In the particle in the box problem, the particle is not in both boxes at the same time as some would have you believe. It is a set definition situation with the two boxes being part of a set that also contains a particle. Set and subset differences are explored. Atomic electron orbitals can be mimicked by roulette wheel probability; thus ELECTRONIC ROULETTE. 0 and 00 serve as boundary limits and are on opposite sides of the central core - a point that quantum physics ignores. Considering a stray marble on the floor as part of the roulette wheel menage is taking assumptions a bit too far. Likewise, the attraction between a positive and negative charge at distance does not make the negative charge part of the positive charge's orbital system. This, of course, is contrary to the stance of current quantum physics methodology that carries this orbital association a bit too far.
In the particle in the box problem, the particle is not in both boxes at the same time as some would have you believe. It is a set definition situation with the two boxes being part of a set that also contains a particle. Set and subset differences are explored. Atomic electron orbitals can be mimicked by roulette wheel probability; thus ELECTRONIC ROULETTE. 0 and 00 serve as boundary limits and are on opposite sides of the central core - a point that quantum physics ignores. Considering a stray marble on the floor as part of the roulette wheel menage is taking assumptions a bit too far. Likewise, the attraction between a positive and negative charge at distance does not make the negative charge part of the positive charge's orbital system. This, of course, is contrary to the stance of current quantum physics methodology that carries this orbital association a bit too far.
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.
With the exception of the latter half of chapter 6 we have, up to this point, been discussing single-particle quantum mechanics. This is easy (although you may not think so). One can certainly gain a lot of insight and understanding from a single-particle theory. However, in real life there are very few (no) situations in physics in which a single-particle theory is able to paint the whole picture. Any real physical process involves the interaction of many particles. In fact even that is a vast simplification. Really any physical process involves the interaction of all particles. Even the gravitational attraction due to an electron at the other end of the universe is felt by an electron on earth. To describe all particles in a calculation is, of course, absurd. You would have to include the particles of the paper you are writing the calculation down on and the particles in your brain thinking about the many-body problem. However, many-body theory on a more limited scale is feasible. In this chapter we discuss two ways of going beyond the one-electron approximation. Actually, we are not going very far beyond the one-electron approximation and you will see what is meant by that soon. This chapter is essentially divided into two halves.
Introduction I give in this chapter the quantum-mechanical description of an atom. I look in some detail at the hydrogen atom, which is the simplest atom we have, with just one proton and one electron. From the picture that emerges, we shall be able to see how to describe all other atoms. Before continuing, I need to explain what I mean when I say that the proton and electron have equal and opposite electric charges. It is because of these charges that the atom is stable, and does not spontaneously break up into the two separate particles. Electric charge and potential energy If I take a ball in my hand and then let go, it drops to the floor. This is because of the gravitational interaction between the ball and the earth. The ball had gravitational potential energy when I held it, and when I release it, this energy is progressively converted to kinetic energy as it falls. Gravitation makes two bodies move towards each other, and not away from each other: the gravitational force is always attractive. As a result of many experiments done over the years, we find that there is another kind of force in nature, somewhat analogous to, but in one aspect different from, gravitation. I take two idealized examples, and use them to show the features of this force. First I consider two electrons that I imagine to be pinned down a distance r apart.
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.
This chapter talks about the structure of the atom. The positive and negative charges in atoms are separated from one another. Anyone who wishes to hold the opposite point of view would maintain that the electrons spread themselves out or at least possess continuity and can mix with the positive charges. Such an assumption would be quite arbitrary, and there is no single known fact which supports it. However, there are numerous phenomena which quite definitely contradict it. On many occasions the authors have succeeded in separating the two types of electricity from one another—in fact they might say that all phenomena which have led to the discovery of the electron, and which have just been mentioned, support this view. In Bohr's theory of the atom, where the charges, radii of the orbits, and the velocities of the electrons are all fixed, it is possible to calculate as a matter of course the magnetic moments due to the individual electrons.