Physics
Bound State
A bound state is a physical system where the constituent particles are confined to a finite region of space by attractive forces. The particles in a bound state have lower energy than when they are separated, and they cannot escape from the region of confinement without the addition of energy. Examples of bound states include atoms, molecules, and nuclei.
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5 Key excerpts on "Bound State"
eBook - PDF- Mark Fox(Author)
- 2018(Publication Date)
- Cambridge University Press(Publisher)
Atomic physics proceeds by a series of approximations that make this problem tractable. Before we set about this task, it is first necessary to cover a number of important basic concepts and definitions. 1.1 Quantized Energy States in Atoms The first basic concept we need is that of Bound States . Atoms are held together by the attractive force between the positively charged nucleus and 3 4 Preliminary Concepts Large distance Electron Electron bound E negative Necleus (a) (b) Electron free E = 0, when v =0 --+ + v Figure 1.1 (a) UnBound State with the electron far from the nucleus. The electron moves freely with velocity ( v ) independent of the presence of the nucleus. (b) Bound electron state with negative energy. the negatively charged electrons: the electrons are bound to the atom, rather than being free to move though space. In the limit where the electron is very far away from the nucleus, the attractive force is negligible; the electron is free to move with velocity ( v ) without any influence from the nucleus, as illustrated schematically in Figure 1.1 (a). It is natural to define the energy ( E ) of this free (or unbound ) state as being zero when v = 0. When the electron moves closer to the nucleus, it begins to experience an attractive force, leading to the formation of a stable Bound State as illustrated in Figure 1.1 (b). The energy of the Bound State is lower than that of the free electron since it requires energy to pull the electron away from the nucleus. The amount of energy required is called the binding energy of the electron. With our definition of E = 0 corresponding to the unBound State, the absolute energy ( E ) of the Bound State must be negative, with the binding energy equal to − E = | E | . The early understanding of the atom was built around the solar system analogy, that the planets orbit around the sun under the influence of the attractive gravitational force.
eBook - PDF- David Griffiths(Author)
- 2008(Publication Date)
- Wiley-VCH(Publisher)
The quark model, though, changed everything. Sud- denly the hadrons themselves were Bound States all mesons were two-quark systems, and all baryons were three-quark systems. With this discovery the theory of Bound States became an important component of elementary particle physics. 143 144 5/Bound StateS The analysis of a Bound State is simplest when the constituents travel at speeds substantially less than c, for then the apparatus of nonrelativistic quantum mechanics can be brought to bear. Such is the case for hydrogen and for hadrons made out of heavy quarks (c, b, and t). The more familiar light-quark states (made out of u, d, and s) are much more difficult to handle, because they are intrinsically relativistic, and quantum field theory (as currently practiced) is not well suited to bound-state problems.
eBook - PDF- Vishnu S. Mathur, Surendra Singh(Authors)
- 2008(Publication Date)
- Chapman and Hall/CRC(Publisher)
5 Bound StateS OF SIMPLE SYSTEMS 5.1 Introduction We shall now apply the time-independent Schr¨ odinger equation to study the Bound States of simple systems. 1. A free particle in a box with sharp boundaries. 2. Particle moving in a one-dimensional harmonic potential well (a simple harmonic oscillator). 3. Two-body system with mutual central interaction between its constituents. Under this heading we shall consider (i) the problem of the Hydrogen (or Hydrogen-like) atoms and (ii) the Bound State of the neutron-proton system (the deuteron). 4. A particle in three-dimensional (a) square well potential (b) harmonic oscillator potential. As discussed in Chapter 4, a physically acceptable wave function in the coordinate space satisfies the following conditions: (i) Continuity : The wave function must be single-valued and both the wave function and its first derivative must be continuous even if the potential has a (finite) discontinuity at some point. When the potential has infinite discontinuity the first derivative may be discontinuous and the wave function may have a kink. (ii) Boundary Conditions: For a Bound State of the system, the wave function must be finite everywhere and decrease to zero as r → ∞ . This condition follows from the requirement that Bound State wave function must be normalized to unity in order to maintain the probabilistic interpretation of ψ ( r ). 5.2 Motion of a Particle in a Box Consider a particle which moves freely inside a cubical box of dimension L and volume V = L 3 . If we choose the origin of the coordinates at one corner of the cube as in Fig. 5.1, then the particle confinement to the cube means that the walls are infinite potential steps so that the wave function must vanish at the boundary surfaces x = 0 , x = L ; y = 0 , x = L ; z = 0 , and x = L shown in Fig.
eBook - PDFQuantum Mechanics
A Modern Development
- Leslie E Ballentine(Author)
- 1998(Publication Date)
- WSPC(Publisher)
(10.49) This result was first presented by Wolsky (1974). Examples As a first example, we apply (10.49) to the ground state of the hydrogen atom (10.32), for which we have denotes the deuteron state. Then (10.49) implies that (D|T re i|I>) > 28.4 MeV. (One MeV is 10 6 eV. For comparison, the binding energy of the hydrogen atom is 13.6 eV.) Since the binding energy of the deuteron is known to be 2.2 MeV, it follows that the average potential energy must satisfy (£>|W|JD) < —30.6 MeV. When nuclear forces were not yet understood this was a very valuable piece of information. 10.4 Some Unusual Bound States All of the Bound States considered so far have the property that the total energy of the state is less than the value of the potential energy at infinity. The system remains bound because it lacks sufficient energy to dissociate. This same property characterizes classical Bound States. However, it is possible in quantum mechanics to have Bound States that do not possess this property, and which therefore have no classical analog. Let us choose the zero of energy so that the potential energy function vanishes at infinity. The usual energy spectrum for such a potential would be a positive energy continuum of unBound States, with the Bound States, if any, occurring at discrete negative energies. However, Stillinger and Herrick (1975), following an earlier suggestion by Von Neumann and Wigner, have constructed potentials that have discrete Bound States embedded in the positive energy continuum.
eBook - PDF- Andrzej Deloff(Author)
- 2003(Publication Date)
- World Scientific(Publisher)
They are de-fined in the usual way: their wave functions must be square integrable and the corresponding S-matrix has a pole in the upper half of the k-plane. Of course, a nuclear Bound State cannot be formed unless the nuclear at-traction has reached certain critical value. The second case (ii) is more complicated because the Coulomb potential taken alone has infinite num-ber of Bound States and in the (ii) case it is supplemented by a short ranged potential which may be capable of supporting nuclear Bound States when the Coulomb potential is turned off. Although all Bound States result then from the combined attraction provided by these two potentials but never-theless we can distinguish two different types of Bound States: (a) atomic levels shifted from Coulombic positions by the nuclear interaction, and, (b) Bound States which may, or may not, exist which we shall call quasi-nuclear Bound States. Similarly as in (i) and (iii), all Bound States are necessarily associated with poles of the S-matrix that are located in the upper half of the k-plane. In the case (ii), however, when the strength of the nuclear potential is reduced to zero (s — > 0) the Bound State poles either tend to occupy positions on the imaginary axis as appropriate for the unperturbed Coulomb spectrum, or move into the lower half of the k-plane. If in the limit s — > 0 the pole resumes a position on the imaginary axis we have to do with the case (a) and if it moves into the lower half of the k-plane we have a quasi-nuclear Bound State (b). It should be noted that knowing just the position of the pole is not enough to tell to which category it belongs either (a), or (b). To answer this question the limiting procedure s —>■ 0 has Bound States and Low-Energy Scattering 77 to be effected in order to examine the motion of the pole in the k-plane.
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