One of the classic first problems in quantum theory textbooks is that of the particle trapped in a box. Despite the apparent artificiality of the situation, it has quite a few interesting features that we can quickly apply to some more physically plausible situations. The particle in a box is literally an object, such as an electron, trapped inside a ‘box.’ In a realistic physical example, this ‘box’ would be a volume of space bound by some constraining force acting on the particle. For example, an atom is a ‘box’ as far as the orbiting electrons are concerned, with the electrostatic force between the positive protons in the nucleus and the negative electrons providing the constraint. The first move is to assume that the wave function for any particle in the box will have separate spatial and temporal dependencies. One of the key steps in obtaining our wave function was the quantization of the momentum.
One of the classic first problems in quantum theory textbooks is that of the particle trapped in a box. Despite the apparent artificiality of the situation, it has quite a few interesting features that we can quickly apply to some more physically plausible situations. The particle in a box is literally an object, such as an electron, trapped inside a ‘box.’ In a realistic physical example, this ‘box’ would be a volume of space bound by some constraining force acting on the particle. For example, an atom is a ‘box’ as far as the orbiting electrons are concerned, with the electrostatic force between the positive protons in the nucleus and the negative electrons providing the constraint. The first move is to assume that the wave function for any particle in the box will have separate spatial and temporal dependencies. One of the key steps in obtaining our wave function was the quantization of the momentum.
An atom consists of a nucleus, containing protons and neutrons, with various electrons swarming around the outskirts. The electrons are held in the vicinity of the nucleus as they are negatively charged and hence attracted by the positively charged protons. Although atoms are complicated objects, there are certain situations where they can be treated as simple lumps of matter. Their internal structure has no relevance to the details of what is happening. This sort of situation is exemplified by experiments in which atoms are deliberately smashed into one another at comparatively low energy and as such provide an ideal start in a study of the quantum physics of identical particles. When we are dealing with atoms, there is something more to being ‘identical’ than we can appreciate with larger-scale objects. Atoms of the same type are really identical, measurably identical if you like, which can make a considerable difference in some physically important situations.
Abstract The simple ‘particle in a box’ (Piab) is introduced in this chapter so that the reader can get familiar with applying the quantum recipe and atomic units. The PiaB is introduced in its one, two, and three dimensional variants, which demonstrates the use of the separation of variables technique as a strategy to solve the Schrodinger equation for a particle with two or three degrees of freedom. It is shown that the confinement of the particle causes the energy to be quantized. The one-dimensional PiaB is then applied to treat the electronic spectra of cyanine dyes and their absorption colors. The chapter then introduces more general setups with finite potential wells, in order to introduce the phenomenon of quantum tunnelling and to discuss more generally with the unintuitive ‘quantum behavior’ of particles such as electrons. Scanning tunnelling and atomic force microscopes are also discussed briefly.
The behavior of classical macroscopic particles – from baseballs to planets – is governed by Newtonian mechanics as we have described in the first chapters of this book. The behavior of very small particles, on the other hand, is governed by an entirely different collection of equations and rules that go under the name of “quantum mechanics” and these particles are referred to as quantum particles. These rules have analogies to classical mechanics, but are fundamentally different and somewhat mysterious as we discussed in Chapter 8. The fundamental principles of quantum mechanics cannot be proven to be correct. They were developed based on the results of experimental observation, by analogy with classical mechanics, and through the consideration of logical implications of the formalisms used. There is no way to derive the fundamental elements of quantum mechanics, any more than one could predict from first principles that Newtonian forces should add as vectors. On the other hand, quantum mechanics is among the most rigorously tested theories in science, and the predictions of quantum mechanics have, so far, always been in agreement with experiment. In this chapter, we will present quantum mechanics as a series of postulates and examine their implications in application to simple one-dimensional systems. In the next chapter we will examine quantum effects in real systems including atoms, molecules, crystals, and artificially engineered nano-structures. Postulates of quantum mechanics QM Postulate 1: A quantum particle is described by a wave function Ψ( x , y , z , t ) which is a function of spatial coordinates and time. All physically determinable quantities can be derived from the wave function Ψ( x , y , z , t ) ; quantities which cannot be found from the wave function are not physically meaningful. Note that some things that one could ask about a quantum particle, for example the exact position and simultaneous exact momentum of a particle, are not derivable from the wave function and are therefore considered indeterminate (or uncertain ). These are things which cannot ever be known about a particle, and, as such, they are meaningless to contemplate.
The wave function Before beginning to describe the quantum theory of measurement, we shall discuss briefly the main concepts of quantum mechanics. Readers familiar with quantum mechanics may skip this section and go on to section 2.2. Consider, as a pedagogical aid, a classical particle that can move only along the x direction. For example, it might be a bead on an infinite needle. If we know at any moment of time the values of two numbers, the particle's position and momentum, then we can compute from them all other features of the particle's motion: the forces acting on it, its resulting acceleration, etc. In other words, this pair of numbers, the position and momentum, completely define the state of the classical particle at the chosen moment of time. Similarly, for three-dimensional motion the state is defined by six numbers: the three spatial coordinates x, y, z and the three corresponding components of the momentum p x , p y , p z . A quantum object, with its dual particle-like and wave-like properties, is much more complex than a classical particle. Interferometric experiments (chapter I) show that the object can be spread out over some region of space, or for the bead on a needle, over an interval of x .
Abstract This chapter covers the rudiments of quantum theory, including the dual nature of light and matter, the Uncertainty Principle and probability concept, the electronic wavefunction, and the probability density function. Numerical examples are given to show that given the electronic wavefunction of a system, the probability of finding an electron in a volume element around a certain point in space can be readily calculated. Finally, the electronic wave equation, the Schrödinger equation, is introduced. This discussion is followed by the solution of a few particle-in-a-box problems, with the ‘box’ having the shape of a wire (one-dimensional), a cube, a ring, or a triangle. Where possible, the solutions of these problems are then applied to a chemical system.
In the early 1900s more and more experimental evidence was accumulated indicating that microscopic particles show wave-like properties in certain situations. These particle-wave features are very evident, e.g., in measurements where electrons are diffracted from a double slit to propagate toward a screen where they are detected. Based on the classical averaging of particles discussed in Section 1.2.2, one expects that the double slit only modulates the overall intensity, not the spatial distribution. Experimentally, however, one observes a nearly perfect interference pattern at the screen implying that the electrons exhibit wave averaging features such as discussed in Section 2.1.2. This behavior, originally unexpected for particle beams, persists even if the experiment is repeated such that only one electron at a time passes the double slit before it propagates to the detection screen. Thus, the wave aspect must be an inherent property of individual electrons and not an ensemble effect. Another, independent argument for the failure of classical physics is that the electromagnetic analysis of atoms leads to the conclusion that the negatively charged electron(s) should collapse into the positively charged ion because the electron–ion system loses its energy due to the emission of radiation. As we will see, this problem can be solved by including the wave aspects of particles into the analysis. In particular, as discussed in Section 2.3.3, waves can never be localized to a point without increasing their momentum and energy beyond bounds.