Indeterminacy principle

STATISTICAL MECHANICS. One major problem of physios involves the prediction of the macroscopic properties of matter in terms of the properties of the molecules of which it is composed. According to the ideas of classical physics, this could have been accomplished by a determination of the detailed motion of each molecule and by a subsequent superposition or summation of their effects. The Heisenberg indeterminacy principle now indicates that this process is impossible, since we cannot acquire sufficient information about the initial state of the molecules. Even if this were not so, the problem would be practically insoluble because of the extremely large numbers of molecules involved in nearly all observations. Many successful predictions can be made, however, by considering only the average, or most probable, behavior of the molecules, rather than the behavior of individuals. This is the mediod used in statistical mechanics. 

The uncertainty principle is sometimes called the indeterminacy principle. 

This is known as the Heisenberg indeterminacy principle (also called uncertainty principle by some authors). It has to do with precision and not with accuracy. This situation has already been met in Section 1.2 when referring to the double-slit experiment. 

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. 

To make a rough estimate of the precision allowed by the indeterminacy principle, let us take as our spread of values in the position. Ax, the wavelength of our probe A. This choice means that we can locate the particle somewhere between two crests of the wave. Let us take as our estimate of the spread in the momentum, Ap, the value of the momentum itself, p that is, we know p to within p. Their product is therefore AxAp = h, but because we have asserted that this is the best we can do, we write AxAp > ft. A better choice for the spread in both variables is one standard deviation or the root mean square deviation from a series of measurements. For this choice, the result becomes... 

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. 

State the Heisenberg indeterminacy principle and use it to establish bounds within which the position and momentum of a particle can be known (Section 4.4, Problems 35 and 36). 

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. 

As a result of the random nature of radioactive decay, it appears that a sort of indeterminacy principle operates in quantitative microautoradiography. One can measure the amount of flux through large areas rather precisely or one can measure the location of a point source within a few microns, but one cannot locate and measure both quantities with precision. For sintering studies, however, this does not detract from the value of the technique. If diffusion of the oxygen tracer over appreciable distances is important, the measurement of concentration as a function of distance can be carried out satisfactorily. Conversely, if the tracer has not spread out, its position can be found. 

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