The Vacuum State

We introduce first the concept of the vacuum state. The vacuum state is simply an abstract state which is empty. That is, it contains no electron. It is denoted by  

The concept of the vacuum state seems to be rather abstract, and in fact, it is. One often asks whether the vacuum state has any physical meaning or significance. The answer is no the vacuum state does not describe any real physical system. It is merely an abstract mathematical entity which will turn to be useful throughout the following development. In general, one must realize that only expectation values may have any real physical significance in quantum theory the rest is just mathematics. 

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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 problem now is to find the corresponding Hamiltonian, t Hooft shows that the most obvious construction, obtained by rewriting U(t+l,t) as a product of cyclic elements, unfortunately does not work because at the end of the calculation there is no way to uniquely define the vacuum state. Given a cellular automaton with a local unitary evolution operator U = WgUg and the commutator [Ug, Ug ] 0 if [ af — af j> d for some d > 0, the real problem is therefore to find a Hamiltonian... 

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

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 ... 

These fluctuations will affect the motion of charged particles. A major part of the Lamb shift in a hydrogen atom can be understood as the contribution to the energy from the interaction of the electron with these zero point oscillations of the electromagnetic field. The qualitative explanation runs as follows the mean square of the electric and magnetic field intensities in the vacuum state is equal to... 

It should be noted that by virtue of Eq. (9-693), A x) is self-adjoint within the indefinite metric. The vacuum state can now be characterized by the relation... 

In order to interpret the above results, consider the expectation value of the total energy density in the vacuum state, i.e., of the hamiltonian density, Eq. (10-12). There is a contribution J u(x)Al(x) from the external field and a contribution m<0 j (a ) 0)ln 4 (a ) from the induced current, hence to lowest order... 

As no confusion can arise, we shall often in the subsequent exposition omit the in or out subscript and denote the vacuum state by 0>. 

In arriving at Eq. (11-249) we have made use of Eq. (11-241), of the (pseudo)vector character of the surface element dau(x) and of the invariance of the vacuum state expressed by Eq. (11-239). We now insert into the right-hand side of Eq. (11-249) the expansion of iftin(x) in terms of operators, and find... 

In 1995, one of the authors (A.K.) introduced the state of a molecule embedded in a perfect conductor as an alternative reference state, which is almost as clean and simple as the vacuum state. In this state the conductor screens all long-range Coulomb interactions by polarization charges on the molecular interaction surface. Thus, we have a different reference state of noninteracting molecules. This state may be considered as the North Pole of our globe. Due to its computational accessibility by quantum chemical calculations combined with the conductor-like screening model (COSMO) [21] we will denote this as the COSMO state. 

A different view of the OMT process is that the molecule, M, is fully reduced, M , or oxidized, M+, during the tunneling process [25, 26, 92-95]. In this picture a fully relaxed ion is formed in the junction. The absorption of a phonon (the creation of a vibrational excitation) then induces the ion to decay back to the neutral molecule with emission (or absorption) of an electron - which then completes tunneling through the barrier. For simplicity, the reduction case will be discussed in detail however, the oxidation arguments are similar. A transition of the type M + e —> M is conventionally described as formation of an electron affinity level. The most commonly used measure of condensed-phase electron affinity is the halfwave reduction potential measured in non-aqueous solvents, Ey2. Often these values are tabulated relative to the saturated calomel electrode (SCE). In order to correlate OMTS data with electrochemical potentials, we need them referenced to an electron in the vacuum state. That is, we need the potential for the half reaction ... 

Hence Eip is a new eigenfunction with eigenvalue (E + Etl). By applying the operator E n times, starting from the vacuum state (i/j0,E0), the nt 1 excited eigenfunction... 

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... 

This condition applies not only to the vacuum state, but to any arbitrary state which can only mean that aj/zt = 0. To incorporate this condition into the previous set of commutation rules for Bose fields, it is sufficient to change the negative into positive signs, such that... 

When the vacuum is relieved with pure nitrogen, the moles of oxidant are the same as in the vacuum state and the moles of nitrogen increase. The new (lower) oxidant concentration is... 

For a two-electron system in 2m-dimensional spin-space orbital, with and denoting the fermionic annihilation and creation operators of single-particle states and 0) representing the vacuum state, a pure two-electron state ) can be written [57]... 

It was found earlier that a sudden frequency change during an electronic Franck-Condon transition leads to special quantum mechanical statistics, called squeezing [2-9], of the molecular vibrations [10-12], A state is termed squeezed if some of its characteristics have less noise than the corresponding quantum noise of the vacuum state. The concept of squeezing turned out to be very fruitful in basic research and implies a lot of promising practical possibilities. 

This definition is consistent with the definition of overlap between two Slater-determinants having the same number of electrons. The overlap between Slater determinants having a different number of electrons is not defined. The extension to have a well-defined, but zero, overlap between two occupation number vectors with different numbers of electrons is a special feature of the Fock-space formulation that allows a unified description of systems with a different number of electrons. As a special case of Eq. (1.3), the vacuum state is defined to be normalized... 

An annihilation operator times the vacuum state still vanishes so the effect of an annihilation operator times an occupation number vector becomes... 

The corresponding transformation of the spin-orbitals is obtained by multiplying (3 26) with an a or P spin function. When we make the transformation from one set of spin-orbitals to the other, the annihilation and creation operators will change. The following relations are easily established, by operating with the creation operator in the primed space on the vacuum state ... 

We have thus shown the relations (3 28) for the transformation of the annihilation and creation operators to a new spin-orbital basis. We can use these relations to express an arbitrary Slater determinant in the new basis in terms of the determinants in the original basis. In order to do so, we generate the Slater determinant by applying a sequence of creation operators on the vacuum state ... 

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... 

In principle, an A-electron wave function can always be represented as a certain combination of creation operators acting on the vacuum state. Specifically, for the one-determinant wave function (13.1) we have... 

In the theory of many-electron atoms, the particle-hole representation is normally used to describe atoms with filled shells. To the ground state of such systems there corresponds a single determinant, composed of one-electron wave functions defined in a certain approximation. This determinant is now defined as the vacuum state. In the case of atoms with unfilled shells, this representation can be used for the atomic core consisting only of filled shells. Then, the excitation of electrons from these shells will be described as the creation of particle-hole pairs. 

The utilization of the particle-hole representation to describe the wave functions of unfilled shells yields no substantial simplifications. As has been noted above, these wave functions can be expressed in terms of linear combinations of one-determinant wave functions. Selection of one of these determinants as the vacuum state yields a linear combination of one-determinant wave functions, now described using the particle-hole representation. 

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... 

The rank of the operator q> in quasispin space will only be determined by the number of electrons N and can be ignored as a characteristic. In particular, operator - a tensor of rank one in quasispin space - acting on the vacuum state gives rise to the two-electron wave function with v = 0... 

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 connection between appropriate quantities for partially and almost filled shells can be established by another method - by going over from the particle representation to the hole one. According to the results of Chapters 2, 13 and 14, the wave function of the completely filled shell is at the same time the vacuum state for holes... 

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