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chapter1_v0 - 1 CENTRAL QUANTUM CONCEPTS 1.1 Introduction...

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1 CENTRAL QUANTUM CONCEPTS 1.1 Introduction: The first step in learning quantum mechanics is understanding why simpler classical theories are inadequate for describing matter at the level of electrons, atoms, and molecules. The classical theory used is Newtonian mechanics (F=ma) which describes motion of particles in terms of trajectories. The ability to launch satellites into orbits around the earth is a great example of the predictive power of Newtonian mechanics at the macroscopic level. As we shall see in Sec 1.2 below, this predictive power evaporates at the level of electrons, where they behave in ways that can seem wave-like rather than like classical particles. Classically, electromagnetic waves are described by Maxwell’s equations , which account for many of the properties of electric and magnetic fields due to charged particles. From the design of aerials for effective radio transmission to the new field of photonics, Maxwell’s equations offered new predictive power as a result of unifying electrical and magnetic phenomena. At the level of light interacting with atoms and molecules, Maxwell’s equations prove grossly inadequate. We shall see, for instance, in Sec 1.2.1, that in this regime, light actually can behave as if it were a stream of particles! In fact, the experiments which show that particles at the atomic level have wave- like character and that electromagnetic waves have particle-like character hint at something larger. They suggest that a proper description of both light and matter at the atomic scale should be unified such that both particle and wave properties can be manifested. Quantum mechanics is the proper description for most purposes in chemistry and related fields such as biology, condensed matter and molecular physics, materials science etc. The remainder of this introductory chapter aims to give you some feel for what the contents and rules of quantum mechanics should be. With this in the back of your mind, we will then set up the structure and postulates (rules) precisely in Chapters 2 and 3. The starting point is to carefully review the properties of traveling waves (Sec. 1.3) and what happens when they are added together (Sec. 1.4). A single traveling wave naturally represents a free particle with definite momentum and indefinite position. On the other hand we’ll see that adding together traveling waves of slightly different wavelength will make the momentum less definite but can yield a sum (or “wavepacket”) that has localized position. Later on we’ll see that this inability to have precise knowledge of position and momentum together is a manifestation of Heisenberg’s famous uncertainty principle. The behavior of these traveling waves (sometimes called matter waves) is then used to sketch the postulates (or rules) of quantum mechanics in Sec 1.5. It is important to understand that while the postulates can be motivated based on physical arguments (as in this chapter) or given a precise mathematical statement (as in Chapter 2) they cannot be derived. Quantum mechanics, as defined by its postulates, is only useful insofar as it
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This note was uploaded on 02/22/2010 for the course CHEM N/A taught by Professor Head-gordon during the Spring '09 term at Berkeley.

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chapter1_v0 - 1 CENTRAL QUANTUM CONCEPTS 1.1 Introduction...

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