Creation and Annihilation Operators

6 Key excerpts on "Creation and Annihilation Operators"

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  Dynamics of Classical and Quantum Fields

  An Introduction

  • Girish S. Setlur(Author)
  • 2013(Publication Date)
  • CRC Press

    (Publisher)

  Chapter 8 Creation and Annihilation Operators in Fock Space In this chapter, we discuss the notion of Creation and Annihilation Operators. These operators correspond to addition or removal of excitations of a system of particles or particles themselves. We show that these operators may be used to rewrite many-body Hamiltonians in a compact form where information about Bose or Fermi statistics is encoded in the Hamiltonian itself, which greatly reduces the effort re-quired in studying its properties. 8.1 Introduction to Second Quantization The older term for rewriting quantum formalisms using Creation and Annihilation Operators in place of position and momentum operators is ‘Second Quantization’. Second Quantization does not mean quantizing twice! We start by discussing the simple harmonic oscillator. Consider the Hamiltonian, H = p 2 2 m + 1 2 m ω 2 x 2 = E 0 p 2 2 mE 0 + m ω 2 x 2 2 E 0 . (8.1) This looks like H = E 0 ( X 2 + Y 2 ) = E 0 ( Y + iX )( Y -iX ) -iE 0 [ X , Y ] where Y = p √ 2 mE 0 and X = q m 2 E 0 ω x for some appropriate choice of E 0 . Set a † = Y + iX ; a = Y -iX (8.2) since X and Y are Hermitian. We are going to choose E 0 by demanding that, [ a , a † ] = 1. We see that [ X , Y ] = q m 2 E 0 ω i ~ √ 2 mE 0 but [ a , a † ] = -2 i [ X , Y ] = ~ ω E 0 = 1 189 190 Field Theory Figure 8.1: Shows a chain of masses and springs meant to illustrate the concept of a field. or E 0 = ~ ω . Thus, H = E 0 ( Y + iX )( Y -iX ) -iE 0 [ X , Y ] = ~ ω a † a + 1 2 ~ ω . (8.3) Here, a † creates an excitation or a quantum of energy ~ ω . The quantity N = a † a measures the number of quanta or excitations in a state this operator acts on. Next we consider a chain of harmonic oscillators. It is described by, H = N ∑ j = 1 p 2 j 2 m + 1 2 m ω 2 N -1 ∑ j = 1 ( x j + 1 -x j ) 2 . (8.4) We wish to rewrite this using Creation and Annihilation Operators.

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  Quantum Mechanics for Scientists and Engineers

  • David A. B. Miller(Author)
  • 2008(Publication Date)
  • Cambridge University Press

    (Publisher)

  The quantized wave field model introduced in the previous Chapter, of course, has both wave and particle characters, as required. 384 Chapter 16 Fermion operators creation operators does not fundamentally add anything to the quantum mechanics we already have for fermions (at least, at the level to which we consider the quantum mechanics of electrons and similar fermion particles here). It does, however, give us a very convenient way of writing the quantum mechanics and also gives us a formalism that is of the same type as the boson Creation and Annihilation Operators we created previously. This approach naturally includes the Pauli exclusion principle so we do not have to add it in some ad hoc fashion. Once we work in systems with many fermions, the use of fermion Creation and Annihilation Operators is almost essential from a practical point of view so as to keep track of the fermion character of systems with many particles. Even when we are only considering a single fermion, the Creation and Annihilation Operators give a particularly simple notation that we can use to describe other operators, such as the Hamiltonian. This kind of description is particularly useful for processes that involve collisions of fermions with one another (e.g., as in electron-electron scattering) or the interaction of fermions and bosons (e.g., as in optical absorption and emission or electron-phonon scattering). 16.1 Postulation of fermion annihilation and creation operators The approach we take here is simply to postulate annihilation and creation operators for fermions, giving them the required properties. Later, we use them to rewrite operators involving interactions with fermions. The key property these operators require, in comparison to the boson operators, is that they correctly change the sign of the wavefunction upon exchange of particles.

