Vacuum state, second quantization

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. 

Many-electron wave functions in second-quantization form can conveniently be represented in an operator form. To this end, we shall introduce the vacuum state 0), i.e. the state in which there are no particles. We shall define it by... 

Since the wave functions with N > v can be found from the wave functions with N = v using (16.1), in the second-quantization representation it is necessary to construct in an explicit form only wave functions with the number of electrons minimal for given v, i.e. IolQLSMq = —Q). But even such wave functions cannot be found by generalizing directly relation (15.4) if operator cp is still defined so that it would be an irreducible tensor in quasispin space, then the wave function it produces in the general case will not be characterized by some value of quantum number Q v). This is because the vacuum state 0) in quasispin space of one shell is not a scalar, but a component of a tensor of rank Q = l + 1/2... 

Using relationships (16.5) and (16.6) and taking into account the ten-sorial structure of the vacuum state we find that the second-quantized operators q>(lv)v(LS), that produce from the vacuum the function with N = v = 2 and N = v = 3, have the form... 

The notation concerns are easily overcome by the following simple construct bearing the name of second quantization formalism.21 Let us consider the space of wave functions of all possible numbers of electrons and complement it by a wave function of no electrons and call the latter the vacuum state vac). This is obviously the direct sum of subspaces each corresponding to a specific number of electrons. It is called the Fock space. The Slater determinants eq. (1.137) entering the expansion eq. (1.138) of the exact wave function are uniquely characterized by subsets of spin-orbitals K = k,, k2,..., fc/v which are occupied (filled) in each given Slater determinant. The states in the list are the vectors in the carrier space of spin-orbitals (linear combinations of the functions of the (pk (x) = ma (r, s) basis. We can think about the linear combinations of all Slater determinants, may be of different numbers of electrons, as elements of the Fock space spanned by the basis states including the vacuum one. 

Fiow does Wick s theorem help us in evaluating matrix elements of second-quantized operators Recall that any matrix element of an operator may be written as a vacuum expectation value by simply writing its left- and right-hand determinants as operator strings acting on the vacuum state, I ). The... 

In a non-relativistic theory we would now continue by adding a second quantized operator for two-body interactions. In the relativistic case we need to step back and first consider the interpretation of the eigenvalues of the Hamiltonian. Dirac stated that positrons could be considered as holes in an infinite sea of electrons . In this interpretation the reference state for a system with neither positrons nor electrons is the state in which all negative energy levels are filled with electrons. This vacuum state... 

Here (P)0 7 ) is the second-quantized operator producing the relevant normalized wave function out of vacuum, i.e. it simply generalizes the relationship (15.4) to the case of tensors with an additional (isospin) rank. Since the vacuum state is a scalar in isospin space (unlike quasispin space), the expressions for wave functions and matrix elements of standard quantities in the spaces of orbital and spin momenta can readily be generalized by the addition of a third (isospin) rank to two ranks in appropriate formulas of Chapter 15. 

Goldstone used a second quantized particle-hole formalism based on an arbitrary choice of vacuum state. The interaction representation, which is intermediate between the Schrddinger and Heisenberg pictures, was employed and the energy was evaluated by the Gell-Mann-Low formalism78 with Hamiltonian... 

A final comment about wavefunctions. We already have analytical expressions for wavefunctions of the harmonic, Morse, and Poschl-Teller potentials in the one-dimensional case. They can be obtained in terms of the single coordinate in a conventional differential approach. Instead, in the algebraic framework (or in a second-quantization scheme, more generally speaking), wavefunctions are expressed in terms of boson annihilation-creation operators acting on the Fock vacuum state, as shown in Eq. (2.29). In the specific case of U(2), for example, the basis state N,m) can be written (created) as... 

In the second-quantized operators (31) and (32), the summation over the particle indices i,j,... runs over all the electron states of the (complete one-electron) spectrum. If these operators act to the right upon the reference state, i.e. the many-electron vacuum of the particle-hole formalism, some of these (strings of) creation and annihilation operators create excitations while other gives simply zero, i.e. no contribution. For the pure vacuum, in particular, the behavior of the second-quantized operators can be read off quite easily because the creation operators appear left of the annihilation operators in expressions (31) and (32), respectively. 

All the properties of Slater determinants are contained in the anticommutation relations between two creation operators (Eq. (2.194)), between two annihilation operators (Eq. (2.208)), and between a creation and an annihilation operator (Eq. (2.217)). In order to define a Slater determinant in the formalism of second quantization, we introduce a vacuum state denoted by >. The vacuum state represents a state of the system that contains no electrons. It is normalized. 

Let us turn now to the problem of creating more than two particles consider a many-electron one-determinantal wave function in its second quantized representation. As is easily seen from the previous example for the two-electron case, a many-electron one-determinantal wave function is constructed by successive application of creation operators on the vacuum state. It will be useful though to consider this problem from a somewhat different point of view. 

Besides the apparent similarities. Table 8.1 illustrates also the obvious formal differences between bras and kets and their second quantized counterparts. Namely, the corresponding symbols are mathematically very different. The bra and ket vectors are elements of a linear vector space over which quantum-mechanical operators are defined, while the creation and annihilation operators are defined over the abstract space of particle number represented wave functions serving as their carrier space. This carrier space leads to the concept of the vacuum state, which has no analog in the bra-ket formalism. Moreover, an essential difference is that the effect of second quantized operators depends on the occupancies of the one-electron levels in the wave function, since no annihilation is possible from an empty level and no electron can be created on an occupied spinorbital. At the same time, the occupancies of orbitals play no role in evaluating bra and ket expressions. Of course, both formalisms yield identical results after calculating the values of matrix elements. 

Hartree-Fock wave function, and, in fact, the most general derivation of the Hartree-Fock equations is possible through the Brillouin theorem which can be proved directly from the variation principle (Mayer 1971,1973,1974). We shall not prove here the complete equivalence of the Hartree-Fock equations and Eq. (11.1), it will be shown only that the Brillouin theorem is fulfilled for the Hartree-Fock wave function. The proof will make use of second quantization which helps us to evaluate the matrix element easily. To this goal, Eq. (11.1) should be rewritten in the second quantized notation. The ground state is simply represented the Fermi vacuum ... 

In second quantization. Slater determinants are expressed as products or strings of creation operators aj, working on the vacuum state... 

The existence of a convenient reference state means that when using second-quantization methods we do not need to start always from the vacuum state vac) we can instead use... 

However, before going into a detailed discussion of various relativistic Hamiltonians we will introduce an alternative form of the electronic Hamiltonian (3.4), which is useful for wavefunction-based correlation methods. It is obtained by switching to a particle-hole formalism and then introducing normal ordering. In the second-quantization formalism creation and annihilation operators refer to some specific set of (orthononnal) orbitals, and Slater determinants in Hilbert space translate into occupation-number veetors in Fock space. The annihilation operators in equation 3.4 by definition give zero when acting on the vacuum state... 

Second quantization treats operators and wave functions in a unified way - they are all expressed in terms of the elementary creation and annihilation operators. This property of the second-quantization formalism can, for example, be exploited to express modifications to the wave function as changes in the operators. To illustrate the unified description of states and operators afforded by second quantization, we note that any ON vector may be written compactly as a string of creation operators working on the vacuum state (1.2.4)... 

A tensor operator working on the vacuum state generates a set of spin eigenfunctions with total and projected spins S and M (provided the tensor operator does not annihilate the vacuum state). We may prove this assertion in the following way. Since the second-quantization spin-component... 

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