Operator, creation

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. 

There is another commonly used notation known as second quantization. In this language the wave function is written as a series of creation operators acting on the vacuum state. A creation operator aj working on the vacuum generates an (occupied) molecular orbital i. 

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. 

Operating on the vacuum state with a succession of creation operators, on the other hand, permits us to build up a system with any desired population ... 

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... 

Whether this concept can stand up under a rigorous psychological analysis has never been discussed, at least in the literature of theoretical physics. It may even be inconsistent with quantum mechanics in that the creation of a finite mass is equivalent to the creation of energy that, by the uncertainty principle, requires a finite time A2 A h. Thus the creation of an electron would require a time of the order 10 20 second. Higher order operations would take more time, and the divergences found in quantum field theory due to infinite series of creation operations would spread over an infinite time, and so be quite unphysical. 

Similarly the state o p> is a state with energy-momentum eigenvalues Pu + K(h = Vk2 + w2). These facts further validate the interpretation of Ok and ojf as destruction and creation operators for a particle of momentum k and energy Vk2 + m2. 

We define the configuration space Heisenberg creation operator by the equation... 

Next we establish the connection of the previous formalism with the Fo< space description of photons. From the interpretation of a>(k) as the number operator for photons of momentum k polarization A, and of cA(k) and cA(k) as destruction and creation operators for... 

The proof is by induction. It is clearly true for two factors since then it reduces to the definition of the contraction symbol. Furthermore, it is sufficient to prove the theorem under the assumption that Z is a creation operator and that all the operators UV XY are destruction operators. If UV- - -XY are all destruction operators and Z is a creation operator, we may then add any number of creation operators to the left of all factors on both sides of Eq. (10-196) within the N product, without impairing the validity of our theorem, since the contraction between two creation operators gives zero. If on the other hand Z is a destruction operator and UV - - - are creation operators, then Eq. (10-196) reduces to a trivial identity... 

Creation operator, 505 representation of, 507 Critical value, 338 Crystallographic point groups irreducible representations, 726 Crystallographic symmetry groups construction of mixed groups, 728 Crystal, eigenstates of, 725 Crystal symmetry, changes in, 758 Crystals... 

D. V. Averim and K. K. Likharev developed a theory for describing the behavior of small tunneling junctions based on electron interactions. They had started from previous work on Josephson junctions (Likharev and Zorin 1985, Ben-Jacob 1985, Averin and Likharev 1986b) and established the fundamental features of the single-charging phenomena. Their work is based on a quantization theory and handles the tunneling phenomenon as a perturbation, described by annihilation and creation operators of a Hamiltonian. 

Here, n denotes a number operator, a creation operator, c an annihilation operator, and 8 an energy. The first term with the label a describes the reactant, the second term describes the metal electrons, which are labeled by their quasi-momentum k, and the last term accounts for electron exchange between the reactant and the metal Vk is the corresponding matrix element. This part of the Hamiltonian is similar to that of the Anderson-Newns model [Anderson, 1961 Newns, 1969], but without spin. The neglect of spin is common in theories of outer sphere reactions, and is justified by the comparatively weak electronic interaction, which ensures that only one electron is transferred at a time. We shall consider spin when we treat catalytic reactions. 

The analytical description of high-frequency line shapes becomes possible in the low-temperature limit, i.e., at n((uK) exp -p/jcoK l, which represents an experimentally important case. In this situation, the Wick coupling for the operators of low-frequency modes in expression (A3.19) involves only the terms in which the annihilation operator is to the left of the creation operator in all but one operator pair. Then Eq. (A3.19) can be written as ... 

The functions fk and are the counterparts of the so-called destruction (annihilation) and creation operators in the Heisenberg-Dirac picture. It is noted in anticipation that these operators occur as the solutions a,k(t) = lulkt of the Hamiltonian equation... 

To establish the appropriate commutation rules the new creation operator is applied twice to the vacuum state 0 in order to create two particles in the same state g by forming 0. Since this two-particle state is not... 

In terms of the creation operator of second quantization each energy level has an eigenfunction... 

Defining the time- and temperature-dependent annihilation and creation operators through the Bogoliubov transformation... 

The main idea of TFD is the following (Santana, 2004) for a given Hamiltonian which is written in terms of annihilation and creation operators, one applies a doubling procedure which implies extending the Fock space, formally written as Ht = H H. The physical variables are described by the non-tilde operators. In a second step, a Bogolyubov transformation is applied which introduces a rotation of the tilde and non-tilde variables and transforms the non-thermal variables into temperature-dependent form. This formalism can be applied to quite a large class of systems whose Hamiltonian operators can be represented in terms of annihilation and creation operators. 

The salesperson uses the Ordering system s Order creation operation. 

In (4.3), all the occupation numbers remain unchanged, except nk. Likewise, the creation operator c is defined by... 

Because the creation operators (4 (E) for the eigenfunctions of El diagonalize the Hamiltonian, we have... 

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