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  Many-Body Methods in Chemistry and Physics

  MBPT and Coupled-Cluster Theory

  • Isaiah Shavitt, Rodney J. Bartlett(Authors)
  • 2009(Publication Date)
  • Cambridge University Press

    (Publisher)

  Obviously, the ordinary form 54 3.2 Creation and Annihilation Operators 55 of the Hamiltonian, ˆ H = N  µ=1 ˆ h µ + N  µ<ν ˆ v µν (3.3) where ˆ h µ and ˆ v µν are the one-particle and two-particle terms in ˆ H and µ, ν are particle labels, does not satisfy this requirement because N appears in the summation limits. 3.2 Creation and Annihilation Operators 3.2.1 Definitions We begin by considering the representation of a normalized Slater determi- nant Φ = Φ ijk...z ≡ Aφ i φ j φ k · · · φ z ≡ |φ i φ j φ k · · · φ z  ≡ | ijk · · · z  , (3.4) where A is the antisymmetrizer and each φ is a spinorbital in our one-particle basis. The various forms of (3.4) are equivalent notations for the same function, and we shall normally use one of the last two forms. The Slater determinant (SD) Φ is represented in second-quantized form by specifying the occupancies (or occupation numbers ) n 1 , n 2 , . . . of the basis spinorbitals φ 1 , φ 2 , . . . in the determinant. Obviously, n i (Φ) =  0 if φ i is empty (not present) in the SD Φ 1 if φ i is occupied (present) in the SD Φ (i = 1, 2, . . .) . (3.5) The determinant itself (and various operators on it) are represented in terms of a set of Creation and Annihilation Operators. The notation for these varies: creation operator for spinorbital φ i , ˆ X † i , ˆ a † i ; ˆ c † i , ˆ i † ; annihilation operator for spinorbital φ i , ˆ X i , ˆ a i , ˆ c i , ˆ i . Here we shall use ˆ a † i , ˆ a i and, when there is no possibility of confusion, ˆ i † , ˆ i. They are defined in terms of their action on SDs: ˆ a † i |jk · · · z  = | ijk · · · z  , ˆ a i | ijk · · · z  = |jk · · · z  . (3.6) It is convenient to arrange the spinorbitals in an SD in lexical order as | ijk · · · z  , where i < j < k < · · · < z , 56 Second quantization and therefore it is necessary to determine the effects of the creation and an- nihilation operators when the affected orbital is not (or is not being placed) at the beginning of the SD.

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  Lectures On Quantum Mechanics

  • Gordon Baym(Author)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  Again we can introduce Creation and Annihilation Operators defined. by a 0 | n 0, n 1 〉 = n 0 | n 0 – 1, n 1 〉 a 0 † | n 0, n 1 〉 = n 0 + 1 | n 0 + 1, n 1 〉 a 1 | n 0, n 1 〉 = n 1 | n 0, n 1 – 1 〉 a 0 † | n 0, n 1 〉 = n 1 + 1 |[--=PLGO-SEP. ARATOR=--]n 0, n 1 + 1 〉. (19-12) a 0 destroys a particle in the state φ 0, a 1 † creates a particle in the state φ 1, etc. It is trivial to show that from (19-12) that [ a 0, a 0 † ] = 1, [ a 1, a 1 † ] = 1 and furthermore that the “0” operators commute with the “1” operators [ a 0, a 1 ] = 0, [ a 0 †, a 1 † ] = 0 [ a 0, a 1 † ] = 0, [ a 0 †, a 1 ] = 0, since for bosons it makes no difference in what order one performs an operation such as adding a particle to one level and removing one from the other level. Again, all the states. |n 0, n 1 〉 can be constructed from the “vacuum” |0, 0〉 by acting with a 0 † and a 1 † repeatedly: | n 0, n 1 〉 = (a 1 †) n 1 n 1 ! (a 0 †) n 0 n 0 ! | 0, 0 〉. (19-13) The operator a 0 †a 0 is the operator for the number of particles in the state φ 0 and a 1 † a 1 measures the number of particles in the state φ 1. Then N = a 0 † a 0 + a 1 † a 1 (19-14) is the total number operator: N | n 1, n 2 〉 = (n 1 + n 2) | n 1, n 2 〉. (19-15) For fermions occupying the two levels φ 0 and φ 1 (again with their spins up, say) there are four possible states |n 0,. n 1 〉: | 0, 0 〉, | 0, 1 〉, | 1, 0 〉, | 1, 1 〉. We first introduce the Creation and Annihilation Operators a 1 † and a 1 defined by the. operations a 1 † | 0, 0 〉 = | 0, 1 〉, a 1 † | 1, 0 〉 = | 1, 1 〉 a 1 † | 0, 1 〉 = a 1 † | 1, 1 〉 = 0 (19-16) a 1 | 0, 0 〉 = a 1 | 1, 0 〉 = 0 a 1 | 0, 1 〉 = | 0, 0 〉, a[--=PLGO-SEPARATOR=--. ]1 | 1, 1 〉 = | 1, 0 〉. (19-17) These operators create or destroy particles with the single particle wave function φ 1

