Superposition Theorem

6 Key excerpts on "Superposition Theorem"

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  Sneaking a Look at God's Cards

  Unraveling the Mysteries of Quantum Mechanics - Revised Edition

  • Giancarlo Ghirardi, Gerald Malsbary(Authors)
  • 2021(Publication Date)
  • Princeton University Press

    (Publisher)

  79 C H A P T E R F O U R The Superposition Principle and the Conceptual Structure of the Theory The assumption of superposition relationships between the states leads to a mathematical theory in which the equations that define a state are linear in the unknowns. In consequence of this, people have tried to establish analogies with systems in classical mechanics, such as vibrating strings or membranes, which are governed by linear equations and for which, therefore, a superposition principle holds. Such analogies have led to the name “Wave Mechanics” being sometimes given to quantum mechanics. It is important to remember, however, that the superposition that occurs in quantum mechanics is of an essentially different nature from any occurring in the classical theory, as is shown by the fact that the quantum superposition principle demands indeterminacy in the results of observations in order to be capable of a sensible physical interpretation. The analogies are thus liable to be misleading. —Paul Adrien Maurice Dirac W e can now deepen our analysis of the most innovative point of the new theory: the superposition principle. In the preceding chapters we have shown that the formal structure of the theory is such as to permit the “sum-mation” of quantum states. In particular, our attention has been drawn to the fact that, for instance, the polarization state |45° > of a photon is the “sum” of the states |V > and |H > (for “vertical” and “horizontal,” respec- tively). Analogously, it was asserted that the state of a particle with upward spin along the direction of the x axis is the “sum” of the states correspond-ing to upward and downward spin along the z axis.

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  Foundations of Physics

  • Steve Adams(Author)
  • 2019(Publication Date)
  • Mercury Learning and Information

    (Publisher)

  15

  Superposition Effects

  15.0   Superposition Effects

   

  Superposition is what happens when two or more waves are present at the same point. This might occur because they originate from different sources or because of reflection. If the waves are of the same type, the resultant disturbance at that point is the vector sum of the disturbances from each individual wave. This is called the principle of superposition. We can determine the effects of superposition graphically, by adding phasors or by calculation. Many important phenomena are linked to superposition, including interference and diffraction and the formation of standing (stationary) waves.

  15.1   Two-Source Interference

   

  If waves of the same type with equal wavelength, frequency, and amplitude are emitted from two sources placed a short distance apart (comparable to a few wavelengths), then a regular interference pattern is formed. The pattern consists of regions where the waves reinforce to produce maximum intensity (constructive interference) and regions where they cancel to produce minimum intensity (destructive interference).

  The famous double slit experiment carried out by Thomas Young in 1801 provided strong evidence for the wavelength of light and enabled Young to calculate its wavelength. A similar setup with sound can be used to demonstrate superposition patterns and, in a modified form, to create noise-cancelling headphones.

  In order for stable clear interference effects to be created, the two sources must be coherent.

  Coherent sources maintain a constant phase relationship.

  This means that they must be the same type of wave and have the same wavelength and frequency. The sources do not have to be in phase, but the phase difference between them must be constant. They must also have comparable amplitudes; if one wave has a much greater amplitude than the other, then variations in intensity will be hard to detect.

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  The Art of Modeling in Science and Engineering with Mathematica

  • Diran Basmadjian, Ramin Farnood(Authors)
  • 2006(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  299 6 Solution of Linear Systems by Superposition Methods We have previously (in Section 2.1.2 ) drawn the reader’s attention to the impor-tant superposition principle, which is one of the principal tools listed in dealing with linear systems. It states, in essence, that the general solution of a linear differential equation can be composed of the sum of all independent particular solutions. Its application in that section was restricted to ordinary differential equations (ODEs), and in Section 2.3.2 , we used it successfully to compose solutions of linear ODEs with constant coefficients from the sum of exponential functions of the so-called eigenvalues (D-operator method). Later, in Section 2.3.4 , we extended the method to linear systems with variable coefficients and saw the emergence of infinite sums of particular solutions, a feature we shall encounter again at the PDE level. We now extend and amplify this principle and apply it to the solution of partial differential equations (PDEs). We make the following distinctions. 1. Superposition by addition: This is the standard definition of the term as it was originally applied at the ODE level. Superposition of simple flows, which is the first item to be taken up, is the classical example of this method. 2. Superposition by integration: This procedure views the integration as one infinite set of superpositions by addition. Integration is with respect to time or distance, and its most fruitful application is found in the treatment of time-varying sources and boundary conditions. The use of Green’s functions, to be taken up in Chapter 7, also falls within this category. 3. Superposition by multiplication: This is a somewhat unconventional category. It is not generally viewed as a superposition method, but has all the necessary features of one and is, therefore, included here.

