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Oxford University Press eBooks

The quantum mechanics of motion

Peter Joshua Atkins, R. George Ratcliffe, Mark R. Wormald, Julio de Paula

Published Mar 10, 2023• 0 citations
Publisher

Abstract

This chapter pays particular attention to the quantum mechanics of motion. It is not possible to explain the structures and properties of atoms and molecules, where the role of electrons is central, without using quantum mechanics. Many of the processes of biochemistry involve the transfer of electrons and protons from a donor to an acceptor, and this migration can be understood only in terms of quantum theory. Here, the chapter introduces three fundamental types of motion — translation, rotation, and vibration — along with their characteristic quantum mechanical behaviours. It shows how various unexpected properties emerge, such as the ability of particles to penetrate into classically forbidden regions, the quantization of energy and angular momentum, and the existence of energy that cannot be removed from the system.

DOI
10.1093/hesc/9780198830108.003.0040

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This chapter pays particular attention to the quantum mechanics of motion. It is not possible to explain the structures and properties of atoms and molecules, where the role of electrons is central, without using quantum mechanics. Many of the processes of biochemistry involve the transfer of electrons and protons from a donor to an acceptor, and this migration can be understood only in terms of quantum theory. Here, the chapter introduces three fundamental types of motion — translation, rotation, and vibration — along with their characteristic quantum mechanical behaviours. It shows how various unexpected properties emerge, such as the ability of particles to penetrate into classically forbidden regions, the quantization of energy and angular momentum, and the existence of energy that cannot be removed from the system.
This chapter evaluates translational motion, motion through space, which is one of the fundamental types of motion treated by quantum mechanics. According to quantum theory, a particle constrained to move in a finite region of space is described by only certain wavefunctions and can possess only certain energies. That is, quantization emerges as a natural consequence of solving the Schrödinger equation and the conditions imposed on it. The solutions also expose a number of non-classical features of particles, especially their ability to tunnel into and through regions where classical physics would forbid them to be found. Light particles are more able to tunnel through barriers than heavy ones. The chapter also considers the occurrence of degeneracy, which is a consequence of the symmetry of the system.
This chapter addresses quantum mechanics and spectroscopy. Two key ideas emerge from the theory of quantum mechanics: the first is that the electron is described by a wavefunction, and the second is that the energy of the electron is quantized. The wavefunction is important since its square gives the probability density of the electron. Quantization means that the energy can only have certain discrete values, rather than being allowed to vary continuously. The chapter looks at how these wavefunctions and energy levels arise in quantum mechanics, and how they can actually be computed in some simple cases. This will involve setting up and then solving the famous Schrödinger equation. Quantum mechanics is not restricted to the description of electrons in atoms and molecules, but can also be used to describe the motion of molecules in space (translation), as well as the rotation and vibration of molecules.
This chapter considers the quantum mechanics of translation and vibration. Both types of motion can be solved in certain cases, and both are important not only in their own right but also because they form a basis for the description of more complicated types of motion. It mentions translational motion, which has the advantage of introducing in a simple way many of the striking features of quantum mechanics. The chapter describes certain features of wavefunctions that are common to all the problems. The combination of features of wavefunctions with the solution of the Schrödinger equation results in one of the most characteristic features of quantum mechanics, the quantization of energy.
This chapter focuses on the concept of rotational motion. It explains that energy and angular momentum for a particle rotating in two- or three-dimensions are quantized and satisfied cyclic boundary conditions, which imply the quantization of angular momentum's magnitude and z-component. The electronic structure of atoms showcases rotational motion because electrons orbit around nuclei and spin on their axis. The chapter details the solutions for the Schrödinger equation and the vector model of angular momentum. It also discusses the concept of space quantization revolving around the quantum mechanical result of a rotating body that may not take up an arbitrary orientation concerning some specified axis.
Quantum mechanics was developed as a response to the inability of the classical theories of mechanics and electromagnetism to provide a satisfactory explanation of some of the properties of electromagnetic radiation and of atomic structure. As a result, a theory has emerged whose basic principles can be used to explain not only the structure and properties of atoms, molecules and solids, but also those of nuclei and of “elementary” particles such as the proton and neutron. Although there are still many features of the physics of such systems that are not fully understood, there are presently no indications that the fundamental ideas of quantum mechanics are incorrect. In order to achieve this success, quantum mechanics has been built on a foundation that contains a number of concepts that are fundamentally different from those of classical physics and which have radically altered our view of the way the physical universe operates. This book aims to elucidate and discuss the conceptual basis of the subject as well as explaining its success in describing the behaviour of atomic and subatomic systems.
Chapter 4 deals with Quantum Mechanical (QM) phenomena. The first section describes the first attempts at quantization of energy, momentum, angular momentum, etc. We start with Plank's solution to the black-body problem, introducing the concept of photons: discrete packets of electromagnetic energy. We show how Plank's description of black-body radiation is perfectly fit by the cosmic microwave radiation, providing the strongest evidence for the Big-Bang at the origin of the Universe. We then describe Bohr's solution to the paradoxical stability of atoms, introducing quantization of angular momentum (or action) and correctly predicting the energy levels of the hydrogen atom. We show how Einstein used Plank's idea to explain the photo-electric effect (an effect exploited in CCD cameras and solar cells) and how he used Bohr's understanding of energy levels in atoms to provide an intuitive explanation of Plank's law. Einstein insight opened the way to the field of spectroscopy and the development of the laser. In the second section, following the lead of Dirac, we show that QM is equivalent to Linear Algebra in the space of complex functions (which describe the state of the system). We show that Schrödinger's equation is a natural consequence of that description, once the relation between energy and frequency established by Plank (for photons) and de Broglie (for particles) is taken into account. In the context of the chiral ammonia molecule we discuss the queerness of QM which describes physical systems as superposition of states (such as left and right chirality state of a molecule, dead and alive cat). We show that this queerness is nonetheless supported by experiments and is behind much of our technology: masers and lasers, atomic clocks, semiconductors (diodes, transistors, etc.). In the following section we introduce the representation of operators in QM (the equivalent of matrices in linear algebra): momentum, position, energy, etc. which correspond to measurements on the system. We show that the non-commutability of some operators (common with matrix operations) leads to Heisenberg uncertainty principle: the position and momentum of a particle cannot be determined with absolute precision. We discuss the solution of Schrödinger's equation for free particles demonstrating their equivalence with EM phenomena: particles undergo refraction and diffraction, exemplifying the particle/wave duality inherent to QM systems. We discuss QM tunneling and its use in the Scanning Tunneling Microscope. In the following section we study Dirac's equation: the successful attempt by Dirac to unify QM with Einstein's relativity, with the resulting prediction of anti-particles and spin. We then solve for the energy levels of electrons in the hydrogen atom (from Schrödinger's equation) and show that the existence of spin profoundly affects the behavior of electrons in an atom: they cannot share the same energy level, a rule known as Pauli's exclusion principle. Pauli's principle is central to the understanding of the chemical properties of atoms and fully explains Mendeleev's periodic table of the elements. In the next section we discuss along with two exactly soluble systems (diatomic rotational and vibrational levels), various approximations used to study systems (e.g. molecules) for which there are no analytical solutions. In particular, we introduce Pauling's variational approach, Hückel molecular orbital theory and time independent perturbation theory. We apply those approaches to determine the energy levels in alkenes, in multi-electron atoms and in systems under constant electric or magnetic field (such as in Magnetic Resonance Imaging). In the last section we discuss time dependent perturbation theory. We use it to establish the rules allowing for transitions between energy levels and revisit Einstein description of absorption/emission phenomena.