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  The Quantum Revolution

  A Historical Perspective

  • Kent A. Peacock(Author)
  • 2007(Publication Date)
  • Greenwood

    (Publisher)

  In the early years of quantum field theory, the process of turning field variables into Creation and Annihilation Operators was called second quantization. The move from ordinary quantization to second quantiza- tion is a move from quantum mechanics of a single particle to that of many particles. The term “second quantization” is now largely a historical curiosity, but the procedure is not. Physically, the creation and annihilation of particles is a consequence of Einstein’s equivalence of mass and energy in combination with the laws of quantum mechanics. If a particle possesses a certain amount of energy in ad- dition to its rest mass, it can emit a particle or particles with that amount of energy, so long as all conservation laws are respected. For instance, the elec- trical charges and spins have to add up. In quantum mechanics, if something can happen (according to conservation laws) then there is an amplitude for it to happen, and therefore a probability (sometimes vanishingly small, sometimes not) that it will happen. Hence, so long as there is enough energy in a system to allow for the creation of particles or their annihilation and transformation into other particles, then it will sooner or later occur. One of the first successes of early QED in the hands of Dirac and others was that it gave methods for calculating the amplitudes for spontaneous and in- duced emission that Einstein had identified in 1917, and thus allowed for the most general derivation of Planck’s radiation law yet found. Early QED was most successful in dealing with free fields, fields that are not in the presence of matter; describing how the electromagnetic field interacts with matter proved to be a much harder task. Early in the 1930s Dirac predicted another bizarre quantum phenome- non, vacuum polarization. It is a consequence of pair creation. Consider an electron—a bare electron—sitting in the vacuum.

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  Statistical Mechanics

  A Set Of Lectures

  • Richard P. Feynman(Author)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  CHAPTER 6 Creation and Annihilation Operators 6.1 A SIMPLE MATHEMATICAL PROBLEM In this chapter we shall describe an operator formalism that has widespread applications in quantum mechanics, notably in dealing with harmonic oscillators and in describing many-particle systems. We begin by formulating and solving the following simple problem: Suppose an operator a satisfies |> ,0 +] = 1. (6.1) The problem is to find the eigenvalues of the Hermitian operator a+a , and to relate the eigenvectors. (Note: a+ denotes the Hermitian conjugate of a , and [A , B ] is, of course, the commutator AB — BA.) We first note that, if |a) is a normalized eigenvector with a +aoc ) = a|a>, (6.2) then a = { ||2 > 0. (6.3) That is, the eigenvalues are all real and nonnegative. Using the identity [ AB , C] = A[B, C] + [A, C~B , we observe that [, a+a , a = [a+, a]a = —a, (6.4) [< a+a , a +] -I- a+[a, a +] = a +; (6.5) or, equivalently, (< a+a)a = a(a+a — 1), (6.4') (a+a)a+ = a+(a+a + 1). (6.5') From Eq. (6.4') we have, for an eigenvector |a>, (i a+a)a(x ) = a(a+a — l)|a) = a{ a — l)|a) = (a — l)a|a). (6.6) Therefore a|a) is an eigenvector with eigenvalue a — 1, unless a || = Va. (6.7) Similarly, ||a+1a) || = Va + 1. (6.8) Now, suppose that c^cc) # 0 for all n. Then by repeated application of Eq. (6.6), o|a) is an eigenvector of a +a with eigenvalue a -n. This contradicts Eq. (6.3), because a — n < 0 for sufficiently large n. Therefore we must have ctd ) # 0 but fl+1|a> = 0 (6.9) for some nonnegative integer n. Let |a — ri) = d 1 ri)/

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