Physics
Two Particles
Two particles refer to a pair of objects that interact with each other through forces such as gravity or electromagnetism. In physics, the study of two-particle systems is important for understanding the behavior of larger systems and for developing theories that explain the fundamental forces of nature. The behavior of two particles can be described using mathematical models such as the Schrödinger equation.
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7 Key excerpts on "Two Particles"
- Paul Teller(Author)
- 2020(Publication Date)
- Princeton University Press(Publisher)
C H A P T E R T W O From Particles to Quanta QUANTUM THEORIES, from their inception, have used particle and wave ideas in combinations quite unlike those occurring in classical mechanics. Early interpreters worried: How can one and the same entity be in some ways a particle and in other ways a wave? Are particle and wave concepts not, after all, mutually exclusive? The ploy of calling quantum entities 'wavicles' just gave an amusing name to the problem instead of solving it. To address this problem one needs to recognize that wave and par-ticle concepts are neither univocal nor unanalyzable primitives. If we analyze these concepts, perhaps we will find components that will fit to-gether consistently. 1 Conventional one-particle quantum mechanics al-ready gives up exact space-time trajectories, at least in its descriptions. Inexact space-time trajectories will not figure in our deliberations, so I'll move immediately to another feature of particle concepts, namely, the idea that particles are concrete, substantial entities that can have properties and can be named. To have a word for our problem, I'll talk about the idea that each particle is, is composed of, or in some way involves a nameable substance. IDEAS OF SUBSTANCE In discussing a two-particle system in conventional quantum mechanics one usually begins by declaring that one of the particles will be referred to as 'particle 1' and the other as 'particle 2'. What preconceptions about things thus named might one bring to the theory? Because we are interested in our pretheoretical preconceptions, let's use the more general term 'objects' instead of 'particles', referring to two objects, 1 and 2. Many people find that the use of such label-names suggests that the thing named has an identity that is independent of its properties. There seems (to some) to be a sense in which we can talk about this object 'Chapter 5 will continue this project, only begun in the present chapter.
eBook - ePub- Gregory Choppin, Jan-Olov Liljenzin, Jan Rydberg, Christian Ekberg, JAN RYDBERG(Authors)
- 2013(Publication Date)
- Academic Press(Publisher)
Annual report 1961 , CERN).This has created a scientific area called elementary particle physics . It is quite different from nuclear physics, which is concerned with composite nuclei only. A principal objective of elementary particle physics has been to group the particles together according to their properties to obtain a meaningful pattern, which would describe all particles as parts of some few fundamental building blocks of nature. One step in this direction is to study how the elementary particles interact with each other, i.e. what kind of forces are involved.Scientists have long doubted that all the particles produced with masses between the electron and the proton (loosely referred to as mesons , i.e. “intermediate”), and with masses greater than the proton (referred to as baryons , “heavy”) really are “elementary”. It was proposed that they have a substructure or constitute excited states of each other. Are they waves or particles since they serve as carriers of force? At this point it is important to understand what is meant by “particle” in nuclear physics.2.2 Forces of Nature
Considering what an immense and incredible diverse assembly the universe is – from the cosmos to man and microbes – it is remarkable that scientists have been able to discover with certainty only four basic forces, which govern the attraction and repulsion of all physical objects of nature. Let us consider these forces of nature in a qualitative way from the weakest to the strongest, see Table 2.1 .Table 2.1 Forces of nature in order of strength and their exchange particles† Not yet detected.The first and weakest force of nature is that of gravity . The gravitational force, F g , beween two masses, m 1 and m 2 , at a distance r is given by eqn (2.1) , where G is the universal constant of gravity (G = (6,6742 ± 0,0007) × 10–11 N m2 kg–2
eBook - PDFDynamics of Particles and the Electromagnetic Field
(With CD-ROM)
- Slobodan Danko Bosanac(Author)
- 2005(Publication Date)
- WSPC(Publisher)
