Complex Scalar Products

Complex Scalar Product Spaces (a.k.a. Hilbert Spaces)... 

The natural mathematical setting for any quantum mechanical problem is a complex scalar product space, dehned in Dehnition 3.2. The primary complex scalar product space used in the study of the motion of a particle in three-space is called (R ), pronounced ell-two-of-are-three. Our analysis of the hydrogen atom (and hence the periodic table) will require a few other complex scalar product spaces as well. Also, the representation theory we will introduce and use depends on the abstract nohon of a complex scalar product space. In this chapter we introduce the complex vector space dehne complex scalar products, discuss and exploit analogies between complex scalar products and the familiar Euclidean dot product and do some of the analysis necessary to apply these analogies to inhnite-dimensional complex scalar product spaces. 

Physicists often refer to complex scalar product spaces as Hilbert spaces. The formal mathematical definition of a Hilbert space requires more than just the existence of a complex scalar product the space must be closed a.k.a. complete in a certain technical sense. Because every scalar product space is a subset of some Hilbert space, the discrepancy in terminology between mathematicians and physicists does not have dire consequences. However, in this text, to avoid discrepancies with other mathematics textbooks, we will use complex scalar product. ... 

In Section 3.1 we introduce Lebesgue equivalence and define the complex vector space in Section 3.2 we define complex scalar products in... 

A second bit of caginess in the introduction is our statement that (R ) is the primary complex scalar product space used in the study of a particle in three-space. Beware the passive voice We used it here to gloss over the fact that 7,2 (R3) is not the set of all states of the particle. The fact that we want /g3 2 = 1 is only part of the story. Because the only numbers we can measure physically are of the form, we cannot distinguish between... 

We start with the definition of a complex scalar product (also known as a Hermitian inner product, a complex inner product or a unitary structure on a complex vector space. Then we present several examples of complex scalar product spaces. 

Mathematicians should note that we have taken the physicists convention in criterion 1 in many mathematics texts, the dehnition requires Unearity in the first argument. See Exercise 3.4. A complex vector space with a complex scalar product defined on it is known as a complex scalar product space. The complex scalar product is sometimes called a unitary structure on the space. 

For example, for any natural number n there is a natural complex scalar product on the n-dimensional complex vector space C" defined by... 

There are other complex scalar products on C as well. In fact, for any set... 

Recall the vector space P" of homogeneous polynomials of degree n in two variables defined in Section 2.2. We will find it useful (see Proposition 4.7) to define the following complex scalar product on P ... 

One complex scalar product on C[ -1, 1], the complex vector space of continuous functions on [—1, 1], is... 

Finally, we introduce another complex scalar product space necessary to our analysis. 

Euclidean-style Geometry in Complex Scalar Product Spaces 85... 

Complex scalar products arise naturally in quantum mechanics because there is an experimental interpretation for the complex scalar product of two wave functions (as we saw in Section 1.2). Students of physics should note that the traditional brac-ket notation is consistent with our complex scalar product notation—just put a bar in place of the comma. The physical importance of the bracket will allow us to apply our intuition about Euclidean geometry (such as orthogonality) to states of quantum systems. 

Since a complex scalar product resembles the EucUdean dot product in its form and definition, we can use our intuition about perpendicularity in the Euclidean three-space we inhabit to study complex scalar product spaces. However, we must be aware of two important differences. Eirst, we are dealing with complex scalars rather than real scalars. Second, we are often dealing with infinite-dimensional spaces. It is easy to underestimate the trouble that infinite dimensions can cause. If this section seems unduly technical (especially the introduction to orthogonal projections), it is because we are careful to avoid the infinite-dimensional traps. 

For example, the constant function 3 and the function cos rtx are perpendicular in the complex scalar product space C[—1, 1] since... 

Unitary operators are also known as complex orthogonal operators. If we use the standard basis and the standard complex scalar product on C , then the columns of the matrix of a unitary operator are all mutually perpendicular and have length one. In other words, a transformation T C" C" is unitary if and only if T T = I. 

We define complementary subspaces of complex scalar product spaces. 

Definition 3.6 Suppose B is an arbitrary subset of a complex scalar product space V. Then the perpendicular space to B in V is... 

In EucUdean space, orthonormal bases help both to simplify calculations and to prove theorems. Unitary bases, also called complex orthonormal bases, play the same role in complex scalar product spaces. To define a unitary basis for arbitrary (including infinite-dimensional) complex scalar product spaces, we first define spanning. 

If V is finite dimensional, then Definition 3.7 is consistent with Definition 2.2 (Exercise 3.13). In infinite-dimensional complex scalar product spaces. Definition 3.7 is usually simpler than an infinite-dimensional version of Definition 2.2. To make sense of an infinite linear combination of functions, one must address issues of convergence however, arguments involving perpendicular subspaces are often relatively simple. We can now define unitary bases. 

Definition 3.8 Suppose V is a complex scalar product space and B is a subset ofV. Suppose that B satisfies the following ... 

Eor example, if we consider C" with the standard complex scalar product, then the set ck k = 1,..., , where Ck denotes the vector whose kth entry is 1 and all of whose other entries are 0, is a unitary basis of V. A more sophisticated example (left to readers in Exercise 3.14) is that the set of functions... 

Proposition 3.2 Suppose V is a finite-dimensional complex scalar product space. Suppose T V —> V is a linear operator. Then T is unitary if and only if the columns of its matrix in any unitary basis form a unitary basis. 

Definition 3.9 Suppose V and W are finite-dimensional complex scalar product spaces, and let ( , > v and , denote their complex scalar products. Suppose T.V W is a linear transformation, that is, suppose T Hom (V, VP). Then the adjoint of T is the unique linear transformation T W V such that for all v e V and all w e W we have... 

Does this T have the desired property By the bilinearity of the complex scalar products (condition 1 of Definition 3.2), it suffices to check the condition on basis elements. For any j and any k we have... 

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