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  Mechanical Vibrations

  • J. P. Den Hartog(Author)
  • 2013(Publication Date)
  • Dover Publications

    (Publisher)

  CHAPTER 8

  SYSTEMS WITH VARIABLE OR NON-LINEAR CHARACTERISTICS

  8.1. The Principle of Superposition. All the problems thus far considered could be described by linear differential equations with constant coefficients, or, physically speaking, all masses were constant, all spring forces were proportional to the respective deflections, and all damping forces were proportional to a velocity. In this chapter it is proposed to consider cases where these conditions are no longer true, and, on account of the greater difficulties involved, the discussion will be limited to systems of a single degree of freedom. The deviations from the classical problem Eq. (2.1) , page 26 , are twofold.

  First, in Sees. 8.2 to 8.4 , we shall consider differential equations which are linear but in which the coefficients are functions of the time. In the remainder of the chapter non-linear equations will be discussed. The distinction between these two types is an important one. Consider the typical linear equation with a variable coefficient:

  which describes the motion of a system where the spring constant varies with the time. Assume that we know two different solutions of this equation:

  Then C1 φ1 (t) is also a solution and

  is the general solution of Eq. (8.1) . Any two known solutions may be added to give a third solution, or

  The principle of superposition holds for the solutions of linear differential equations with variable coefficients.

  The proof of this statement is simple.

  Multiply the first equation by C1 and the second by C2 and add:

  This shows that [C1 φ1 (t) + C2 φ2 (t)] fits the differential Eq. (8.1) and therefore is a solution.

  In mechanical engineering it is usually the elasticity that is variable (Eq. 8.1). There is, however, one important case where the mass is variable with time (Fig. 5.10 , page 184 ). This case can be discussed on the same mathematical basis as that of variable elasticity, provided damping is absent. We have

  where m(t) is the variable mass. Dividing by m(t

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  Foundations of Physics

  • Steve Adams(Author)
  • 2023(Publication Date)
  • Mercury Learning and Information

    (Publisher)

  15

  SUPERPOSITION EFFECTS

  15.0 SUPERPOSITION EFFECTS

  Superposition is what happens when two or more waves are present at the same point. This might occur because they originate from different sources or because of reflection. If the waves are of the same type the resultant disturbance at that point is the vector sum of the disturbances from each individual wave. This is called the “principle of superposition.” We can determine the effects of superposition graphically, by adding phasors or by calculation. Many important phenomena are linked to superposition including interference and diffraction and the formation of standing (stationary) waves.

  15.1 TWO-SOURCE INTERFERENCE

  If waves of the same type with equal wavelength, frequency, and amplitude are emitted from two sources placed a short distance apart (comparable to a few wavelengths) then a regular interference pattern is formed. The pattern consists of regions where the waves reinforce to produce maximum intensity (constructive interference) and regions where they cancel to produce minimum intensity (destructive interference).

  The famous double slit experiment carried out by Thomas Young in 1801 provided strong evidence for the wavelength of light and enabled Young to calculate its wavelength. A similar set up with sound can be used to demonstrate superposition patterns and, in a modified form to create noise-canceling headphones.

  In order for stable clear interference effects to be created the two sources must be coherent. Coherent sources maintain a constant phase relationship.

  This means that they must be the same type of wave, and have the same wavelength and frequency. The sources do not have to be in phase but the phase difference between them must be constant. They must also have comparable amplitudes; if one wave has a much greater amplitude than the other then variations in intensity will be hard to detect.

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  Electrical, Electronics, and Digital Hardware Essentials for Scientists and Engineers

  • Ed Lipiansky(Author)
  • 2012(Publication Date)
  • Wiley-IEEE Press

    (Publisher)

  x  + 7 is the equation of a straight line.

  3.2.1  Circuits Superposition

  Let us now apply the superposition property to electric circuits. Assume that we are given an electrical circuit that can contain any number of resistors, in the black box represented in Figure 3.3 . Two external voltage sources are applied to the circuit. We also refer to these two voltage sources as the circuit excitations. The output of the circuit is referred to as the circuit response.

  Figure 3.3  Electrical linear circuit with two external voltage sources: v 1 and v 2 .

  If we have a linear circuit where x is the excitation and y  = f (x ) is its response, the superposition property tells us that

  Given:

  y 1  = f (v 1 ), where y 1 is the response of the circuit due to excitation v 1 and

  y 2  =  f (v 2 ), where y 2 is the response of the circuit due to excitation v 2 .

  The sum of the circuit responses y 1  + y 2  = f (v 1 ) + f (v 2 ) equals the response of the sum of the circuit excitations y 1  + y 2  = f (v 1  + v 2 ).

  Moreover, thanks to the linearity of the circuit, we can also calculate the response of the circuit to excitation v 1 while excitation v 2 is inhibited. This yielding the response y 1 for v 2  = inhibited. Similarly, we can calculate the response of the circuit y 2 when excitation v 1 is inhibited. Finally, adding the individually found responses we obtain

  (3.13)  

  Equation (3.13) provides the complete response of the circuit due to noninhibited excitations or the response of the circuit due to both excitations applied simultaneously.

  When the excitation is a voltage source v , inhibiting the excitation means to replace the voltage source with a short circuit (v  = 0). When the excitation is a current source i

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