Quantum Indeterminacy

As the subject of quantum theory has now been broached, it is perhaps opportune to say something about quantum uncertainty at this point. The discovery toward the end of the nineteenth century that the laws of physics were in many cases based on statistics came as a considerable surprise. However, this surprise was nothing compared to the profound sense of shock experienced by the scientific community as a whole when the propositions of quantum theory first became widely known. In fact, the reverberations of this shock are still keenly felt today, and the ongoing debate about what it all means has remained lively and controversial. Feynman even went as far as to admonish us not to keep asking But 

The heavy superstructure of modern quantum mechanics rests largely upon a set of mathematical relationships published in 1927 by Heisenberg. These relationships are now usually referred to collectively as the uncertainty principle. Heisenberg showed that in any quantum-mechanical system, pairs of dynamical variables for particles can be simultaneously and sharply defined only if their operators commute. This means only if their operators H and K satisfy the equation 

In cases where Eq. (5) is not satisfied, some uncertainty will always be 

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I am for the moment ignoring the hypothesis that quantum indeterminacy may be the source of the brain indeterminacy necessary for philosophically-real free will, (back)... 

This means there is no place for the study of choice (free will) in physics because physics has an altogether different subject matter extra-mental material objects undergoing physical changes. Free will is not an object of study in the same sense that a neutrino is. If a physicist were to suggest the only types of existents are material objects and physical phenomena, or if he were to suggest free will can be located behind quantum indeterminacy, or if he... 

Heisenberg s principle of uncertainty (or indeterminacy) was based in the Dirac-Jordan transformation theory (see Kragh, Dirac, 44) P. A. M. Dirac, "The Physical Interpretation of the Quantum Dynamics,"... 

It is, perhaps, less known that the concepts of complementarity and indeterminacy also arise naturally in the theory of Brownian motion. In fact, position and apparent velocity of a Brownian particle are complementary in the sense of Bohr they are subject to an indeterminacy relation formally similar to that of quantum mechanics, but physically of a different origin. Position and apparent velocity are not conjugate variables in the sense of mechanics. The indeterminacy is due to the statistical character of the apparent velocity, which, incidentally, obeys a non-linear (Burgers ) equation. This is discussed in part I. 

Formally, replacing vd by v and D by h/2m, Eq. (12) goes over into the quantum-mechanical indeterminacy relation. However, the same substitution does not transform the diffusion equation into the Schrodinger equation, but only if the time or the mass is made imaginary. This last difference reflects the fact that the diffusion equation describes an irreversible process, while the Schrodinger equation concerns a reversible situation. Incidentally, there is a classical analog to the relation... 

Boltzmann s expression for S thereby reduces the description of the molecular microworld to a statistical counting exercise, abandoning the attempt to describe molecular behavior in strict mechanistic terms. This was most fortunate, for it enabled Boltzmann to avoid the untenable assumption that classical mechanics remains valid in the molecular domain. Instead, Boltzmann s theory successfully incorporates certain quantal-like notions of probability and indeterminacy (nearly a half-century before the correct quantum mechanical laws were discovered) that are necessary for proper molecular-level description of macroscopic thermodynamic phenomena. 

That is the very beauty of quantum mechanics. It is a perfectly deterministic theory for calculating the development of a physical system, once you know it at a given time. But because only the absolute square of the wave-function is accessible to experiment, it is not possible for us to know the state precisely at such an initial time, and hence all the further development of the system is to the human observer clouded with imcertainty relations and similar indeterminacy, despite the fact that the system itself knows perfectly well what it is doing and follows a uniquely given path of development in time and space. 

If two operators do not commute with each other, then the observables they represent cannot be determined simultaneously with an arbitrarily small indeterminacy. These observables are said to be incompatible (also called complementary), and are the object of the indeterminacy principle introduced in Chapter 1. We will keep to this usual expression of the Heisenberg indeterminacy principle, although, as we found in Chapter 1, this means the impossibility of preparing a state for which two incompatible properties can be determined with arbitrarily small indeterminacies. The fact that some operators do not commute with each other represents one of the main differences between classical and quantum mechanics. 

