The integral in (6.1) is Fourier Transform; and therefore its lower and upper limits are minus infinity and plus infinity respectively. As we have already mentioned infinity is not meaningful in nature, therefore existence of infinity was Heisenberg’s second assumption. We show that, even if you assume (6.1), changing infinity to a finite number will eliminate uncertainty defined by (6.2), that is, the right hand side of (6.2) can be made arbitrarily small.
We also show that Uncertainty Principle violates a very fundamental concept or result in mathematics. This theory says that all continuous functions defined over finite or infinite interval has infinite dimensions. This is a very well accepted concept in QM, but it is not so in engineering. For more details please review the paper [Das, 2013-1] and search for the blog site ‘uncertaintyprincipleiswrong’ all one word. Do not hesitate to add your comments and you will get a feedback from this author.
To understand why (6.1) will lead to (6.2), we will explain Equivalence Principle, a very fundamental concept in mathematics, which we all know but probably we never took it seriously, noticed it, or thought about it.
3. Equivalence Principle
It is very fundamental to recognize that a function can be represented in many equivalent ways. For example the following are equivalent.
(a) It can be described by providing a table of values of x coordinates at fixed intervals and corresponding values of y coordinates, as numerical data;
(b) The above table of data can be interpolated and then represented by a smooth graph in the x-y plane;
(c) It is also possible to represent the same graph using an algebraic equation of y as a function of x; like y= 3x+x2;
(d) It can be represented by a combination of sinusoidal or exponential functions, like Fourier or Taylor series;
It is clear that all of the above representations of the numeric data in (a), are equivalent. This various representation methods may be called equivalence principle. In fact Ohanian [Ohanian] on page 33 admits that – “The free-particle Schrodinger equation does not produce any solutions except those that can be directly constructed by superposition – the equation gives us nothing new”. Any information derived from any one of the above items can be derived from all other representations. They only enhance the conveniences. Schrodinger equation is nothing but the derivative of a sine function.
A better and sophisticated look at the numerical data in (a) will definitely allow us to visualize the graph. Similarly we may be able to visualize the spectrum from the graphical representation (b) of the numeric data. The information content of the time function and its spectrum is same, because one can be derived from the other. Thus Fourier Transform cannot give any new information that is not present in the numeric data in (a) or the graphical content in (b). An experienced eye will be able to observe the harmonic contents of FT in the graph, although not numerical values. It will be possible to get the fine characteristics of the data from any one of the above (a-g) equivalent representations.
Thus there is no reason to believe, as is done in quantum mechanics, see Ohanian [Ohanian] pages 32-36, that if the time function represents the position of the particle then the corresponding spectrum function will represent the momentum. Position and momentum are two independent variables of a dynamical system. It was very unfortunate that Heisenberg linked them via the FT pair (6.1). FT pair cannot give any new information as explained by the equivalence principle. As a consequence of this FT property, Heisenberg has to introduce uncertainty principle causing more confusion. This position-momentum relationship (6.1) via FT does not satisfy engineering or physical intuitions. Both time and spectrum functions are different representations of same thing; on the other hand both position and momentum are independent properties of a particle. The uncertainty principle (6.2) is thus a mathematical consequence of equivalence principle and cannot be a law of nature.
about that engineering. Clearly in modern times this will not be the design of an experiment by any stretch of mind of any system engineer. Two unknowns cannot be found out by one measurement. One measurement will produce only one equation in two unknowns. At least two equations will be necessary to solve for both p and q variables. Heisenberg seems to believe that only one measurement will give values for both p and q. This is an assumption he used unconsciously. In reality, a significantly large volume of dynamic data should be collected, for a long period after hitting the electron, all simultaneously, and then eliminate all unknowns by least square curve fitting algorithm of dynamical systems, something like Kalman Filtering.
It is almost unbelievable that how much accuracy we can achieve using modern technology and with such simultaneous measurements. Simultaneity is a law of nature, more we encompass it better results we get. GPS satellites are about 20,000 km above earth. Yet we can measure distances on the surface of earth at the accuracy of sub-millimeter level, see [Hughes], in geodetic survey, by measuring the satellite distances. In one sense then we can measure a distance of 20,000 km at the accuracy of sub-millimeter. It is about one part in one trillion. This approach uses only non-military GPS signals. So we can think, how accurate the results can be, with the exact military signals from new generation of GPS satellites and receivers. Thus at this modern time, in retrospect, it is difficult to understand why Heisenberg thought about such an experiment involving one measurement to identify two variables.