Symmetrization Postulate

7 Key excerpts on "Symmetrization Postulate"

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  Quantum Mechanics

  A Modern Development

  • Leslie E Ballentine(Author)
  • 1998(Publication Date)
  • WSPC

    (Publisher)

  The Symmetrization Postulate states that: 17.3 The Symmetrization Postulate 475 (a) Particles whose spin is an integer multiple of h have only symmetric states. (These particles are called bosons.) (b) Particles whose spin is a half odd-integer multiple of h have only anti-symmetric states. (These particles are called fermions.) (c) Partially symmetric states do not exist. (Nevertheless they give rise to the name paraparticles.) The superselection rule deduced in Sec. 17.2 is trivialized by the symmetriza-tion postulate, since obviously no interference between symmetry types is possible if only one symmetry type exists. The three parts of this postulate cannot be deduced from the other principles of quantum mechanics, so we shall examine their consequences and the empirical evidence that supports them.

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  Frontiers of Molecular Spectroscopy

  • Jaan Laane(Author)
  • 2011(Publication Date)
  • Elsevier Science

    (Publisher)

  The Symmetrization Postulate In the Standard Model of theoretical physics, electrons are fundamental particles and they are identical to each other. This means that for an atom or a molecule any permutation of the space and spin coordinates of the electrons commutes with the Hamiltonian and is therefore a symmetry operation. For a system containing n electrons, the symmetry group of all possible n ! electron permutations is called the symmetric group and is denoted S n [10]. The class structure of a symmetric group is easy to determine because all permutations of the same shape (i.e., consisting of the same number of independent transpositions, independent cycles of three, independent cycles of four, etc.) are in the same class. For example, in the group S 5 there are seven classes, and the elements in these classes have the following shapes: (1) The number of classes in the symmetric group S n is given by the partition number of n, i.e., the number of ways of writing n as the sum of integers. For example, for n = 5 we can write (2) so that the partition number of 5 is 7. The parallel between this partitioning of the number 5 and the shapes of the permutations in each class in Equation (1) is obvious. Because the number of irreducible representations in a group is equal to the number of classes, we can immediately deduce the number of irreducible representations in S n once we have the partition number of n. For example, the group S 5 has seven irreducible representations; the character table of the symmetric group S 5 is given in Table 1. Table 1. The character table [ a ] of the symmetric. group S 5 E (12) (12)(34) (123) (12)(345) (1234) (12345) 1 10 15 20 20 30 24 D (0) : 1 1 1 1 1 1 1 : 1 −1 1 1 −1 −1 1 D (1) : 4 2 0 1 −1 0 −1 : 4 −2 0 1 1 0 −1 D (2) : 5 1 1 −1 1 −1 0 : 5 −1 1 −1 −1 1 0 D ¯ (3) = : 6 0 −2 0[--=PLGO-SEPARATOR

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  Quantum Mechanics

  A Modern Development

  • Leslie E Ballentine(Author)
  • 2014(Publication Date)
  • WSPC

    (Publisher)

  bosons .)

  (b) Particles whose spin is a half odd-integer multiple of ħ have only antisymmetric states. (These particles are called fermions .)

  (c) Partially symmetric states do not exist. (Nevertheless they give rise to the name paraparticles .)

  The superselection rule deduced in Sec. 17.2 is trivialized by the Symmetrization Postulate, since obviously no interference between symmetry types is possible if only one symmetry type exists. The three parts of this postulate cannot be deduced from the other principles of quantum mechanics, so we shall examine their consequences and the empirical evidence that supports them.

  [[ Many books contain arguments that purport to derive one or more parts of the Symmetrization Postulate from other principles of quantum mechanics. A typical argument begins with the assertion that permutation of identical particles must not lead to a different state . (This is stronger than the principle of indistinguishability , which asserts only that such a permutation must not lead to any observable differences.) Hence it is asserted that an allowable state vector must satisfy

  Pij

  |Ψ〉 = c |Ψ〉. Since (

  Pij

  )2 = 1, it follows that c = ± 1, and so the argument concludes that only symmetric and antisymmetric states are permitted.

  Implicit in this argument is the assumption that a state must be represented by a one-dimensional vector space, i.e. by a state vector with at most its overall phase being arbitrary. But this is equivalent to excluding by fiat the partially symmetric state vectors, which belong to multidimensional invariant subspaces. So it is practically equivalent to the assumption of (c), which was to be proven. If one drops the assumption that a state must be represented by a one-dimensional vector space, the conclusion no longer follows. Consider an n -dimensional permutation-invariant subspace. (Examples with n = 2 were given in Sec. 17.1 .) From it we construct a state operator ρ

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  Quantum Mechanics

  Non-Relativistic Theory

  • L D Landau, E.M. Lifshitz(Authors)
  • 1981(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  2 conventionally denoting the three coordinates and the spin projection for each particle. Then we must have

  where α is some real constant. By repeating the interchange, we return to the original state, while the function ψ is multiplied by e

  2i α

  . Hence it follows that e

  2i α

  = 1, or e

  i α

  = ± 1. Thus

  We thus reach the result that there are only two possibilities: the wave function is either symmetrical (i.e. it is unchanged when the particles are interchanged) or antisymmetrical (i.e. it changes sign when this interchange is made). It is obvious that the wave functions of all the states of a given system must have the same symmetry; otherwise, the wave function of a state which was a superposition of states of different symmetry would be neither symmetrical nor antisymmetrical.