Chapter 6 Interact ion of Two Particles 6.1 Forces with Time Delay In the previous chapters the emphasis was on the single particle dynamics. Few examples were discussed in which more particles are involved, but by a suitable transformation it was possible to reduce those systems to a single particle dynamics. Based on this observation the impression is that everything what is being discussed is easily extendable to a two-particle system, without gaining any new insight. This is a very dangerous view because some tacit assumptions are made in these transformations that simplify analysis considerably, from which one draws such a conclusion. If a truly strict analysis is made then extension of a single particle dynamics to a two or many particle dynamics is far from being straightforward. In fact without making some very radical steps this problem is not solvable. The basic feature of a two (or more) particle system is that the force between the particles should be included. Therefore two types of forces now determine the motion of each particle: one is the external force and the other is resulting from the other particles. While the external force is arbitrary, the force between the particles cannot be because Axiom 3 must be satisfied. For Two Particles, e.g. particle 1 and 2, it says that the force that acts on particle 1 and originates in the particle 2 must be equal and opposite in sign to the force that acts on the second and originates in the first particle. This means that the force between the particles must depend on their relative separation, and not on the absolute coordinates of particles with respect to some coordinate system. For Two Particles the potential, from which the force is derived, must therefore be a function of the distance between them. For the system with larger number of particles this dependence may be of a more complicated kind, but always in terms 97
eBook - PDFQuantum Themes: The Charms Of The Microworld
The Charms of the Microworld
- Thanu Padmanabhan(Author)
- 2009(Publication Date)
- World Scientific(Publisher)
The third category of particles is called “vector bosons”. These particles play the important role of mediating interactions between other particles. According to quantum theory, all forces arise due to the exchange of some fundamental particles. Consider, for example, the electrical repulsion be-tween two electrons. In quantum theory it is described by a process shown schematically in the top left frame of Fig. 4.6. (We will discuss this con-cept in detail in the next section.) We assume that the first electron emits a particle (in this case, called a photon) which is absorbed by the sec-ond electron. It is this transfer of a photon which actually produces the electrical repulsion between the Two Particles. In fact, the simplest of the vector bosons is a photon, which is the quantum of the electromagnetic radiation. All electromagnetic interactions between charged particles can be thought of as arising due to exchange of photons. Other forces of inter-action are mediated by some other vector bosons. The weak nuclear force between quarks or leptons is mediated through the exchange of certain par-ticles called “W” and “Z” bosons and the strong nuclear force between the quarks is mediated by a set of 8 particles called “gluons” (see Fig. 4.6). The gravitational interaction is thought to arise out of certain particles called gravitons. Of all these particles, photons are the most familiar and we shall discuss them at length in the next section; the W and Z bosons have ac-tually been produced in the laboratory while the gluons are seen in some of the particle interactions indirectly. Thus the existence of these particles is well founded. The gravitons have not been seen directly and should be thought of as a theoretical speculation at present especially since we do not yet have sensible model for quantum gravity; see Sec. 6.1. Thus, according to our current understanding, we can interpret all forces of nature using the picture of quarks, leptons and vector bosons.
eBook - PDF- Donald H. Perkins(Author)
- 2000(Publication Date)
- Cambridge University Press(Publisher)
However, there are two signs of electric charge, the world is electrically neutral, and the enormously greater electrical force on a proton due to all other protons in the Earth is exactly cancelled by that due to the electrons. To summarise the results of the last few sections, we list in Table 2.2 the fundamental interactions and their principal characteristics. 2.10 The interaction cross-section The strength of a particular interaction between Two Particles is specified by the interaction cross-section , defined as follows. Imagine a two-body to two-body reaction of the form a + b → c + d (2.13) 52 2 Interactions and fields Table 2.2. Fundamental interactions (Mc 2 = 1 GeV) Gravitational Electromagnetic Weak Strong field boson graviton photon W ± , Z gluon spin-parity 2 + 1 − 1 − , 1 + 1 − mass, GeV 0 0 M W = 80 . 2 0 M Z = 91 . 2 range, m ∞ ∞ 10 − 18 ≤ 10 − 15 source mass electric ‘weak ‘colour charge charge’ charge’ coupling G N M 2 4 π ¯ hc α = e 2 4 π ¯ hc G ( Mc 2 ) 2 ( ¯ hc ) 3 α s ≤ 1 constant = 5 × 10 − 40 = 1 137 = 1 . 17 × 10 − 5 typical cross-10 − 33 10 − 39 10 − 30 section, m 2 typical 10 − 20 10 − 10 10 − 23 lifetime, s in which a well-defined parallel beam of particles of type a impinges normally on a target of thickness dx containing n b particles of type b per unit volume, and c and d are the product particles. If the density of particles in the incident beam is n a , the flux through the target will be φ = n a v i (2.14) particles per unit area and per unit time, where v i is the velocity of the incident beam relative to the target. If each of the target particles has an effective cross-section σ , the probability that any particle a will hit a target particle is σ n b dx , since this is the fraction of the target area obscured by the b particles (see Figure 2.9).
