Conjugate quantities

Plotting Oq xct vs then leads to a straight line with an intercept equal to the Gibbs energy difference between the f.c.c. and b.c.c. forms of Cr, at the temperature where measurements were made (Fig. 6.5), while the slope of the line yields the associated regular solution interaction parameter. The lattice stability and the interaction parameter are conjugate quantities and, therefore, if a different magnitude... 

This constant h has the physical dimensions of an action , ML2T l. Any two quantities, the product of which has the physical dimensions of an action, are conjugate quantities, for instance... 

We consider the complex-conjugated quantities in order to calculate the loss factor %" or e" as determined by positive imaginary components of % or . ... 

Action. This technical term is a historic relic of the 17th century, before energy and momentum were understood. In modern terminology, action has the dimensions of energyxtime. Planck s constant has those dimensions, and is therefore sometimes called Planck s quantum of action. Pairs of measurable quantities whose product has dimensions of energyxtime are called conjugate quantities in quantum mechanics, and have a special relation to each other, expressed... 

Ax /A C -C 1, the position of particle 2 is centered at x, and its uncertainty is Ax. Similar relations hold also for the momentum of particle 2 after the momentum of particle 1 is measured. Thus, either of the two conjugate quantities of particle 2 can be predicted with arbitrarily high precision. Of... 

The extent to which statistical concepts enter the picture as we go from the micro- to the macroworld is not at all at our disposal. For example, quantum mechanics as we know it today teaches us that it is impossible in principle to obtain complete information about a microscopic entity (i.e., the precise and simultaneous knowledge of an electron s location and momentum, say) at any instant in time. On account of Heisenberg s Uncertainty Principle, conjugate quantities like, for instance, position and momentum can only be known with a certain maximum precision. Quantum mechanics therefore already deals with averages only (i.e., expectation values) when it comes to actual measurements. 

The relation thus formulated is capable of immediate generalization. Consider in the first place, as an example with one degree of freedom, the case already treated above (p. 100), that of the rotator. Here the co-ordinate is the azimuth q== (f>y to which belongs, as canonically conjugated quantity, the angular momentum (or, in other words, the moment of momentum) p. In the free rotation p is constant, i.e. independent of the angle turned through. Thus... 

Now we find the norm C of the Boltzmann distribution W = C exp[-/ (T)], where h(T) represents dependence of the normalized energy h = H(kBT) 1 on the phase variables T. We may choose for these variables the energy h and its canonically conjugated quantity— initial time cp0. The latter is involved in the law of motion additively with the time variable q>. According to definition, C is inversely to the statistic integral st ... 

Eq. (10.2.2) is simply the inverse Fourier transform of Eq. (10.2.1). Note that in Fourier analysis q and r are conjugate quantities that is, to describe properties at large values of r we usually require only the small q Fourier components. 

In one dimension, the units of phase space are Joule-sec. This is often referred to as a unit of action. According to the uncertainty principle, energy and time or momentum and position are conjugate quantities which cannot be simultaneously and precisely known, that is, Ap Aq s tiH. Hence, the smallest allowable unit in phase space must be on the order of h, so that the quantum phase space is divided up into units of h. 

Main Effects When a main quantity is changed, this affects the conjugated quantities as well. The type of effect in which one observes the reciprocal dependency of two associated quantities is called main effect. The main effect of an increase... 

Density Fluctuationa A formal sUUistical mechanics in non-equilibrium states has been developed by Fischer and Wendorff (13) in terms of the afHnity A as a conjugate quantity to an ordering parameter. Their result is ... 

As in classical physics, the conjugate quantity to the position operator is — according to the last section—the momentum operator. For this reason, velocity operators play a minor role in quantum mechanics. However, Heisenberg s... 

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