Complex numbers waves

A term that is nearly synonymous with complex numbers or functions is their phase. The rising preoccupation with the wave function phase in the last few decades is beyond doubt, to the extent that the importance of phases has of late become comparable to that of the moduli. (We use Dirac s terminology [7], which writes a wave function by a set of coefficients, the amplitudes, each expressible in terms of its absolute value, its modulus, and its phase. ) There is a related growth of literatm e on interference effects, associated with Aharonov-Bohm and Berry phases [8-14], In parallel, one has witnessed in recent years a trend to construct selectively and to manipulate wave functions. The necessary techifiques to achieve these are also anchored in the phases of the wave function components. This bend is manifest in such diverse areas as coherent or squeezed states [15,16], elecbon bansport in mesoscopic systems [17], sculpting of Rydberg-atom wavepackets [18,19], repeated and nondemolition quantum measurements [20], wavepacket collapse [21], and quantum computations [22,23], Experimentally, the determination of phases frequently utilizes measurement of Ramsey fringes [24] or similar" methods [25]. 

Making use of the polar representation of a complex number, the nuclear wave function can be written as a product of a real amplitude, A, and a real phase, S,... 

The bra n denotes a complex conjugate wave function with quantum number n standing to the of the operator, while the ket m), denotes a wave function with quantum number m standing to the right of the operator, and the combined bracket denotes that the whole expression should be integrated over all coordinates. Such a bracket is often referred to as a matrix element. The orthonormality condition eq. (3.5) can then be written as. 

System Complex ions Wave numbers, cm 1 Spectrum Ref. 

A final example is the concept of QM state. It is often stated that the wave function must be square integrable because the modulus square of the wave function is a probability distribution. States in QM are rays in Hilbert space, which are equivalence classes of wave functions. The equivalence relation between two wave functions is that one wave function is equal to the other multiplied by a complex number. The space of QM states is then a projective space, which by an infinite stereographic projection is isomorphic to a sphere in Hilbert space with any radius, conventionally chosen as one. Hence states can be identified with normalized wave functions as representatives from each equivalence class. This fact is important for the probability interpretation, but it is not a consequence of the probability interpretation. 

Thus, the two wave functions can at most differ by a unimodular complex number e1. It can be shown that the only possibilities occurring in nature are that either the two functions are identical (symmetric wave function, applies to particles called bosons which have inte-... 

It should be recognized that the discrete Fourier coefficients G(x, y, co) are represented by complex numbers. The real part Re(G(x, y, to)) of the complex number represents the amplitude of the cosine part of the sinusoidal function and the imaginary part Im(G(x, y, co)) represents the amplitude of the sine wave. 

Most of the graphs we deal with are trees- that is, they do not have any self tracing loops. Choose a vertex which will be the root . In the example we consider here we shall take vertex 5 as the root (see figure l.(c)). A wave which propagates from the root along a certain branch is reflected, and this reflection can be expressed by a reflection from the vertex which is next to the root. Once we know the reflection coefficient, which because of unitarity is a complex number with unit modulus, we can construct the SB(k) matrix. In the present cases, when the valency of the root is 3, we get... 

In order to describe the material properties as a function of frequency for a body that behaves as a Maxwell model we need to use the constitutive equation. This is given in Equation (4.8), which describes the relationship between the stress and the strain. It is most convenient to express the applied sinusoidal wave in the exponential form of complex number notation ... 

So far we have seen that if we begin with the Boltzmann superposition integral and include in that expression a mathematical representation for the stress or strain we apply, it is possible to derive a relationship between the instrumental response and the properties of the material. For an oscillating strain the problem can be solved either using complex number theory or simple trigonometric functions for the deformation applied. Suppose we apply a strain described by a sine wave ... 

Recognizing that many chemistry students do not have a strong background in physics, I have introduced most of the chapters with some essential physics, concerning waves, mechanics, and electrostatics. I have also tried to keep the mathematical level at a minimum, consistent with a proper understanding of what is necessary. Basic calculus and an understanding of the properties of elementary trigonometical and exponential functions are assumed but I have not used complex numbers. Each chapter ends with some simple problems. 

