Functions product of two

The spacial distribution of electron density in an atom is described by means of atomic orbitals Vr(r, 6, (p) such that for a given orbital xp the function xj/ dv gives the probability of finding the electron in an element of volume dv at a point having the polar coordinates r, 6, 0. Each orbital can be expressed as a product of two functions, i e. 0, [Pg.1285]

To develop those ideas, we seek the solution to equation (16a) as a product of two functions, one of which T = T tj) depends only on t = tj and the other X = X x ) only on x = under the approved decomposition y[x, t) = X x) T t). Substituting this expression into (16a) and taking into account that... 

However, such an equality is impossible, since by changing one of arguments, for example, R, the first term varies while the second one remains the same, and correspondingly the sum of these terms cannot be equal to zero for arbitrary values of R and 0. Therefore, we have to conclude that neither term depends on the coordinates and each is constant. This fact constitutes the key point of the method of separation of variables, allowing us to describe the function C/ as a product of two functions, each of them depending on one coordinate only. For convenience, let us represent this constant in the form +m, where m is called a constant of separation. Thus, instead of Laplace s equation we have two ordinary differential equations of second order ... 

A and B are constants and they are independent of time. Bearing in mind that = X + iy, it is simple to find functions x t) and y t), which describe a motion of the pendulum on the earth s surface. In accordance with Equation (3.88) a solution is represented as a product of two functions. The first one characterizes a swinging of the pendulum with the angular velocity p, which depends only on the gravitational field and the length /, while the second is also a sinusoidal function and its period is defined by the frequency of the earth s rotation and the latitude of the point, (Foucault s pendulum). In order to understand the behavior of the pendulum at the beginning consider the simplest case when a rotation is absent, co — 0. Then, we have... 

Thus, the function (t) is a product of two functions one of them is a decaying exponential, but the other is a sinusoidal function with a frequency p. For instance, if K<free vibrations are close to a function described by a sinusoid slightly decaying with time, and their frequency is approximately coq. 

We see that for this special case the composite wave is the product of two functions one only of the distance x and the other only of the time t. The composite wave (x, t) vanishes whenever cos kx is zero, i.e., when kx = jr/2, 2)71/2, 5tc/2,. .., regardless of the value of t. Therefore, the nodes of P(x, i) are independent of time. However, the amplitude or profile of the composite wave changes with time. The real part of P(x, /) is shown in Figure 1.3. The solid curve represents the wave when cos.cot is a maximum, the dotted curve when coscot is a minimum, and the dashed curve when cos cot has an intermediate value. Thus, the wave does not travel, but pulsates, increasing and decreasing in amplitude with frequency co. The imaginary part of I (x, t) behaves in the same way. A composite wave with this behavior is known as a standing wave. 

The first step in the solution of the partial differential equation (2.6) is to express the wave function (x, /) as the product of two functions... 

This partial differential equation may be readily separated by writing the wave function (R, r) as the product of two functions, one a function only of the center of mass variables X, Y, Z and the other a function only of the relative coordinates x, y, z... 

In other words the differential of a product of two functions is equal to the first function times the differential of the second, plus the second times the differential of the first. Numerous examples of this principle will be encountered in the exercises at the end of this chapter, as well as in following chapters. The other rules presented above can easily be modified accordingly. 

In this the three-dimensional can be transformed to work just with the radial coordinate, r. For this purpose, the wave function is written as a product of two functions. The first one depends only on the radial coordinate and the other on the angular coordinates denoted by R(r) and Y(6,4>), respectively. In the solution of the hydrogen atom, it is found that R(r) depends on the quantum number... 

A useful formula for the polynomial L x) can be obtained by finding a new representation for the confluent hypergcometric function on the right hand side of equation (42.2). By Leibnitz s theorem for the n-th derivative of a product of two functions we have... 

But in fact, Y (or the more general time-dependent wave function Y, which is the product of two functions, one involving the time alone and the other the coordinate alone) was difficult to interpret physically, because the idea of the... 

The overlap integral is the simplest and serves as a basis for the rest of the integrals. The product of two functions is... 

Since the scalar product of two functions in direct space is equal to the scalar product of their Fourier transforms in reciprocal space, we have... 

Table 8.2. Analogies between inner product of two vectors and inner product of two functions.

It is also useful to define the transformed operator L whose operation on a function f is L f = L[Peqf). This operator coincides with the time reversed backward operator, further details on these relationships may be found in Refs. 43,44. L operates in the Hilbert space of phase space functions which have finite second moments with respect to the equilibrium distribution. The scalar product of two functions in this space is defined as (f, g) = (fgi q. It is the phase space integrated product of the two functions, weighted by the equilibrium distribution P The operator L is not Hermitian, its spectrum is in principle complex, contained in the left half of the complex plane. 

Convolution has an interesting property with respect to differentiation. The first derivative of the convolution product of two functions may be given by the convolution of either function with the derivative of the other. Thus, if... 

The convolution operation is a way of describing the product of two overlapping functions, integrated over the whole of their overlap, for a given value of their relative displacement (Bracewell 1978 Hecht 2002). The symbol is often used to denote the operation of convolution. The convolution theorem states that the Fourier transform of the product of two functions is equal to the convolution of their separate Fourier transforms... 

The operators just described will leave the scalar product of two functions of the function space unchanged (O,/, O,/,) = (/,-./ ). Such operators are said to be unitary and they can always be represented by unitary matrices (see 6-4). The proof that the 0M are unitary follows from considering... 

This momentum equation is a linear parabolic partial differential equation (for constant p) that can be solved by the method of separation of variables. In this approach the solution can be found to be a product of two functions as w(t, r) = f t)g r). The solution is represented as an infinite series that can be readily evaluated at any time or value of r. Such a solution is available for a variety of boundary conditions, including time-oscillating rotation rates. At this point, however, we choose to proceed with a numerical solution. 

We shall now explain the importance of direct products in the solution of problems in molecular quantum mechanics. Whenever we have an integral of the product of two functions, for example,... 

From the foregoing discussion of the integrals of products of two functions it is easy to derive some important rules regarding integrands that are products of three, four, or more functions. The case of a triple product is of particular importance. In order for the integral... 

The wave functions for the electron in the hydrogen atom are all products of two functions. First there is the radial function R(n, r), which depends on the principal quantum number n and the coordinate r. Then there is the... 

Suppose we are required to solve a differential equation of the form of equation (7.11), in which dy/dx is equal to the product of two functions, each of which depends only on one of the variables ... 

The concept of limit seems to be essential in the understanding and the present teaching of Calculus. In this article, however, we show how to structure and use differential calculus without introducing this concept. The crucial idea in this development is to use Leibniz rule for the derivative of a product of two functions as one of the postulates, rather than as a derived theorem. Within this approach, the idea of limit could be introduced belatedly and only in order to define concepts such as continuity and differentiability in a more rigorous fashion. 

The rule to calculate the derivative of a product of two functions was first introduced by Leibniz [5, 6], The crux of our presentation is to take the multiplication rule as an initial postulate, rather than as a derived result. Leibniz rule for the derivative of a product of functions is not privy of calculus. It also appears when calculating commutators of matrices or linear operators . ..,BC = B[...,C] + [...,B]C. There is no need to invoke the concept of limit in this case, or when dealing with Lie brackets, or other derivations. The ultimate justification for this choice of initial postulate is given a posteriori in terms of the logarithmic function [7]. 

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