If we try to measure A precisely and make AA nearly zero, then the spread in AB will have to increase to satisfy this condition. Trying to determine the angular position of the orbiting electron described by the de Broglie wave discussed earlier is a perfect illustration. Indeterminacy is intrinsic to the quantum description of matter and applies to all particles no matter how large or small. 

At last, we can resolve the paradox between de Broglie waves and classical orbits, which started our discussion of indeterminacy. The indeterminacy principle places a fundamental limit on the precision with which the position and momentum of a particle can be known simultaneously. It has profound significance for how we think about the motion of particles. According to classical physics, the position and momentum are fully known simultaneously indeed, we must know both to describe the classical trajectory of a particle. The indeterminacy principle forces us to abandon the classical concepts of trajectory and orbit. The most detailed information we can possibly know is the statistical spread in position and momentum allowed by the indeterminacy principle. In quantum mechanics, we think not about particle trajectories, but rather about the probability distribution for finding the particle at a specific location. 

Wave-particle duality accounts for the probabilistic nature of quantum mechanics and for indeterminacy. Once we accept that particles can behave as waves, then we can apply the resnlts of classical electromagnetic theory to particles. By analogy, the probability is the sqnare of the amplitnde. Zero-point energy is a con-seqnence of the Heisenberg nncertainty relation all particles bound in potential wells have finite energy even at the absolnte zero of temperature. 

The function Dxp is composed of Dira functions D = 2b gb 5b where 2b gb = 1 gb => 0 and the 5s are forbidden by Planck s law of the finiteness of the quantum of action (which may be formulated as the indeterminacy principle). This may be corrected for by the so-called Wigner transformation which transforms the expectation value linear form to U = / Hx x DXx where the coordinates x occur twice, independently, so that H and D become matrices. Since the Wigner transformation must lead to indeterminacy, it is closely related to a Fourier transformation and both matrices H and D become hermitian (Bopp, 1961). The dyads of hermitian matrices may be written j/x i//x and we see that their contribution u to the expected energy becomes u = / jj Hi//, therefore if we choose i//x as eigenvectors of H we see that we have in fact only discrete possible states in agreement with indeterminacy. 

In summary, on the one hand, classical mechanics was able to presume that the constructive properties were attributes of matter even if the experiments that were necessary for their determination were not accepted. On the other hand, in quantum physics, this was no longer possible due to the limitation of Heisenberg s indeterminacy relation, for any couple and conjugated variables. Weyl accepted it as a fundamental insight, different from Heisenberg s mathematical characterization of the commutation relation. In the case of electrochemistry and electrocatalysis, the fundamentals of... 

This result confirms the quantum behavior of the quantum motion that is not entirely encompassed by the (semi) classical turning points domain, even its major part lays there, with the rest being dispersed by tunneling process (see the Postulate I discussion), being however a new manifestation of the indeterminacy (Heisenberg) relationship that impedes the position and momentum to be with the same precision simultaneously determined. Further aspects of the quantum vibrational motions are to be discussed in the next sections and chapter of this volume, whereas the application to various molecular systems will be systematically presented in the Volume 3 of this five-volume set. 

This source of energy was discovered in quantum theory, and the principle, as it affects this discussion, can be stated roughly as follows if the electrons in a classical valency structure can be delocalized whilst obeying the Pauli exclusion principle, then the increased indeterminacy in their positions results in the molecule having a lower energy in the ground state than would be expected from the classical structure. 

For instance, the quantum chemical literature is full of comparisons between computed harmonic frequencies and observed fundamental ones. While this may be acceptable in situations where an anharmonic force field calculation (see Anharmonic Molecular Force Fields) is technically impossible, the anharmonicity a>i — v, of a fundamental may easily range from 2 to 200 cm- and therefore such a comparison is essentially meaningless for benchmark quality results. Yet for polyatomics, the experimentally derived harmonic frequencies are often associated with large uncertainties (cf. the C2H4 example at the end of this article) due to indeterminacies, and often the only meaningful comparison will be between computed and observed fundamentals, requiring the ab initio calculation of the anharmonic part of the force field. 

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