Inadequate descriptions of atoms and molecules by the methods of classical physics led researchers to propose new ways to describe physical reality, giving birth to a totally new science, quantum mechanics. The methods of quantum mechanics are based on the introduction of a wave function, whose physical meaning is related to the probability of finding a certain particle, at a certain time in a volume element, positioned between x and x + dx in the x = direction, between y and y + dy in the y = direction, and between z and z + dz in the z = direction at certain time t. This wave function Ψ satisfies the Schrödinger equation,
(−2mℏ2∇2+v)Ψ=EΨ,ℏ=2πh, or for short, HΨ = EΨ, where H, the Hamiltonian operator, is defined by the expression
H=−2mℏ2∇2+V; h is Planck’s constant; ∇2 is the sum of the partial second derivatives with respect to x, y, and z; m is the mass of the particle; and V is the potential energy of the system. The Hamiltonian H represents the quantum equivalent of the sum of the kinetic energy and potential energy, with V being the potential energy operator and
2m−ℏ2∇2 the kinetic energy operator. Finally, E is the total energy of the system and is a number, not an operator.
This chapter describes a particle confined to a finite region of space that can possess only certain discrete energies. It also looks at the corresponding wavefunctions. It analyzes the quantization of energy that emerges as a natural consequence of solving the Schrödinger equation and the conditions imposed on it. It also introduces a striking non-classical feature of small particles, which describes their ability to penetrate into and through regions where classical physics would forbid them to be found. The chapter talks about the translational energy levels of a particle which are confined to a finite region of space and are quantized. It explores how quantum mechanics reveal the ability of a particle to penetrate into and through regions where classical physics would forbid it to be found.
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 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.