Complex numbers modulus

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]. 

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. 

When expressed in polar coordinates, the quantity r is the magnitude or absolute value or modulus and (f> is the argument or phase of the complex number. It follows immediately that... 

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... 

These two experiments are fundamentally different in the nature of the applied deformation. In the case of the relaxation experiment a step strain is applied whereas the modulus is measured by an applied oscillating strain. Thus we are transforming between the time and frequency domains. In fact during the derivation of the storage and loss moduli these transforms have already been defined by Equation (4.53). In complex number form this becomes... 

We denote the set of complex numbers by C. Readers should be familiar with complex numbers and how to add and multiply them, as described in many standard calculus texts. We use i to denote the square root of —1 and an asterisk to denote complex conjugation if x and y are real numbers, then (x + iy) = X — iy. Later in the text, we will use the asterisk to denote the conjugate transpose of a matrix with complex entries. This is perfectly consistent if one thinks of a complex number x + iy as a one-by-one complex matrix ( x + iy ). See also Exercise 1.6. The absolute value of a complex number, also known as the modulus, is denoted... 

The distance from the origin 0 to the point Pisr (a2 + A1)A, which is called the norm or modulus of the complex number [a, A], The product of two complex numbers is as follows ... 

In a rheomety experiment the two plates or cylinders are moved back and forth relative to one another in an oscillating fashion. The elastic storage modulus (G - The contribution of elastic, i.e. solid-like behaviour to the complex dynamic modulus) and elastic loss modulus (G" - The contribution of viscous, i.e. liquid-like behaviour to the complex modulus) which have units of Pascals are measured as a function of applied stress or oscillation frequency. For purely elastic materials the stress and strain are in phase and hence there is an immediate stress response to the applied strain. In contrast, for purely viscous materials, the strain follows stress by a 90 degree phase lag. For viscoelastic materials the behaviour is somewhere in between and the strain lag is not zero but less than 90 degrees. The complex dynamic modulus ( ) is used to describe the stress-strain relationship (equation 14.1 i is the imaginary number square root of-1). 

Fig. 1 a,b. Strain amplitude dependence of the complex dynamic modulus E E l i E" in the uniaxial compression mode for natural rubber samples filled with 50 phr carbon black of different grades a storage modulus E b loss modulus E". The N numbers denote various commercial blacks, EB denotes non-commercial experimental blacks. The different blacks vary in specific surface and structure. The strain sweeps were performed with a dynamical testing device EPLEXOR at temperature T = 25 °C, frequency f = 1 Hz, and static pre-deformation of -10 %. The x-axis is the double strain amplitude 2eo... 

An alternative set of terms is best introduced by noting that a complex number can be represented as in Fig. 11-15 by a point P (with coordinates x and y) or by a vector OP in a plane. Since dynamic mechanical behavior can be represented by a rotating vector in Fig. 11-13, this vector and hence the dynamic mechanical response is equivalent to a single complex quantity such as G (complex modulus)... 

Complex flnids are viscoelastic when the fluid stiU maintains internal stress after the external shear stress has ceased. The internal stress decays with time the time reqnired for the fluid to recover to the initial state is called the relaxation time. For this case the shear modulus (G ) is a complex number ... 

Introducing the modulus (= absolute value in complex number language) of the surface dilational modulus K° as... 

The second restriction of this type of smdy is the determination of the stmcture factor. This is because this term is a complex number, whose phase has a random value in the general case. The determination of these stmcture factors is achieved based on the measurements of the intensities diffracted by each of the families of planes, but, as we showed in Chapter 1, the diffracted intensity is proportional to the square modulus of the stmcture factor. Therefore, the measurements do not make it... 

Determine the modulus and argument of a complex number, and denote its location on an Argand diagram... 

Figure 2.3 An Argand diagrann showing the complex numbers Zi= - 1 + i and Z2=1 - i with modulus V2 and arguments 3n/4 and -n/A, respectively...

It may seem odd to think of the exponential function, z = e , as periodic because it is clearly not so when the exponent is real. However, the presence of the imaginary number i in the exponent allows us to define a modulus and argument as 1 and 6, respectively. If we represent the values of the function on an Argand diagram, we see that they lie on a circle of radius, r= 1, in the complex plane (see Figure 2.4). Different values of 6 then define the location of complex numbers of modulus unity on the circumference of the circle. We can also see that the function is periodic, with period 2% ... 

Ey is called the modulus of a complex number, z. Unlike real numbers. 

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