Identity state functions

We commonly speak about our ordinary state of consciousness as if it were one. Given our discussions of identity states, we now can see that ordinary consciousness is a collection of identity states. Since most identity states function within the allowable limits of consensus consciousness, they are not recognized as altered states in the way some obviously... 

The terms AG, AH, and AS are state functions and depend only on the identity of the materials and the initial and final state of the reaction. Tables of thermodynamic quantities are available for most known materials (see also Thermodynamic properties) (11,12). 

This list of postulates is not complete in that two quantum concepts are not covered, spin and identical particles. In Section 1.7 we mentioned in passing that an electron has an intrinsic angular momentum called spin. Other particles also possess spin. The quantum-mechanical treatment of spin is postponed until Chapter 7. Moreover, the state function for a system of two or more identical and therefore indistinguishable particles requires special consideration and is discussed in Chapter 8. 

Obviously, the state function (jc) is not an eigenfunction of H. Following the general procedure described above, we expand in terms of the eigenfunctions n). This expansion is the same as an expansion in a Fourier series, as described in Appendix B. As a shortcut we may use equations (A.39) and (A. 40) to obtain the identity... 

While capillary pressure can be determined independently through experiments implementing a series of equilibrium states, this can be very time consuming, particularly if the entire capillary pressure function is to be reconciled. Furthermore, as there can be difficulties in re-establishing identical states of initial saturation, it is most desirable to determine capillary pressure and relative permeability functions simultaneously, from the same experiment. 

Now, we assume that the functions, tcoj, j = 1,. .., N are such that these uncoupled equations are gauge invariant, so that the various % values, if calculated within the same boundary conditions, are all identical. Again, in order to determine the boundary conditions of the x function so as to solve Eq. (53), we need to impose boundary conditions on the T functions. We assume that at the given (initial) asymptote all v / values are zero except for the ground-state function /j and for a low enough energy process, we introduce the approximation that the upper electronic states are closed, hence all final wave functions v / are zero except the ground-state function v /. ... 

A state function is a property of a system that has a value that depends on the conditions (state) of the system and not on how the system has arrived at those conditions (the thermal history of the system). For example, the temperature in a room at a given time does not depend on whether the room was heated up to that temperature or cooled down to it. The difference in any state function is identical for every process that takes the system from the same given initial state to the same given final state it is independent of the path or process connecting the two states. Whereas the internal energy of a system is a state function, work and heat are not. Work and heat are not associated with one given state of the system, but are defined only in a transformation of the system. Hence the work performed and the heat... 

The concept of a state function can be quite difficult, so let us consider a simple example from outside chemistry. Geographical position has analogies to a thermodynamic state function, insofar as it does not matter whether we have travelled from London to New York via Athens or flew direct. The net difference in position is identical in either case. Figure 3.4 shows this truth diagrammatically. In a similar way, the value of A U for the process A C is the same as the overall change for the process A -> B -> C. We shall look further at the consequences of U being a state function on p. 98. 

Figure 3.4 If geographical position were a thermodynamic variable, it would be a state function because it would not matter if we travelled from London to New York via Athens or simply flew direct. The net difference in position would be identical. Similarly, internal energy, enthalpy, entropy and the Gibbs function (see Chapter 4) are all state functions...

The state function property of the enthalpy should be kept in mind for the next move of our discussion. In figure 2.1 we have decomposed reaction 2.1 in a series of steps whose net effect must correspond to the overall reaction. This means that the correct value for Asin//(2) is the solution enthalpy of 1 mol of oxygen in the (ethanol + water) mixture described—and not the solution enthalpy of the gas in pure water. Unfortunately, solution enthalpy data in organic liquid mixtures are not abundant in the chemical literature. So, either we are lucky to find them, we have the equipment to measure them in the laboratory, or we assume that the values will be identical to the ones in the pure solvent. The validity of this assumption depends on the system under discussion and on the accuracy needed for the final result, but in the present case it seems fair. Leaving further discussion to section 2.5, we shall take Asin//(2) = -12 4 kJ mol-1 [17],... 

As we have written it in Equation (8), the DMRG wave function contains redundant variational parameters. This means that the set of variational tensors f/ni... ij/Hk in the DMRG wave function is not unique, because we can find another set of tensors whose matrix product yields an identical state. This redundancy is analogous to the redundancy of the orbital parametrization of the Hartree-Fock determinant. In the case of the DMRG wave function, we can insert a matrix T and its inverse between any two variational tensors and leave the state invariant... 

If we take a large number of identical systems, each with the same state function A, and measure A in each system, we shall in general get different results in different systems. (This is one of the striking differences between classical and quantum mechanics.) The average value of A, denoted by (A), is postulated to be given by... 

Remember that any state of consciousness is a system the parts interact with each other to form a particular pattern. Thus, changes in other subsystems that might not be directly involved in one of the four psi transmission routes we have examined may still have important effects on psi functions. Consider, for instance, the functioning of our Sense of Identity subsystem. We all possess a variety of identities that change rapidly with various situations and emotions, but when a particular identity is functioning, it tends to organize the rest of our mental functioning into a consistent pattern. 

