Quantum mechanics probabilistic interpretation

The derivation given above shows clearly that the kinetic laws are actually no deterministic laws, but have rather a probabilistic character. In the common sense, it is strongly engraved that by the development of quantum mechanics the interpretation of states as probabilities is forced. In fact, the interpretation of states emerging from nondeterministic laws is sound even in the classical statistical mechanics, in addition, like here, in chemical kinetics. 

III. Experimental observation of Quantum Mechanics. Only this final section should address the rules that govern interpretations of experiments measuring properties of QM systems with macroscopic devices. This includes probability interpretation, uncertainty relations, complementarity and correspondence. Then experiments can be discussed to show how the wave functions manipulated in section I can be used to predict the probabilistic outcome of experiments. 

E. Barbosa and F. Gonzalez, Antiphoton and the Probabilistic Interpretation of Quantum Mechanics, thesis, Univ. National, Bogota, 1995 presented at Colombian Congress of Physics, Bogota, 1999. 

Equation (3.32a) implies normalization, and Eq. (3.32b) contains the essential probabilistic interpretation of the projections onto the momenta or coordinates. The last condition, a natural consequence of the definitions of the density matrix and the Wigner-Weyl transform, explicitly eliminates the singular distributions allowed in Eq. (3.31d). That is, although the completeness of the quantum PijXp, q) basis permits the construction of 8 function distributions, they make, unlike classical mechanics, no natural appearance in quantum mechanics wherein eigenfunctions of LQ are square integrable and such singular distributions are explicitly excluded in Eq. (3.32d). 

Correlation energy probabilistic approach. Recalling the quantum mechanical interpretation of the wave function as a probability amplitude, we see that a product form of the many-body wave function corresponds to treating the probability amplitude of the many-electron system as a product of the probability amplitudes of individual electrons (the orbitals). Mathematically, the probability of a composed event is only equal to the probability of the individual events if the individual events are independent (i.e., uncorrelated). Physically, this means that the electrons described by the product wave function are independent.30 Such wave functions thus neglect the fact that, as a consequence of the Coulomb interaction, the electrons try to avoid... 

A major difference between quantum and classical mechanics is that classical mechanics is deterministic while quantum mechanics is probabilistic (more correctly, quantum mechanics is also deterministic, but the interpretation is probabilistic). Deterministic means that Newton s equation can be integrated over time (forward or backward) and can predict where the particles are at a certain time. This, for example, allows prediction of where and when solar eclipses will occur many thousands of years in advance, with an accuracy of meters and seconds. Quantum mechanics, on the other hand, only allows calculation of the probability of a particle being at a certain place at a certain time. The probability function is given as the square of a wave function, P t,i) = P (r,f), where the wave function T is obtained by solving either the Schrodinger (non-relativistic) or Dirac (relativistic) equation. Although they appear to be the same in Figure 1.2, they differ considerably in the form of the operator H. 

The postulates of quantum mechanics, especially the probabilistic interpretation of the wave function given by Max Bom, limit the class of functions allowed class Q or quantum"). Any wave function conforms to the following ... 

Schiodinger s fonnulation of quantum mechanics, Bom proposed the probabilistic interpretation of the wave function. Despite such seminal achievements, the Nobel Pri s in the thirties were received by his colleagues. Rnally, when in 1954 Bom obtained the Nobel Pri for Ns fundament research in quantum mechancs. especially for his statistical interpretation of the wave-functiorf, there was a gresit relief among his famous ftierKfs. 

Since the equation of motion, Eq. (4.16), contains a first derivative in time, the quantum state Y(r, t) at time t is solely determined by the initial state Y(r, 0), which, mathematically speaking, fixes the only integration constant. However, this fact demonstrates that the time evolution of a quantum mechanical system as described by Eq. (4.16) is a purely deterministic process, because a quantum mechanical state evolves uniquely according to the linear differential equation of motion at time t out of a state at an earlier time. The probabilistic character of quantum mechanics solely arises from the measurement process for which no satisfactory and consistent mathematical description is available (Bom s interpretation aims at coping with this problem see also section 4.3). 

Moreover, employing the techniques of second quantization prohibits the direct interpretation of the field operators ip as usual quantum mechanical wave functions, since superpositions of states with variable numbers of particles are not compatible with the simple probabilistic interpretation of the wave function. In order to restore this feature, it has become the standard procedure of quantum chemistry to return to a first quantized formulation based on suitable generalizations of the original Dirac equation Eq. (7.9), which will be the subject of the next chapter. 

FIGURE 12.12 Max Bom (1882-1970). Not only did he develop the probabilistic interpretation of the wavefunction, but he also devised a quantum-mechanical description for molecules. 

It would be wrong to interpret this work as an effort to refute the importance of quantum theory for chemistry. It does the opposite, but questions the methodology that developed from a naive interpretation of three-dimensional wave mechanics to confirm the electron-pair model of Lewis and the molecular structure theory of van t Hoff. Even in terms of the probabilistic interpretation of wave mechanics, a rigid three-dimensionally structured molecule, with its real molecular orbitals, is undefined. A strategy, based on these concepts and which became known as Quantum Chemistry, amounts to a disastrous misreading of quantum theory and has no predictive power beyond its classical basis. 

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