Dad I understand that light is emitted when an atom decays from an excited state to a lower energy state. Feynman That's right. Dad And light consists of particles called photons. Feynman Yes. Dad So the photon ‘particle’ must be inside the atom when it is in the excited state. Feynman Well no. Dad Well how do you explain that the photon comes out of the atom when it was not there in the first place. Feynman (Physics Teacher 1969) I'm sorry. I don't know I can't explain it to you. INTRODUCTION As was stated in Chapter 1, spectroscopy is a study of the interaction of matter with radiation. In order to understand their mutual interaction, we must therefore characterize both radiation and matter. The starting point is quantum mechanics, because the origin of quantum mechanics was based on unexplained phenomena related to both the nature of electromagnetic radiation and that of matter. In fact, quantum mechanics owes its birth to some puzzling results of spectral and other experiments. We may say that “spectroscopy is quantum mechanics in action.” We therefore begin our study of quantum mechanics with a recapitulation of its postulates. An understanding of the postulates is also essential for our development of the time-dependent perturbation theory and also forms the basis for the connection between theory and experiment. Indeed, spectroscopy is the experimental verification of the laws of quantum mechanics. Quantum mechanics is based on a few postulates (from the Latin postulatum ), meaning a truth that does not need any further proof, because it is obvious by itself, and its success lies in the fact that no experiment till date has proved it wrong. There is no agreement of how many axioms one needs to describe the machinery of quantum mechanics; however, the six given below is an acceptable number.
Dad I understand that light is emitted when an atom decays from an excited state to a lower energy state. Feynman That's right. Dad And light consists of particles called photons. Feynman Yes. Dad So the photon ‘particle’ must be inside the atom when it is in the excited state. Feynman Well no. Dad Well how do you explain that the photon comes out of the atom when it was not there in the first place. Feynman (Physics Teacher 1969) I'm sorry. I don't know I can't explain it to you. INTRODUCTION As was stated in Chapter 1, spectroscopy is a study of the interaction of matter with radiation. In order to understand their mutual interaction, we must therefore characterize both radiation and matter. The starting point is quantum mechanics, because the origin of quantum mechanics was based on unexplained phenomena related to both the nature of electromagnetic radiation and that of matter. In fact, quantum mechanics owes its birth to some puzzling results of spectral and other experiments. We may say that “spectroscopy is quantum mechanics in action.” We therefore begin our study of quantum mechanics with a recapitulation of its postulates. An understanding of the postulates is also essential for our development of the time-dependent perturbation theory and also forms the basis for the connection between theory and experiment. Indeed, spectroscopy is the experimental verification of the laws of quantum mechanics. Quantum mechanics is based on a few postulates (from the Latin postulatum ), meaning a truth that does not need any further proof, because it is obvious by itself, and its success lies in the fact that no experiment till date has proved it wrong. There is no agreement of how many axioms one needs to describe the machinery of quantum mechanics; however, the six given below is an acceptable number.
To date, the physical theory that most accurately describes the behavior of the atomic world is quantum mechanics. Since spectroscopy uses light to probe the atomic and molecular world, it should not be much of a surprise to learn that quantum mechanics is used to describe how spectroscopy works. Nobel Prize-winning physicist Richard Feynmann once said, "If you think you understand quantum mechanics, you don't understand quantum mechanics." This chapter won't bring you to a complete understanding of quantum mechanics, Feynmann's quote notwithstanding. But it will try to point out the important aspects, and hopefully it will at least leave the reader with the understanding that spectroscopy is quantum-mechanically based. Science's job is to try to understand the natural universe. To do this, scientists try to generate models to describe and predict the universe's behavior. (Not the behavior of the entire universe, but small selected parts of it.) If the model and the universe do not agree, there are two choices: change the model, or change the universe. Since all attempts at changing the universe have failed, our only choice is to change the model. Since the 1600s, scientific advance has accelerated at least in part because the proposed models of the universe have been increasingly more acceptable. Perhaps the first true âscientificâ investigation (by the modern definition) was Robert Boyle's investigation of gases in the 1660s. While there were some noteworthy incorrect or improper models developed (phlogiston and vitalism are two that come to mind immediately, but doubtless there are others), incorrect or improper models were ultimately replaced by more viable models. One such group of models was what we now call Newton's laws of motion. In just a few statements, verbal or mathematical, Newton's laws accurately describe the motion of matter. Further, these models withstood the test of multiple examinations, and are accepted as proper models of how matter behaves.
Most introductory courses on physics mention the basics of atomic physics: we learn about the ground state and excited states, and that the excited states have a finite life-time decaying eventually to the ground state while emitting radiation. You might have hoped for a better understanding of these decay processes when you started studying quantum mechanics. If this was the case, you must have been disappointed. Courses on basic quantum mechanics in fact assume that all quantum states have an infinitely long life-time. You learn how to employ the Schrödinger equation to determine the wave function of a particle in various setups, for instance in an infinitely deep well, in a parabolic potential, or in the spherically symmetric potential set up by the nucleus of a hydrogen atom. These solutions to the Schrödinger equation are explained as being stationary, the only time-dependence of the wave function being a periodic phase factor due to the energy of the state. Transitions between different quantum states are traditionally not discussed and left for courses on advanced quantum mechanics. If you have been disappointed by this, then this chapter will hopefully provide some new insight: we learn how matter and radiation interact, and, in particular, how excited atomic states decay to the ground state. Our main tool to calculate emission and absorption rates for radiative processes is Fermi’s golden rule (see Section 1.6.1). We thus start by refreshing the golden rule and specifying its proper use. Before we turn to explicit calculations of transition rates, we use the first part of the chapter for some general considerations concerning emission and absorption, those being independent of the details of the interaction between matter and electromagnetic radiation. In this context we briefly look at master equations that govern the probabilities to be in certain states and we show how to use those to understand properties of the equilibrium state of interacting matter and radiation. We show that it is even possible to understand so-called black-body radiation without specifying the exact interaction Hamiltonian of matter and radiation.