  This result can be immediately generalized to systems consisting of any number of identical particles. For it is clear from the identity of the particles that, if any pair of them has the property of being described by, say, symmetrical wave functions, any other pair of such particles has the same property. Hence the wave function of identical particles must either be unchanged when any pair of particles are interchanged (and hence when the particles are permuted in any manner), or change sign when any pair are interchanged. In the first case we speak of a symmetrical wave function, and in the second case of an antisymmetrical one.

  The property of being described by symmetrical or antisymmetrical wave functions depends on the nature of the particles. Particles described by antisymmetrical functions are said to obey Fermi–Dirac statistics (or to be fermions ), while those which are described by symmetrical functions are said to obey Bose-Einstein statistics (or to be bosons ).†

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  Problems in Quantum Mechanics

  • F. Constantinescu, E. Magyari, J.A. Spiers(Authors)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  in the proper treatment of electromagnetic emission and absorption (of photons) by charged particles, of electron-positron pair creation and annihilation, of /?-decay, etc. Problems 1. Consider a system consisting of two identical particles, each of which has one-particle states represented in coordinate representation by the wavefunctions ipjr), ψ β (τ). Let us define ^>(ri, 2 ) = -—=[v«(ri)^r 2 )-y«(r 2 )v/3(ri)], 2 da) y ^ r i , r 2 ) = — j=-[^a(ri)^(r 2 )+^a(r 2 )^(ri)], V 2 183 Problems in Quantum Mechanics i.e. the corresponding anti-symmetrical and symmetrical wavefunctions, respectively, of the system. Now if the symmetrization rule for identical particles were ignored, the system would in general have the following wavefunction ψ = λψ^+μψ^, |A| 2 +|/i| 2 = 1. (lb) Show that in this case the probability per unit volume of finding a particle at n and another at r 2 depends on λ and on , and discuss this result. 2. Show that the symmetrization and the anti-symmetrization operators are orthogonal projection operators, i.e. that S 2 = S, A 2 = A, SA = AS = 0. 3. Find the state vectors of a system of two spin— particles which are simultaneous eigenvectors of the operators S 2 and S 2 , where S is the total spin operator of the system. Dis-cuss the symmetry of these state vectors. 4. Solve the preceding problem in the case of two particles, each of spin unity. 5. Write down the normalized wavefunctions of a system of three identical bosons, which are in given one-particle states. 6. Show that for a system of two identical particles, each of spin s, the ratio of the num-ber of symmetrical to the number of anti-symmetrical spin states is (s+ l)/s. 7. Show that if the wavefunction of a system of two identical spinless particles is an eigen-function of the orbital angular momentum of relative motion of the two particles, then the quantum number / necessarily has an even (or zero) value.

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  Quantum Chemistry

  • John Lowe(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  CHAPTER 6 POSTULATES AND THEOREMS OF QUANTUM MECHANICS 6-1 Introduction The first part of this book has treated a number of systems from a fairly physical viewpoint, using intuition as much as possible. Now, armed with the concepts already developed, the reader should be in a better position to under- stand the more formal foundation to be described in this chapter. This foundation is presented as a set of postulates. From these follow proofs of various theorems. The ultimate test of the validity of the postulates comes in comparing the theoretical predictions with experimental data. The extra effort required to master the postulates and theorems is repaid many times over when we seek to solve problems of chemical interest. 6-2 The Wavefunction Postulate We have already described most of the requirements that a wavefunction must satisfy, ψ must be acceptable (i.e., single valued, nowhere infinite, continuous, with a piecewise continuous first derivative). For bound states (i.e., states in which the particles lack the energy to achieve infinite separation classically) we require that ψ be square integrable. So far we have considered only cases where the state of the system does not vary with time. For much of quantum chemistry, these are the cases of interest, but, in general a state may change with time, and ψ will be a function of t in order to follow the evolution of the system. Gathering all this together, we arrive at Postulate I Any bound state of a dynamical system ofn particles is described as completely as possible by an acceptable, square-integrable function (# , q 2 , • · ·> #3n> > 2 ,..., ω , t), where the q's are spatial coordinates, ofs are spin coordinates, and t is the time coordinate. * dr is the probability that the space- spin coordinates lie in the volume element dr (= λ dr 2 - · dr n ) at time t, / is normalized.

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  Modern Quantum Mechanics

  • J. J. Sakurai, Jim Napolitano(Authors)
  • 2020(Publication Date)
  • Cambridge University Press

    (Publisher)

  7 Identical Particles This chapter is devoted to a discussion of some striking quantum-mechanical effects arising from the identity of particles. First we present a suitable formalism and the way that nature deals with what appears to be an arbitrary choice. We then consider some applications to atoms more complex than hydrogen like atoms. Next we generalize to systems with many identical particles, and discuss two different formalisms for calculations. Lastly, we will cover one concrete example of a many-particle quantum-mechanical field theory, namely quantizing the electromagnetic field. 7.1 Permutation Symmetry In classical physics it is possible to keep track of individual particles even though they may look alike. When we have particle 1 and particle 2 considered as a system, we can, in principle, follow the trajectory of particle 1 and that of particle 2 separately at each instant of time. For bookkeeping purposes, you may color one of them blue and the other red and then examine how the red particle moves and how the blue particle moves as time passes. In quantum mechanics, however, identical particles are truly indistinguishable. This is because we cannot specify more than a complete set of commuting observables for each of the particles; in particular, we cannot label the particle by coloring it blue. Nor can we follow the trajectory because that would entail a position measurement at each instant of time, which necessarily disturbs the system; in particular the two situations (a) and (b) shown in Figure 7.1 cannot be distinguished – not even in principle. For simplicity consider just two particles. Suppose one of the particles, which we call particle 1, is characterized by |k  , where k  is a collective index for a complete set of observables. Likewise, we call the ket of the remaining particle |k  .

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