As my own appreciation of many aspects of quantum mechanics evolved, I found it useful to start my lectures in this course with a quote from a famous researcher and Nobel laureate (1995, for his work on quantum electrodynamics), the late Professor Richard Feynman, which – taken slightly out of context – reads [6]:
I think I can safely say that nobody understands quantum mechanics.
This rather discouraging statement has to be seen from the viewpoint that, when studying quantum mechanics, one realizes that this theory is not based on axioms, but on postulates – a very unusual fact in the sciences. Furthermore, it replaced deterministic results with probabilistic answers. When exposed to these conundrums, students will naturally ask the question: “Why bother studying quantum mechanics, if I will not understand it anyway?” or worse, “Is quantum mechanics for real, or is it the brainchild of some far‐out mad scientists?” The answer here is also contained in a quote by Feynman:
It doesn't matter how beautiful a theory is, …. If it doesn't agree with experiment, it's wrong.
This statement could also be formulated to imply that a theory that consistently provides answers that agree with the experiment most likely is correct. Thus, although nobody may understand quantum mechanics in its entirety, it gives answers that – over and over – agree with experiments and in fact provides a mechanism and framework for explaining the experimental results.
Quantum mechanics originated in the early decades of the twentieth century, when it was found that some experiment results just could not be explained by existing laws of physics and, in fact, violated established physical dogmas. It was these results that gave rise to the emergence of quantum mechanics that grew out of a patchwork of ideas aimed at explaining these hitherto unexplainable experimental results. These ideas coalesced into the field we now refer to as quantum mechanics. This newly formulated theory was wildly successful in explaining a myriad of physical and chemical observations – from the shape and meaning of the periodic chart of elements to the subject of this book, namely, the interaction of light with matter that is the basis of spectroscopy.
While many aspects of molecular spectroscopy, such as the rotational or vibrational energies of a molecule, can be described in classical terms, the idea that atoms and molecules can exist in quantized, stationary energy states is a direct result of the postulates of quantum mechanics. Furthermore, application of the principles of time‐dependent quantum mechanics explains how electromagnetic radiation of the correct energy may cause a transition between these stationary energy states and produce observable spectra. Thus, the entire field of molecular spectroscopy is a direct result of quantum mechanics and represents the experimental results that confirms the theory. The phenomenal growth of all forms of spectroscopy over the past eight decades has contributed enormously to our understanding of molecular structure and properties. What started as simple molecular spectroscopy such as infrared and Raman vibrational spectroscopy, (microwave) rotational spectroscopy, ultraviolet–visible absorption, and emission spectroscopy has now bloomed into a very broad field that includes, for example, the modern magnetic resonance techniques (including medical magnetic resonance imaging); nonlinear, laser, and fiber‐based spectroscopy; surface and surface‐enhanced spectroscopy; pico‐ and femtosecond time‐resolved spectroscopy, and many more. Spectroscopy is embedded as a major component in material science, chemistry, physics, and biology and other branches of scientific and engineering endeavors. Thus, the quantum mechanical underpinnings of spectroscopy are a major subject that need to be understood in the pursuit of scientific efforts.
References
1 1 Engel, T. and Reid, P. (2010). Physical Chemistry, 2e. Upper Saddle River, NJ: Pearson Prentice Hall.
2 2 Levine, I. (1970). Quantum Chemistry, vol. I&II. Boston: Allyn & Bacon.
3 3 Levine, I. (1983). Quantum Chemistry. Boston: Allyn & Bacon.
4 4 Kauzman, W. (1957). Quantum Chemistry. New York: Academic Press.
5 5 Eyring, H., Walter, J., and Kimball, G.E. (1967). Quantum Chemistry. New Yrok: Wiley.
6 6 Feynman, R. (1964). Probability and Uncertainty: The Quantum Mechanical View of Nature ‐ The Character of Physical Law 1964. Cornell University.
1 Transition from Classical Physics to Quantum Mechanics
At the end of the nineteenth century, classical physics had progressed to such a level that many scientists thought all problems in physical science had been solved or were about to be solved. After all, classical Newtonian mechanics was able to predict the motions of celestial bodies, electromagnetism was described by Maxwell's equations (for a review of Maxwell's equations, see [1]), the formulation of the principles of thermodynamics had led to the understanding of the interconversion of heat and work and the limitations of this interconversion, and classical optics allowed the design and construction of scientific instruments such as the telescope and the microscope, both of which had advanced the understanding of the physical world around us.
In chemistry, an experimentally derived classification of elements had been achieved (the rudimentary periodic table), although the nature of atoms and molecules and the concept of the electron's involvement in chemical reactions had not been realized. The experiments by Rutherford demonstrated that the atom consisted of very small, positively charged, and heavy nuclei that identify each element and electrons orbiting the nuclei that provided the negative charge to produce electrically neutral atoms. At this point, the question naturally arose: Why don't the electrons fall into the nucleus, given the fact that opposite electric charges do attract? A planetary‐like situation where the electrons are held in orbits by centrifugal forces was not plausible because of the (radiative) energy loss an orbiting electron would experience. This dilemma was one of the causes for the development of quantum mechanics.
In addition, there were other experimental results that could not be explained by classical physics and needed the development of new theoretical concepts, for example, the inability of classical models to reproduce the blackbody emission curve, the photoelectric effect, and the observation of spectral “lines” in the emission (or absorption) spectra of atomic hydrogen. These experimental results dated back to the first decade of the twentieth century and caused a nearly explosive reaction by theoretical physicists in the 1920s that led to the formulation of quantum mechanics. The names of these physicists – Planck, Heisenberg, Einstein, Bohr, Born, de Broglie, Dirac, Pauli, Schrödinger, and others – have become indelibly linked to new theoretical models that revolutionized physics and chemistry.
This development of quantum theory occupied hundreds of publications and letters and thousands of pages of printed material and cannot be covered here in this book. Therefore, this book presents many of the difficult theoretical derivations as mere facts, without proof or even the underlying thought processes, since the aim of the discussion in the following chapters is the application of the quantum mechanical principles to molecular spectroscopy. Thus, these discussions should be construed as a guide to twenty‐first‐century students toward acceptance of quantum mechanical principles for their work that involves molecular spectroscopy.
Before the three cornerstone experiments that ushered in quantum mechanics – Planck's blackbody emission curve, the photoelectric effect, and the observation of spectral “lines” in the hydrogen atomic spectra – will be discussed, electromagnetic radiation, or light, will be introduced at the level of a wave model of light, which was the prevalent way to look at this phenomenon before the twentieth century.
1.1 Description of Light as an Electromagnetic Wave
As mentioned above, the description of electromagnetic radiation in terms of Maxwell's equation was published in the early 1860s. The solution of these differential equations described light as a transverse wave of electric and magnetic fields. In the absence of charge and current, such a wave, propagating in vacuum in the positive z‐direction,