Combination of Eqs. 15 and 16 with the expressions in Table 2 thus finally yields the symmetry adapted (t2g)3 multiplets AA2g, 2Eg + 2Tlg, 2T2g described by five quantum numbers SLTMsMry. These resulting states are identical -within a multiplet dependent phase factor - to the state functions published by Griffith [3],... 

Self-observation, observation of others, and psychoanalytic data indicate that various stimuli can produce marked reorgnaizations of ego functioning very rapdily, even though these all remain within the consensus reality definitions of "normal" consciousness. These identity states are much like d-SoCs and can be sutdied in the systems approach framework. They are hard to observe in ordinary life because of the ease and rapidity of transiton, their emotional charge, and other reasons. The isolation of knowledge and experience in various identity states is responsible for much of the psychopathology of everyday life. 

These alterations in functioning that I call identity states can thus be usefully studied with the systems approach to consciousness. 

Second, all these identity states share much psychological functioning in common, such as speaking English, responding to the same proper name, wearing the same sets of clothes. Thse many common properties amke differences difficult to notice. 

Fourth, a person s identification is ordinarily very high, complete, with each of these identity states. He projects the feeling of "i" onto it (the Sense of identity subsystem function discussed in Chapter 8). This, coupled with the culturally instilled need to believe that he is a single personality, causes him to gloss over distinctions. Thus he says, "I am angry," "I am sad," rather than, "A state of... 

The development of an Observer can allow a person considerable access to observing different identity states. An outside observer can often clearly infer different identity states, but a person who has not developed the Observer function well may never notice his many transitions from one identity state to another. Thus ordinary consciousness, or what society values as "normal" consciousness, may actually consist of a large number of d-SoCs, identity states. But the overall similarities between these identity states and the difficulty of observing them, for the reasons discussed above, lead us to think of ordinary consciousness as relatively unitary state. 

An identity state, like a d-SoC, has coping functions. The culture a person is born into actively inhibits some of his human potentials,... 

Being in a particular identity state also functions as limiting stabilization. The identity leads to selective perception to make perceptions congruent with the reigning identity state. Certain kinds of perceptions that might activate other identity states are repressed. The tortured child is perceived as an "enemy agent," not as a "child." This keeps emotional and attention/awareness energy out of empathic processes that, if activated, would undermine and disrupt the "soldier" identity. 

The additivity of changes of enthalpy on the addition of changes of state is identical to a modification of the law that Hess announced in 1840. This law stated that the heat associated with a chemical reaction was independent of the path used in going from the initial to the final state, and thus the heats of chemical reactions were additive. This statement, of course, is not exactly true, because the heat does depend on the path. It is the changes of the state functions such as the enthalpy that are independent of the path. 

The thermodynamic state functions obtained by non-linear regression in both experiments are identical within the error limits and are considered reliable contrary to the parameters of higher complexation that suffer from dramatically increased errors and their cross correlation. 

These results agree with those obtained previously by analytical methods for the same problem [3]. It should be noted that all the contributions to the above RFs come from the identity operator in equation (11) only as the matrix elements arising from the electronic ground state functions are zero for the orbital operator involved. 

We shall discuss adoption of a convention for the sign for work of expansion -(Frames 7, 9, 14 and 15) and use it when we discuss in more detail the gas expansion processes (Frame 9). Also (FIRST LAW OF THERMODYNAMICS - see Frames 2, 8) the internal energy change, A U for the overall process in Figure 1.1 (i.e. gas at Vf and 7j -> gas at Vf and Tf) (being a state function) is identical for both paths between the SAME initial and final states (and so is route independent). 

In Frame 9 we consider the expansion of an ideal gas along an isotherm (or constant temperature curve for which dT = 0) from (Pi, V ) to (Pf, Vf). Whilst the state functions P and V show identical changes in the two expansion processes considered (dP = Pf — Pi) dV = (Vf — V) the work done in the two cases is entirely different. Two routes (paths) are considered ... 

We note here, as was found in the example of the calculation of the work done during the irreversible adiabatic of expansion of a gas (equation 9.4 Frame 9), that under specific conditions a path dependent function can become identically equal to the change in a state function. 

State functions (Frame 1, section 1.5) are amenable to the construction of thermochemical cycles since provided that we can devise alternative routes between the same initial and final states, then the changes in state functions via these two alternative routes will be identical. 

The entropy of the phase transition, AtiansiS , taking place at a fixed temperature, can be calculated using equation (13.14), Frame 13 and since pressure is constant state function, the enthalpy of transition, Alrans// will be identical both for reversible and irreversible changes, so that ... 

Since the heat supplied, Aq at constant pressure, P is equal to the change in enthalpy, AH, which is itself a state function - and is therefore identical for both reversible and irreversible processes - hence we can write ... 

A Cl state function Pi is expressed as a linear combination of the determinantal wavefunctions 0) formed from configurations of identical symmetry and spin,... 

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