Reduction of the wave function

The measurement process is one of the most controversial areas in quantum mechanics. Just how and at what stage in the measurement process reduction occurs is unclear. Some physicists take the reduction of as an additional quantum-mechanical postulate, while others claim it is a theorem derivable from the other postulates. Some physicists reject the idea of reduction [see M. Jammer, The Philosophy of Quantum Mechanics, ley, 1974, Section 11.4 L. E. Ballentine, Am. J. Phys, 55, 785 (1987)]. Ballentine advocates Einstein s statistical-ensemble interpretation of quantum mechanics, in which the wave function does not describe the state of a single system (as in the orthodox interpretation) but gives a statistical description of a collection of a large number of systems each prepared in the same way (an ensemble) in this interpretation, the need for reduction of the wave function does not occur. [See L. E. Ballentine, Am. J. Phys, 40,1763 (1972) Rev. Mod. Phys, 42,358 (1970).] There are many serious problems with the statistical-ensemble interpretation [see Whitaker, pp. 213-217 D. Home and M. A. B. Whitaker, Phys Rep., 210,223 (1992) Problem 10.3], and this interpretation has been largely rejected. 

FIGURE 7.6 Reduction of the wave function caused by a measurement of position. 

The time-dependent perturbation changes the system s state function from exp -iE t/hyif to the superposition (9.122). Measurement of the energy then changes to one of the energy eigenfunctions exp -iE i,t/h)il/ i, (reduction of the wave function, Section 7.9). Tie net result is a transition from stationary state n to stationary state m, the probability of such a transition being... 

The two-component methods, though much simpler than the approaches based on the 4-spinor representation, bring about some new problems in calculations of expectation values of other than energy operators. The unitary transformation U on the Dirac Hamiltonian ho (Eq.4.23 is accompanied by a corresponding reduction of the wave function to the two-component form (Eq.4.26). The expectation value of any physical observable 0 in the Dirac theory is defined as ... 

The unitary transformation of the Dirac Hamiltonian to two-component form is accompanied by a corresponding reduction of the wave function. As discussed in detail in chapters 11 and 12, the four-component Dirac spinor ip will... 

A second approach to achieve a reduction of the 4-component Hamiltonian to an electrons-only Hamiltonian is to introduce approximations by eliminating the small components of the wave function (41-53). Also here, different protocols have been successfully exploited in quantum chemistry. 

It should be stressed that we are discussing here numerically exact results obtained by the solution of the time-dependent Schrodinger equation for an isolated system. No assumptions or approximations leading to decay or dissipation have been introduced. The time evolution of the wave function (t) is thus fully reversible. The obviously irreversible time evolution of the electronic population probabilities in Figs. 2 and 3 arises from the reduction process, that is, the integration over part of the system [in this case, the nuclear degrees of freedom, cf. Eqs. (12) and (13)]. 

The enhancement of the current-function for the anodic wave (I), which involves proton dissociation, upon addition of pyridine is evident [Fig. 1(a) and (b)], whereas the decrease of the current-function for wave (II) conceivably results from the reaction of pyridine with the complex product of the first oxidation (however, in other complexes, namely of the dinuclear dicarbene type [3], this current-function is promoted by base). The second anodic process, which possibly involves a sequence of electron transfer/deprotonadon steps (> 4 electrons, by CPE), is more affected by the temperature and scan rate than the first one (which involves a smaller number of electrons) either a decrease of the temperature [Fig. 1(c), (a)] or an increase of the scan rate leads to a reduction of the current function of wave (II) relative to that of wave (I), by hampering the sequence of the chemical steps involved. Potentiometric titration of the exhaustively electrolyz solution at the second wave indicates the presence of an acidic product (besides the liberated protons), possibly with an acidic methylene group. 

Reduction is probably stepwise when 0 = 90°. (The plot in fact resembles very closely in form the familiar Karplus plot of the magnitude of vicinal proton-proton coupling constants as a function of 0 48>. The half-wave potentials of 43 and 44 are representative the half-wave potential of a related monohalide (45) is given for reference 46>. These investigators did not examine the products of electrochemical reduction of the vicinal dihalides which they studied. If reduction is concerted, the products should... 

Application to anisotropic patterns for the purpose of "isotropization is not reasonable [148], The operation can directly be performed by FIT2D [39], Chap. 11). If IDL or pv-wave shall be used, the library functions DIST() and SHIFT() make the program run fast55. If the pedestrian solution of the algorithm is programmed (e.g., in Excel, Java, Pascal, C,...) the square-root56 should not be drawn in order to avoid further reduction of the program speed. 

Therefore, the principal role of the inclusion of the ionic term in the wave function is the reduction of the kinetic energy from the value in the purely covalent wave function. Thus, this is the delocalization effect alluded to above. We saw in the last section that the bonding in H2 could be attributed principally to the much larger size of the exchange integral compared to the Coulomb integral. Since the electrical effects are contained in the covalent function, they may be considered a first order effect. The smaller added stabilization due to the delocalization when ionic terms are included is of higher order in VB wave functions. 

---