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Darren L. Williams, Ph.D.;
Darren L. Williams, Ph.D.
Chemistry Department,
Sam Houston State University
, Huntsville, Texas, 77341-2117,
Search for other works by this author on:
Victoria S. Jackson, M.S.
Victoria S. Jackson, M.S.
Chemistry Department,
Lonestar College University Park
, Houston, Texas 77070,
Search for other works by this author on:

Spectroscopy Theory in One Dimension is an in-depth introduction to quantum chemistry using classical analytical calculus and symmetry group theory. The book employs the one-dimensional particle in a box and particle on a ring systems to develop the theoretical underpinnings of real-life spectroscopy and quantum mechanics. Wave functions, energy equations, transition intensities, and spectroscopic selection rules for absorption, emission, Rayleigh, and Raman spectra are derived and applied to explore why the sky is blue, why sunsets are red, and how quantum effects depend upon mass, size, temperature, and spectral resolution.

Readers will understand how to:

  • Predict three-dimensional spectroscopic behavior using one-dimensional examples

  • Assign quantum transitions in real spectroscopic data using analytical calculus and group theory

  • Connect the mysterious quantum mechanical wave function to everyday experiences of light and color

Spectroscopy Theory in One Dimension is an ideal and accessible text for students in spectroscopy and quantum mechanics courses. Novice researchers in all physical science fields as well as experienced researchers and technicians needing an in-depth theoretical approach will find it invaluable.

To the creator of all things visible and invisible… Soli Deo Gloria

This project was completed by my father Darrell Williams’ wise words, “the measured task gets done.” I passed this wisdom on to my Master of Science student and co-author Victoria Spenn Jackson when she completed much of the mathematics in this book during her graduate work in the Chemistry Department at Sam Houston State University.

Our work has been questioned, challenged, and honed by the Physical Chemistry I Quantum Mechanics and Spectroscopy students at SHSU over the past 18 years. Their successes in industry and graduate schools across the country speak to the usefulness of teaching spectroscopy theory in one dimension.

Victoria's work on the material in this text was supported by the Welch Foundation Departmental Development Grant X011, and we are grateful for their generous support of the Chemistry Department over many decades.

The Professional and Academic Center for Excellence directed by Dr Todd Primm at SHSU sponsored the Faculty Writing Circles. The writing circles encouraged and aided the writing process with their writing retreats, weekly writing meetings, and instruction in sustained-effort tools such as pomodoro timing.

I thank my book team: colleague David Thompson Ph.D., former students Ashley Williams, Heather Coats M.S., Joe Treviño M.S., Nilan Kamathewatta Ph.D., my sister Patti J. Brown M.S., and current students Andrea Martinez, Brodin Huse, Marie Hailey, Makaul Barbaree, Micaela Badger, and Tommy Gilpin for their constructive comments and reviews.

I am especially grateful to my Ph.D. advisor Dr. Joseph W. Nibler for his in-depth review of the science, for his insightful corrections, and for his informative Foreword.

I thank the many employees and regulars at Sipsy's coffeehouse in Huntsville, TX. They were the first coffeehouse to open after COVID, and I wrote most of this manuscript over two years sitting in their “good chair.”

I acknowledge the active support of my wife Jennifer M. Williams, who frequently told me to “go write” while she finished tasks without me. I have many fond memories of her driving us all over Texas while I wrote on my laptop in the passenger seat.

