Scalar Euclidean

Scalar Fields Consider a continuous field theory in Euclidean space with action... 

In Euclidean space we define squared distance from the origin of a point x by means of the scalar product of x with itself ... 

A weighted Euclidean metric is defined by the weighted scalar product ... 

At any given instant the equation S(x, t) = const, defines a surface in Euclidean space. As t varies the surface traces out a volume. At each point of the moving surface the gradient, VS is orthogonal to the surface. In the case of an external scalar potential the particle trajectories associated with S are given by the solutions mx = VS. It follows that the mechanical paths of a moving point are perpendicular to the surface S = c for all x and t. A family of trajectories is therefore obtained by constructing the normals to a set of... 

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. 

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. 

The complex scalar product lets us dehne an analog of Euclidean orthogonal projections. First we need to dehne Hermitian operators. These are analogous to symmetric operators on R". 

In this section we have extended perpendicularity and orthogonal projections to the context of complex scalar product spaces. In the next section we extend another Euclidean idea—distance. 

Euclidean geometry was originally deduced from Euclid s five axioms. However, it is now known that necessary and sufficient criteria for Euclidean spatial structure can be stated succinctly in terms of distances, angles, and triangles, or, alternatively, the scalar product of the space. We can express these criteria by employing Dirac notation for abstract ket vectors R ) of a given space M with scalar product (R R7). 

Alternatively (and equivalently), we can say that if R,), Rj) are any two vectors in the space, with scalar product (R R7), then M is a Euclidean space if, and only if, they satisfy the Schwarz inequality... 

The criteria (9.23), (9.24), and (9.26) are all rather obvious properties of Euclidean geometry. All of these properties can be traced back to mathematical properties of the scalar product (R Ry), the key structure-maker of a metric space. We therefore wish to determine whether a proposed definition of scalar product satisfies these criteria, and thus guarantees that M is a Euclidean space. 

The essential mathematical requirements for a Euclidean scalar product can be stated as follows (for all possible vectors R ), R7), R ) of Ai) ... 

Problem Prove that the mathematical criteria (9.27a-c) for a proposed scalar product (R R7) are sufficient for the general Schwarz inequality (9.24) in the space M, thereby guaranteeing that M is Euclidean. 

According to the positivity property (9.27c) of a Euclidean scalar property, a vector nr ) is considered zero if and only if it is of zero length, namely,... 

As discussed in Section 9.3, a higher level of mathematical structure is achieved by defining an additional multiplication (X-Y) operation, that is, a rule that associates a (real) scalar with every pair of objects X, Y in the manifold. For Euclidean-like spaces, the scalar product has distributive, commutative, and positivity properties given by... 

It is a remarkable fact that properties (13.4a-c) are necessary and sufficient to give Euclidean geometry. In other words, if any rule can be found that associates a number (say, (X Y)) with each pair of abstract objects ( vectors X), Y)) of the manifold in a way that satisfies (13.4a-c), then the manifold is isomorphic to a corresponding Euclidean vector space. We introduced a rather unconventional rule for the scalar products (X Y) [recognizing that (13.4a-c) are guaranteed by the laws of thermodynamics] to construct the abstract Euclidean metric space Ms for an equilibrium state of a system S, characterized by a metric matrix M. This geometry allows the thermodynamic state description to be considerably simplified, as demonstrated in Chapters 9-12. 

We generally denote scalars by lowercase Greek letters (e.g., P), column vectors by boldface lowercase Roman letters (e.g., x), and matrices by capital italic Roman letters (e.g., H). A superscriptT denotes a vector or matrix transpose. Thus xT is a row vector, xTy is an inner product, and AT is the transpose of the matrix A. Unless stated otherwise, all vectors belong to R , the u-dimen-sional vector space. Components of a vector are typically written as italic letters with subscripts (e.g., xux2,.. . , ). The standard basis vectors in R" are the n vectors ei,e2,. . . , e , where e has the entry 1 in the th component and 0 in all others. Often, the associated vector norm is the standard Euclidean norm, j 2, defined as... 

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