eBook - PDF- Eduard Prugovecki(Author)
- 1982(Publication Date)
- Academic Press(Publisher)
1. Basic Concepts in Scattering Theory of Two Particles 419 spin, etc.), but which do not interact among themselves. For instance, in the case where is given essentially’ by then Hh2’ is taken to be essentially 1 - A1 - 2m1 r = r2- r l , where the domains of definition of the above differential operators are adequately defined,+ as was done in 97 of Chapter IV. However, if both particles move in an external force field, given by a potential Vext(rl , r2), then instead of (1.8) we have and instead of (1.9) we have 1 A1 - -- 2m1 This will be the case when, for example, the system under consideration consists of two spinless charged particles moving in the force field generated by a much heavier particle (such as the heavy nucleus of some atom). Since the heavier particle is practically unaffected by the motion of the lighter particles, such a problem can be treated fairly accurately as a two-body problem in an external field rather than a three-body problem. Consider now the general case when the “free” Hamiltonian Hi2’ and the total Hamiltonian W2) are given by the self-adjoint operators acting in the Hilbert space A?(2) associated with the system. We say that a nonzero vector-valued function ! W ( t ) , t E R1, is a free state in the Schroedinger picture if and only if it satisfies the relation Y(’)(t) = exp( -iH(yt) Y(f)(O), t E R1, and if Y(t)(O) is orthogonal to the closed linear subspace Zit) of bound states in .#(2) of the system with Hamiltonian HA2). In the absence of See the discussion in 97 of Chapter IV on the essential self-adjointness of the t We note that in the preceding two operators, as well as throughout this chapter, we Schroedingeroperator. have adopted a system of units in which fi = 1. 420 V. Quantum Mechanical Scattering Theory external forces both HA’) and its internal energy part Hi:), obtained after subtraction from Hi2) of the center of mass motion, are operators with a pure continuous spectrum so that & A : ) = (0).
eBook - PDF- Lincoln Wolfenstein, Joao P. Silva(Authors)
- 2010(Publication Date)
- CRC Press(Publisher)
11 2 C H A P T E R Waves That Are Particles; Particles That Are Waves A major revolution in our understanding of nature took place in the early twentieth century; we learned that light can have particle-like properties and that particles can have wave-like properties. This is deeply ingrained into the standard model of particle physics. 2.1 PARTICLES VERSUS WAVES This book tells the exhilarating recent history of the search for the funda-mental building blocks of all things and their interactions. When physi-cists mention “point particles,” they may not be talking about fundamental particles at all. Point particles might have some internal structure, but they are so named because, whatever their internal structure might be, it has no bearing on the phenomenon under study. For example, consider a rigid ball sliding down an inclined plane without rolling and without friction. If this experiment is performed in a vacuum (that is, with all the air sucked out), the velocity that the ball has after it slides for 1 in. can be calculated ignoring what the ball is made of. It is even independent of the ball’s mass; it depends exclusively on the slope of the inclined plane. There is an interesting way to describe how this happens. When the ball is placed in a high position, we say that it has the potential to gain speed and we ascribe to it some potential energy. As it accelerates down the 12 ◾ Exploring Fundamental Particles inclined plane, we say that it transforms this potential energy into kinetic energy, from the Greek word kinesis , which means motion. That is, the potential energy the ball had because it was placed in a high position is transformed into the kinetic energy associated with its speed as it moves down the plane. 1 Another interesting quantity is the momentum of this particle. Momentum is an arrow (so-called vector) that has a size equal to the prod-uct of mass with velocity, and it has the direction of the particle’s movement.
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