One of the joys of an academic career is the opportunity to work with creative and interesting students. Over the last 60 years in the Chemistry Department at Oregon State University, I have had the privilege to interact with many such individuals. Not the least of these was Darren Williams, whose intellect and infectious enthusiasm as a student, as a teaching assistant, and as a person were evident from the beginning of his doctoral studies with me. Darren has a quirky sense of humor, an aspect I enjoyed immensely. Once, while serving as a teaching assistant in an advanced Chemistry lab I was teaching, he was asked to obtain photos of all the graduating seniors. He did so, but brought a Groucho Marx nose, moustache, and glasses for each student to wear for the occasion, something each did with delight. Another time, at a Western Spectroscopy Association meeting in Asilomar, CA, he and two other students presented in a poster session their research obtained in our lab using Coherent Anti-Stokes Raman Spectroscopy (CARS). But, they asked me at the last minute if they could bring an additional poster they were preparing with their very latest results. I said OK if it is good science and there is room at the session. Darren assured me that the poster “LOOKS very good.” Turns out it was a bogus poster that mocked the use of acronyms such as CARS, by claiming the development of new spectroscopic techniques (PLANES, TRAINS, and AUTOMOBILES) to detect the delivery of anti-cancer drugs in mice. Replete with spectra, very linear graphs, and complex statistical analysis, only slowly did viewers realize that it was a total spoof. Recognition brought smiles and laughter, results that invariably followed a number of other such Williams's pranks.

Many students have been fortunate to enjoy Darren as a teacher at Sam Houston University, where he is a Professor of Chemistry. This book offers clear evidence of his ability to make complex topics clear and accessible to all. Most physical chemistry and quantum mechanics courses briefly discuss one-dimensional problems such as the particle in a box and on a ring. The focus is usually on the mathematical form of the energy levels and wave functions. In this book, Darren and his co-author, Victoria Jackson, go much further. They discuss in detail the calculation of transition probabilities using the one-dimensional dipole and polarizability operators. The results lead naturally to selection rules and to predicted spectral intensities. Explicit examples are given for electronic, infrared, and Raman spectroscopies, as well as for Rayleigh light scattering. The authors also show how the same selection rules can be derived from symmetry and group theoretical arguments. The result is an exposition that ties many aspects of quantum mechanics together and allows extension to 2- and 3-dimensional problems. This book will be especially helpful to undergraduates taking Physical Chemistry and to beginning graduate students interested in spectroscopy. I heartily recommend it.

Joseph Nibler

Emeritus Professor of Chemistry

Oregon State University

“Quantum is kicking my butt. ” she wrote on social media.

My heart sank and a knot developed in my stomach. I wondered to myself, “I hope it isn't my fault.”

So I reached out to her with a direct message. She had graduated from SHSU the previous May and is now enrolled in a Chemistry Ph.D. program at a top-tier research institution. I was worried that my method of teaching quantum mechanics had not prepared her for graduate school.

She quickly responded and put me at ease. She said all of the students from both large and small universities were struggling with the multivariable calculus. Then, she said, “at least I know where these problems are going,” and that put me at ease.

I have been teaching spectroscopy in one dimension from the wave function to the full simulated spectrum for 18 years and teaching the students to apply their knowledge to three-dimensional systems by analogy. This text was written to document this method of teaching quantum mechanics and spectroscopy.

I hope that this text will serve many roles: (a) as a text for motivated chemistry majors to read prior to (or during) their quantum mechanics and spectroscopy course, (b) as a lecture preparation text for professors giving them many one-dimensional examples of the math of spectroscopy, and (c) as a refresher text for professional spectroscopists who want to resharpen their theoretical understanding of the tools of their trade.

Paul Adrien Maurice Dirac, who shared the 1933 Nobel Prize with Schrödinger, made his now-famous statement in his 1929 paper Quantum Mechanics of Many-Electron Atoms:

“The underlying physical laws necessary for the mathematical theory for a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble.”

In his next sentence, Dirac clearly states the purpose and motivation of this book:

“It therefore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation.”

That is why this book exists—to explain the main features of spectroscopy without too much computation. With these simple models, we discuss why the daytime sky is blue, why the sunsets are red, why diamonds sparkle, and how lasers work.

With this book, we hope you gain a working knowledge of the forest by focusing on one species of tree—the one-dimensional wave function.

Darren L. Williams, Ph.D.

Professor of Physical Chemistry, Sam Houston State University

Huntsville, TX

August 2022

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