Variation of Constants

This formula can be motivated by applying the variation of constants formula to ihtli = -t- (H(y) — H"). The method can be implemented... 

Motor-driven reciprocating compressors above about 75 kW (100 hp) in size are usually equipped with a step control. This is in reality a variation of constant-speed control in which unloading is accomplished in a series of steps, varying from full load down to no load. Three-step eontrol (full load, one-half load, and no load) is usually accomplished with inlet-valve unloaders. Five-step eontrol (fuU load, three-fourths load, one-half load, one-fourth load, and no load) is accomphshed by means of clearance pockets (see Fig. 10-91). On some machines, inlet-valve and clearance-control unloading are used in combination. 

Motor-driven reciprocating compressors above about 75 kW (100 hp) in size are usually equipped with a step control. This is in reality a variation of constant-speed control in which unloading is accomplished in a series of steps, varying from full load down to no load. 

THE TREATMENT OF A TIME-DEPENDENT PERTURBATION BY THE METHOD OF VARIATION OF CONSTANTS... 

There have been developed two essentially different wave-mechanical perturbation theories. The first of these, due to Schrodinger, provides an approximate method of calculating energy values and wave functions for the stationary states of a system under the influence of a constant (time-independent) perturbation. We have discussed this theory in Chapter VI. The second perturbation theory, which we shall-treat in the following paragraphs, deals with the time behavior of a system under the influence of a perturbation it permits us to discuss such questions as the probability of transition of the system from one unperturbed stationary state to another as the result of the perturbation. (In Section 40 we shall apply the theory to the problem of the emission and absorption of radiation.) The theory was developed by Dirac.1 It is often called the theory of the variation of constants the reason for this name will be evident from the following discussion. 

The same probabilities are given directly by our application of the method of variation of constants. The probability of transition to states of considerably different energy as the result of a small perturbation acting for a short time is very small, and we have neglected these transitions. Our calculation shows that the probability of finding the system in the state B depends on the value of t in the way given by Equation 41-12, varying harmonically between the limits 0 and 1. 

Let us consider a system composed of five harmonic oscillators, all with the same characteristic frequency v, which are coupled with one another by weak interactions. The set of product wave functions 4 (a) k(6) (c) (d) k(e) can be used to construct approximate wave functions for the system by the use of the method of variation of constants (Chap. XI). Here (a), ... 

The application of the variation-of-constants treatment shows that if the system at one time is known to have a total energy value close to W°>, where n is a particular value of the quantum number n, then the wave function at later times can be expressed essentially as a combination of the product wave functions for n = n, the wave functions for n n making a negligible contribution provided that the mutual interactions of the oscillators are weak. Let us suppose that the system has an energy value close to 12Ahv, that is, that n is equal to 10. The product wave functions corresponding to this value of n are those represented by the 1001 sets of values of the quantum numbers na, , nc given in Table 49-1. 

This leads to the Dirac variation-of-constants method [10]. Although generall> successful, an unsatisfying feature of this method can be seen when we consider an adiabatically switched-on static perturbation. By the adiabatic theorem [11] the perturbed wave function as t — -)-< has the form... 

Conceptual Problems in Phenomenological Interpretation in Searches for Variation of Constants and Violation of Various Invariances... 

An example of such a problem is a relation of possible time or space variation of fundamental constants and a basic relativistic principle of local position/time invariance (LPI/LTI). Some scientists consider possible variation of constants as violation of LTI. However, that is not correct. 

We should acknowledge that there are two basic possibilities of variations of constants. One is a result of certain violation of LPl, while the other is an observational effect of the interaction with environment, such as the bath of Cosmic Microwave Background (CMB) radiation of photons, neutrinos, gravitons. Dark Matter (DM), matter, etc. 

Here we consider various aspects of a possible interpretation of experiments on variation of constants. 

First of all, we note that the very motto variation of constants is a jargon which is not related to reality. In a sense, the constants cannot vary, because the idea of variation of the constants is based on an assumption, that we can apply conventional equations, but claim that some of their parameters are now adiabatically time- or space-dependent. That cannot be correct. We should completely change the equations. 

When searching for a variation of constants one has to remember about a possible connection with various symmetries related to relativity. 

The other issue is a reason for a variation of constants. There are basically three options. 

Variation of constants could be caused by long-range environment. An example is the phase transition during the early time of the Universe. That has no relation to relativity. 

Since this is a linear differential equation of the first order, it can be solved by the standard method of the variation of constants (see ref. 19... 

As in self-consistent wavefunction functional methods, the linear variation of constants method is commonly employed when solving the KS equations. That is, the <]> values are approximated by... 

The solution for this previous differential equation is more complex. With the condition that at time zero [PJ=0, the solution can be found by using the variation of constants method [70] ... 

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