concrete example.
The Notebook interface of Mathematica provides an interactive experience that holds programs, experimental code, results, and graphics in one place.
Mathematica seamlessly mixes analytic and numerical facilities. For many calculations it allows you to push analytic evaluations as far as possible, and then continue with numerical evaluations by making only minimal changes.
A very large number of algorithms for analytic and numerical calculations is included in the Mathematica kernel and its libraries. Visit My Blog For Daily Very Exclusive Content,We Are Here For You And Without You And Your Support We Can’t Continue
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