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chapter2_v1

# chapter2_v1 - 2 M ATHEM ATICAL FOUND ATIONS 2.1...

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2 MATHEMATICAL FOUNDATIONS 2.1 Introduction: The purpose of this chapter is to give you the mathematical language in which quantum mechanics can be concisely stated. Just like it is difficult to understand the locals in a foreign country if you can’t speak their language, this is an essential piece of background if you want to understand quantum mechanics at more than a superficial level. The best way to absorb this material is to practice using it by working the problems and discussing it with your friends (thought not necessarily at dinner parties…until you’ve got through a few more chapters). Seriously, you should find that while the central topics of this chapter will at first glance seem very unfamiliar (both in names and notation), and therefore possibly very difficult, things will look much better at second (or third) glance. In particular, try looking for two things to help in navigating through the material. First, watch for analogies with mathematics that are already familiar to you, such as vectors, and linear algebra, and matrices and matrix algebra. Second, pay attention to the links between the qualitative concepts of quantum mechanics introduced in the previous chapter and the mathematics introduced here: after all this is why it is introduced! We start in Section 2.2 with a discussion of the fact that valid wave functions belong to a form of vector space, called state space (since the wave function describes the state of a quantum mechanical system). This is essentially no different from the fact that valid vectors ( x,y,z ) belong to Cartesian space. The basic properties of state vectors are summarized and Dirac’s powerful “bra-ket” notation is introduced: it is essentially a generalization of the notion for vectors. The next section (2.3) deals with the very important idea of a basis expansion. A basis expansion is to state space as the x,y,z axes are to real space. We discuss operators in Section 2.4, followed by a discussion of Hermitian conjugation, which is a greatly generalized form of complex conjugation. We then introduce the eigenvalues and eigenvectors of operators in Section 2.5, with particular emphasis on Hermitian operators. They have real eigenvalues and represent observables in quantum mechanics. The eigenfunctions of Hermitian operators form a basis for state space.. 2.2 Abstract vector spaces of bras and kets There are 3 properties that together define a vector space: (1) closure is satisfied, (2) a scalar product exists, and, (3) vectors can be represented in terms of a basis. As the wavefunctions of quantum mechanics belong to a vector space (sometimes called Hilbert space), there will be connections between the basic properties of a vector space, and some of the important characteristics of wavefunctions that have physical implications.

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