Hilbert Space

Hilbert Space - Physics 221A Fall 2010 Notes 1 The...

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Unformatted text preview: Physics 221A Fall 2010 Notes 1 The Mathematical Formalism of Quantum Mechanics 1. Introduction The prerequisites for Physics 221A include a full year of undergraduate quantum mechanics. In this semester we will survey that material, organize it in a more logical and coherent way than the first time you saw it, and pay special attention to fundamental principles. We will also present some new material. Physics 221B will largely consist of new material. We begin by presenting some of the mathematical formalism of quantum mechanics. We will introduce more mathematics later in the course as we need it, but for now we will concentrate on the linear algebra of spaces of wave functions, which are called Hilbert spaces . These notes gather together and summarize most of what you will need to know of this subject, so we can proceed with other matters. In the next set of notes we will turn to the physical postulates of quantum mechanics, which allow us to connect experimental results with the mathematics presented here. Introductory courses on linear algebra are usually limited to finite-dimensional, real vector spaces. Making the vector spaces complex is a small change, but making them infinite-dimensional is a big step if one wishes to be rigorous. We will make no attempt to be rigorous in the following—to do so would require more than one course in mathematics and leave no time for the physics. Instead, we will follow the usual procedure in physics courses when encountering new mathematics, which is to proceed by example and analogy, attempting to gain an intuitive understanding for some of the main ideas without going into technical proofs. Specifically, in dealing with Hilbert spaces we will try to apply what we know about finite-dimensional vector spaces to the infinite-dimensional case, often using finite-dimensional intuition in infinite dimensions, and we will try to learn where things are different in infinite dimensions and where one must be careful. Fortunately, it is one of the consequences of the mathematical definition of Hilbert space that many of their properties are the obvious generalizations of those that hold in finite dimensions, so much of finite-dimensional intuition does carry over. We will use what we know about spaces of wave functions (an example of a Hilbert space) to gain some intuition about where this is not so. 2. Hilbert Spaces To orient ourselves, let us consider a wave function ψ ( x ) for a one-dimensional quantum me- chanical problem. In practice it is common to require ψ ( x ) to be normalized, but for the purpose of 2 Notes 1: Mathematical Formalism the following discussion we will require only that it be normalizable, that is, that integraldisplay | ψ ( x ) | 2 dx < ∞ (1) (which means that the integral is finite). Wave functions that are not normalizable cannot represent physically realizable states, because the probability of finding a real particle somewhere in space must be unity. Nevertheless, wave functions that are not normalizable, such as plane waves, aremust be unity....
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Hilbert Space - Physics 221A Fall 2010 Notes 1 The...

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