Operator, annihilation

The operator annihilates a spin-up electron at p G A, and the operator b annihilates a spin-down electron The electrostatic operator... 

This operator annihilates gi-zn-2 and the final result is therefore... 

Application of pulses, precession and J-coupling modifies the product operators, annihilates some and creates new operators in the expansion according to a defined set of rules. 

An operator annihilates an electron in spin orbital i, while creates an electron in spin orbital a. The creation and annihilation operators satisfy the usual algebra [a, aj]+ = 6pg. Introducing the generic notation for excitation operators and noting that [r, iv] = 0, we may express the coupled-cluster wave function in the standard exponential form [35]... 

Here ak a ) is the annihilation (creation) operator of an exciton with the momentum k and energy Ek, operator an(a ) annihilates (creates) an exciton at the n-th site, 6,(6lt,) is the annihilation (creation) operator of a phonon with the momentum q and energy u) q), x q) is the exciton-phonon coupling function, N is the total number of crystal molecules. The exciton energy is Ek = fo + tfcj where eo is the change of the energy of a crystal molecule with excitation, and tk is the Fourier transform of the energy transfer matrix elements. 

The opposite of a creation operator is an annihilation operator a which removes orbital i from the wave function it is acting on. The a-a product of operators removes orbital j and creates orbital i, i.e. replaces the occupied orbital j with an unoccupied orbital i. The antisymmetry of the wave function is built into the operators as they obey the following anti-commutation relationships. 

The somewhat awkward antisymmetiizing operator necessarily in first quantization is replaced by formal rules for manipulating creation and annihilation operators. 

It was mentioned above that in order for the AND gate shown in figure 3.82-b to operate properly, the gliders in input A must be delayed by a time equal to the distance between the two annihilation reactions. The same is true of the operation of the OR gate, shown in figure 3.82-c. 

It is clear that if all the program counters arc unoccupied (i.e. are in the state I 0 >), nothing at all happens all terms in the Hamiltonian start out with an annihilation operator, and all states thus remain in the state 0 > for all time. If we assume that only one of the sites 0,1,... fc sites is occupied, however, we see that only one site will always be occupied, though not necessarily the same site at different times. If we think of the occupied site, say the first site 0, as a cursor, the Hamiltonian effectively moves the cursor along the program counter sites while the operators Ai operate on the register n. Feynman shows how, by the time the cursor arrives at the final site fc, the n register has been multiplied by the entire set of desired operators Tj, T2,..., A -... 

In the context of fra/u-polyacetylene cjia and c are, respectively, the creation and annihilation operators of an electron with spin projection a in the n-orbital of the nth carbon atom (n= l,...,N) that is perpendicular to the chain plane (see Fig. 3-3). Furthermore, u is the displacement along the chain of the nth CH unit from its position in the undimerized chain, P denotes the momentum of this unit, and M is its mass. 

Thus, the remainder operator, R, is a linear operator that annihilates every function t of the basic set to be used in the approximation. 

Creation and Annihilation Operators.—In the last section there was a hint that the theory could handle problems in which populations do not remain constant. Thus < , < f>s 2 is the probability density in 3A -coordinate space that the occupation numbers are , and the general symmetrical state, Eq. (8-101), is one in which there is a distribution of probabilities over different sets of occupation numbers the sum over sets could easily be extended to include sets corresponding to different total populations N. 

The operator bx annihilates a particle from the A-state, while the operator 6J creates a particle in the A-state, leaving the other states unchanged the total population of the system changes by unity in each case. The numerical factors are chosen so that the product of the two operators in the appropriate order is given by... 

If we operate on the vacuum state with any annihilation operator the result is the null vector ... 

The state w, f>s is an eigenstate of N with eigenvalue N, and N is called the total population operator. Because the vector , Os is a function of the time, it is necessary to specify the time at which the creation or annihilation operators are applied, and in some discussions it may be advisable to indicate the time explicitly in the symbol for the operator. For our present discussion it will be sufficient to keep this time dependence in mind. In an expression such as Eq. (8-109), all the creation operators are applied at the same time, and since they all commute, this presents no logical problem. The order of the operators in the definition Eq. (8-107) is important however the opposite order produces a different operator ... 

When dealing with systems described by antisymmetrical states, the creation and annihilation operators are defined in such a way that the occupation numbers can never be greater than unity. Thus we have a creation operator af defined by... 

At the end of Section 8.16 we mentioned that the Fock representation avoids the use of multiple integrations of coordinate space when dealing with the many-body problem. We can see here, however, that the new method runs into complications of its own To handle the immense bookkeeping problems involved in the multiple -integrals and the ordered products of creation and annihilation operators, special diagram techniques have been developed. These are discussed in Chapter 11, Quantum Electrodynamics. The reader who wishes to study further the many applications of these techniques to problems of quantum statistics will find an ample list of references in a review article by D. ter Haar, Reports on Progress in Physics, 24,1961, Inst, of Phys. and Phys. Soc. (London). 

One speaks of Eqs. (9-144) and (9-145) as a representation of the operators a and o satisfying the commutation rules (9-128), (9-124), and (9-125). The states 1, - , ) = 0,1,2,- are the basis vectors spanning the Hilbert space in which the operators a and oj operate. The representation (9-144) and (9-145) is characterized by the fact that a no-particle state 0> exists which is annihilated by a, furthermore this representation is irreducible since in this representation a(a ) operating upon an n-particle state, results in an n — 1 ( + 1) particle state so that there are no invariant subspaces. Besides the above representation there exist other inequivalent irreducible representations of the commutation rules for which neither a no-particle state nor a number operator exists.8... 

This operator is a destruction operator and has the property that it annihilates the vacuum... 

A state of m particles and n antiparticles can be constructed from the no-particle state 0>, which now is annihilated by both the b and the On operators ... 

In formulating the second-quantized description of a system of noninteracting fermions, we shall, therefore, have to introduce distinct creation and annihilation operators for particle and antiparticle. Furthermore, since all the fermions that have been discovered thus far obey the Pauli Exclusion principle we shall have to make sure that the formalism describes a many particle system in terms of properly antisymmetrized amplitudes so that the particles obey Fermi-Dirac statistics. For definiteness, we shall in the present section consider only the negaton-positon system, and call the negaton the particle and the positon the antiparticle. 

We shall denote the creation and annihilation operators for a negaton of momentum p energy Ep = Vp2 + m2 and polarizations by 6 (p,s) and 6(p,s) respectively. In the following, by the polarization we shall always mean the eigenvalue of the operator O-n, where O is the Stech polarization operator and n some fixed unit vector. We denote the creation and annihilation operators for a positon (the antiparticle) of momentum q energy = Vq2 + m2, polarization t, by d (q,t) and... 

This confirms our interpretation of the operators 6,6 and d,d as creation and annihilation operators for particles of definite momentum and energy. Similar consideration can be made for the angular momentum operator. The total electric charge operator is defined as... 

Definition of Normal Product.—Given a product of free field creation and annihilation operators U,X,- -, FF, we define the operator N as... 

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