Because n > n2, for values of i greater than c, the combination of (9.11) and (9.12) produces the seemingly paradoxical result sin 2 > 1 . This result can be accepted because the electric vector is a complex number and because an electromagnetic wave does not propagate in medium ri but only penetrates into it. The standing wave at the interface then forms an evanescent field at the n2 side of the interface. 

Before we go into the mathematical framework behind wave mechanics, we will review one more mathematical concept normally seen in high school imaginary and complex numbers. As discussed in Section 1.2, for a general quadratic equation ax2 + bx+c =... 

The standard Schrodinger equation for an electron is solved by complex functions which cannot account for the experimentally observed phenomenon of electron spin. Part of the problem is that the wave equation 8.4 mixes a linear time parameter with a squared space parameter, whereas relativity theory demands that these parameters be of the same degree. In order to linearize both space and time parameters it is necessary to replace their complex coefficients by square matrices. The effect is that the eigenfunction solutions of the wave equation, modified in this way, are no longer complex numbers, but two-dimensinal vectors, known as spinors. This formulation implies that an electron carries intrinsic angular momentum, or spin, of h/2, in line with spectroscopic observation. 

The spin state (or wave function ) of this pair is a linear combination of the states aa and PP, with coefficients ci and c2. These coefficients are actually complex numbers, with real parts (a) and imaginary parts (b) ... 

Atomic orbitals can combine and overlap to give more complex standing waves. We can add and subtract their wave functions to give the wave functions of new orbitals. This process is called the linear combination of atomic orbitals (LCAO). The number of new orbitals generated always equals the number of starting orbitals. 

It is often convenient to use a complex number representation for sound waves [2]. The harmonic wave is represented by the complex exponential. 

In the quasi-static case, effective frequency dependent moduli and loss factors may be calculated from Equation 8. With respect to Equation 29, a lossy matrix material implies that k is now a complex number. The new expressions for c and a differ from Equations 31 and 32, but follow straightforwardly. Equation 30 is usually cited only for elastic matrix materials, but, of course, it need not be used to interpret a. The potential problem (also with viscoelastic inclusions) is that the derivation of Equation 30 is based on homogeneous stress waves, whereas in viscoelastic materials one should, strictly speaking, consider inhomogeneous waves. The results obtained from Equation 29 are reasonable in the sense of yielding the expected superposition of scattering and dissipation effects. 

These are acceptable for many purposes, but the complex number fi (= -1) makes them difficult to visualize. Given a set of solutions to the wave equation, which we find inconvenient, it is always permissible to transform these into an equal number of new functions obtained by taking linear combinations of the original ones, provided that orthogonality is upheld, that is, the overlap integral / (i j)... 

When we have a large number of individual waves, like those produced by the scattering of X-rays from families of planes, or from all of the unit cells in a crystal, or from all of the atoms within a unit cell, we are ultimately interested in knowing how all of the waves add together to yield a resultant wave that we can observe, characterize, and use. Waves are more complicated to sum than simple quantities like mass or temperature because they have not only an amplitude, a scaler, but also a phase angle 0 with respect to one another. This must be taken into account when waves are combined. As will be seen below, waves share identical mathematical properties with vectors (and with complex numbers, which are really nothing but vectors in two dimensions). 

As indicated in Figure 4.6, a vector K in the complex plane will have a real component K cos 0, and an imaginary component iK sin 0. The end of the vector K defines a point in the plane, and the coordinates of that point are K cos 0, i K sin 0. Thus we see that a wave in space can be defined not only as a vector in the complex plane, but as a complex number... 

The sum of a large number of waves having the same frequency, then, may be obtained by adding their corresponding vector representatives in the complex plane, which is still awkward, or by simply adding up a list of complex numbers of the form a + ib. We can always recover the resultant amplitude K and phase 0 of the wave from the corresponding... 

At this point the most problematic feature of the process emerges. Inspection of the electron density equation as it was initially stated shows that the coefficient of each term in the summation for p(x, y, z) at any value of x, y, z is T ki The structure factor Ff,u is, as we have seen, a wave. It is a complex number it has an amplitude and a phase. In the final form of the equation we see that this feature persists in the form of the phase angle for each structure factor that must be included in the kernal. To calculate p(x, y, z), then,... 

---