Mathematical Introduction
The aim of this book is to provide you with an introduction to quantum mechanics, starting from its axioms. It is the aim of this chapter to equip you with the necessary mathematical machinery.
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1.1. Linear Vector Spaces: Basics In this section you will be introduced to linear vector spaces. You are surely familiar with the arrows from elementary physics encoding the magnitude and direction of velocity, force, displacement, torque, etc. You know how to add them and multiply them by scalars and the rules obeyed by these operations. For example, you know that scalar multiplication is distributive: the multiple of a sum of two vectors is the sum of the multiples. What we want to do is abstract from this simple case a set of basic features or axioms, and say that any set of objects obeying the same forms a linear vector space. The cleverness lies in deciding which of the properties to keep in the generalization. If you keep too many, there will be no other examples; if you keep too few, there will be no interesting results to develop from the axioms. The following is the list of properties the mathematicians have wisely chosen as requisite for a vector space. As you read them, please compare them to the world of arrows and make sure that these are indeed properties possessed by these familiar vectors. But note also that conspicuously missing are the requirements that every vector have a magnitude and direction, which was the first and most salient feature drilled into our heads when we first heard about them. So you might think that in dropping this requirement, the baby has been thrown out with the bath water. However, you will have ample time to appreciate the wisdom behind this choice as
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you go along and see a great unification and synthesis of diverse ideas under the heading of vector spaces. You will see examples of vector spaces that involve entities that you cannot intuitively perceive as having either a magnitude or a direction. While you should be duly impressed with all this, remember that it does not hurt at all to think of these generalizations in terms of arrows and to use the intuition to prove theorems or at the very least anticipate them. Definition 1. A linear vector space W is a collection of objects 12), ... , IV), ... , I W), ... , called vectors, for which there exists
11 ),
1. A definite rule for forming the vector sum, denoted I V) + IW) 2. A definite rule for multiplication by scalars a, b, ... , denoted al V) with the following features:
• The result of these operations is another element of the space, a feature called closure: IV)+ I W)e'V. • Scalar multiplication is distributive in the vectors: a( IV)+ I W)) = al V)+al W). • Scalar multiplication is distributive in the scalars: (a+b)l V)=al V)+bl V). • Scalar multiplication is associative: a(bl V)) = abl V). • Addition is commutative: I V) + I W) = I W) + I V). • Addition is associative: IV)+ (I W) + IZ)) =(IV)+ I W)) + IZ). • There exists a null vector 10) obeying IV)+ 10) =IV). • For every vector IV) there exists an inverse under addition, 1- V), such that
IV>+ I-V)= IO).
There is a good way to remember all of these; do what comes naturally. Definition 2. The numbers a, b, ... are called the field over which
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