Unlike special relativity, quantum mechanics was not the inspired product of one mind. It developed, in fits and starts, over a period of 25 years (1900–1925), and many cooks stirred the broth. Even now, a century later, quantum mechanics raises profound conceptual questions. Every competent physicist can “do” quantum mechanics, and the predictions it makes are in spectacularly good agreement with experimental results; there can be little doubt that quantum mechanics is “right.” On the other hand, as Richard Feynman once remarked, “nobody understands quantum mechanics.” I'll tell the story in two parts. First the history (how the essential pieces of the puzzle were assembled), and then the implications (what the finished picture has to say about our world). Photons Like relativity, quantum mechanics began with the study of light. In both cases, this historical association is misleading. Relativity, really, has nothing to do with light – to be sure, it involves that magic speed limit, c , and of course the most familiar thing that travels at speed c is light. So light is a convenient vehicle for introducing the subject. But it is perfectly possible to imagine a universe in which there is no such thing as light, and yet relativity might still be valid. Quantum mechanics, too, starts with light – in this case the “quantum” of light (the photon). But, as it turns out, this is just one rather specialized example of much more general principles – in fact, a rather bad example, because photons are by their nature relativistic (they travel at the speed of light), and it would be more natural to begin with the quantum mechanics of slow-moving objects, in analogy with Newtonian mechanics.
Classical mechanics, which gives a fairly accurate description of large systems (e.g., solar system) as also of mechanical systems in our every day life, however, breaks down when applied to small (microscopic) systems such as molecules, atoms and nuclei. For example, (1) classical mechanics cannot even explain why the atoms are stable at all. A classical atom with electrons moving in circular or elliptic orbits around the nucleus would continuously radiate energy in the form of electromagnetic radiation because an accelerated charge does radiate energy. As a result the radius of the orbit would become smaller and smaller, resulting in instability of the atom. On the other hand, the atoms are found to be remarkably stable in practice. (2) Another fact of observation that classical mechanics fails to explain is wave particle duality in radiation as well as in material particles. It is well known that light exhibits the phenomena of interference, diffraction and polarization which can be easily understood on the basis of wave aspect of radiation. But light also exhibits the phenomena of photo-electric effect, Compton effect and Raman effect which can only be understood in terms of corpuscular or quantum aspect of radiation. The dual behavior of light, or radiation cannot be consistently understood on the basis of classical concepts alone or explained away by saying that light behaves as wave or particle depending on the kind of experiment we do with it (complementarity). Moreover, a beam of material particles, like electrons and neutrons, demonstrates wave-like properties (e.g., diffraction). A brief outline of phenomena that require quantum mechanics for their understanding follows.
This chapter starts with the definition of postulates. Here, in quantum mechanics, the postulates concern the atomic and molecular properties, which are quite far removed from everyday experience. The postulates described here make no claim to those happy mathematical adjectives such as ‘rigorous’, ‘complete’, or irreducible. The ‘proof’ of the postulates is their ability to explain the adequate experimental observations. Here, we have incorporated six postulates. The first postulate opines that ‘the state of quantum mechanical objects is described by a wavefunction’, whereas the second postulate says that ‘every physical observable is expressed by a linear operator’. It also describes how a quantum mechanical operator is constructed. The third postulate gives an idea that the measurement of a physical observable will yield a result, which is one of the Eigen values of the corresponding operator for that observable. The fourth postulate gives how the average of a sequence of measurement is connected with an operator. The fifth postulate is called the time-dependent postulate. Here, it gives a relationship between the time-dependent wave function and an operator, whereas the sixth postulate says that the wave function must be antisymmetric (or symmetric) for the exchange of identical fermions. It must be noted that each and every postulate has been explained clearly. The references, solved problems, and questions on concepts have been given at the end of the chapter.
This chapter presents the basic physical ideas that underpin the quantum theory. Historians place the beginnings of the quantum theory at 1900. That year Max Planck published his famous formula for the distribution of energy as a function of frequency in black-body radiation. Curiously, although radiation theory gave birth to quantum mechanics, the interaction of matter and electromagnetic radiation is a rather advanced topic in quantum mechanics and it took almost fifty years for an acceptable formulation to be achieved. One reason for this is that photons are quanta of the electromagnetic field, and the quantum theory of fields requires a relativistic treatment to be entirely successful. The corpuscular nature of light was only half of the input that gave birth to quantum mechanics. The other half concerns matter. Given that the discreteness of photons manifests itself during the emission and absorption process, it is not surprising to find that quantum oddities afflict the behaviour of material particles too.