At the quantum scale, objects such as fundamental particles, atoms, and molecules are modeled as localized wavepackets rather than concrete objects. As such, some quantities that are well-defined at our human scale physics instead become uncertain. For instance, quantum objects do not have a precise position, and instead their position is understood probabilistically and we look to the expected, or average, value of the position. Another quirk of quantum mechanics is that quantities such as position and momentum become operators that act on functions, similar to matrices acting on vectors. As such, quantities that may have commuted classically may not at the quantum level. For instance, position (x) and momentum (p) commute in classical physics (xp = px), but they do not commute at the quantum level (xp ≠ px). For noncommuting operators A and B, the uncertainty principle qualitatively describes how well one can know A and B simultaneously: the more certain one is of the value of A, the less one knows the value B and vice versa. The standard example is that of position and momentum: the Heisenberg uncertainty principle states that the more certain one is of the position of a quantum object, the more uncertain one is about its momentum. The classic uncertainty principles between noncommuting operators are those of position and momentum, the z and x or y components of angular momentum, and number and phase operators. These operators are very rigid and do not provide easy generalization to more general operators. In this project, we used an infinite matrix representation to analyze the uncertainty principle underlying position and momentum which allows for broader generalization. The biggest component of this project was determining eigenvalues of infinite banded matrices which was aided by the use of the mathematical software Mathematica.
At the quantum scale, objects such as fundamental particles, atoms, and molecules are modeled as localized wavepackets rather than concrete objects. As such, some quantities that are well-defined at our human scale physics instead become uncertain. For instance, quantum objects do not have a precise position, and instead their position is understood probabilistically and we look to the expected, or average, value of the position. Another quirk of quantum mechanics is that quantities such as position and momentum become operators that act on functions, similar to matrices acting on vectors. As such, quantities that may have commuted classically may not at the quantum level. For instance, position (x) and momentum (p) commute in classical physics (xp = px), but they do not commute at the quantum level (xp ≠ px). For noncommuting operators A and B, the uncertainty principle qualitatively describes how well one can know A and B simultaneously: the more certain one is of the value of A, the less one knows the value B and vice versa. The standard example is that of position and momentum: the Heisenberg uncertainty principle states that the more certain one is of the position of a quantum object, the more uncertain one is about its momentum. The classic uncertainty principles between noncommuting operators are those of position and momentum, the z and x or y components of angular momentum, and number and phase operators. These operators are very rigid and do not provide easy generalization to more general operators. In this project, we used an infinite matrix representation to analyze the uncertainty principle underlying position and momentum which allows for broader generalization. The biggest component of this project was determining eigenvalues of infinite banded matrices which was aided by the use of the mathematical software Mathematica.
The Heisenberg uncertainty postulates are a convenient way to describe the qualitative properties of quantum mechanics. The response from Heisenberg’s uncertainty principle is that one can create states that require a certain amount of momentum and position localization. The numerical quantities used by the old Newtonian mechanics, position, momentum, and energy are mere “shadows” of what nature describes: operators. The energy value contradicts the Newtonian notion of any energy level being necessary for a harmonic oscillator. The basic postulates of classical statistical mechanics are now applied to quantum statistical mechanics. In classical mechanics, the variables characterize a microstate that evolves according to the Hamiltonian equations. If the standard deviation relative to what we are interested in is very high, it is probably safe to use the expected value as a classical value. A quantum field theory is a relativistic invariant quantum theory where there is a particular form of quantity operators connected with the classical fields.
The quantum mechanical uncertainty principle for position and momentum plays an important role in many treatments on the (philosophical, physical and other) implications of quantum mechanics. Roughly speaking, the more precisely the momentum (position) of a (quantum mechanical) object is given, the less precisely can one say what its position (momentum) is. This quantum mechanical measurement problem is not just an interpretational difficulty, it raises broader issues as well. The measurement (of a property) of a (quantum mechanical) object determines the existence of the measured. In brief, the quantum mechanical uncertainty principle challenges some fundamental principles of Science and especially the principle of causality. In particular, an independently existing (external) objective reality is denied. As we shall see, that the quantum mechanical uncertainty principle for position and momentum is based on the assumption that 1⩽0, which is a logical contradiction.
The quantum mechanical uncertainty principle for position and momentum plays an important role in many treatments on the (philosophical, physical and other) implications of quantum mechanics. Roughly speaking, the more precisely the momentum (position) of a (quantum mechanical) object is given, the less precisely can one say what its position (momentum) is. This quantum mechanical measurement problem is not just an interpretational difficulty, it raises broader issues as well. The measurement (of a property) of a (quantum mechanical) object determines the existence of the measured. In brief, the quantum mechanical uncertainty principle challenges some fundamental principles of Science and especially the principle of causality. In particular, an independently existing (external) objective reality is denied. As we shall see, that the quantum mechanical uncertainty principle for position and momentum is based on the assumption that 1⩽0, which is a logical contradiction.
The uncertainty principle (linked to p,x or E,t) plays an important role in quantum mechanics. For example, an electron with an exact momentum p is described by a wavefunction exp(ipx). This function exists at all x points and so it is suggested in the literature that there is complete uncertainty as to the electron’s position. (More specifically P(x)=exp(-ipx)exp(ipx) normalization constant=constant and so all x points have the same probability.) Thus if p is completely certain, it is said that position x is completely unknown. We note, however, that exp(ipx) consists of cos(px) and sin(px) which are repeated probability distributions (with zero endpoints). Thus even though they exist in all of space, due to repetition which in turn is due to spatial invariance, there is a specific length number, namely the wavelength. Thus we argue that in classical physics there is a set of points (x,p), but in quantum mechanics ( p , x with its dx interval), with the latter being equal to hbar/p. Then the only uncertainty is within the dx interval i.e. the wavelength and spatial density is linked to momentum i.e. there is dynamical probability. In particular, there is a frequency hbar/E and a wavelength hbar/p which define an internal clock and ruler and the two are independent. Thus within a wavelength one has a probabilistic picture (in order to have endpoints defining the dx interval) and no sense of time, yet on average time exists because of hbar/E and the particle moves in space with a constant v. This begs the question: How do time and space x arise in such a scenario? We speculate the following. Assume there exists a property called energy which defines an object. This object, however, may break into two different objects. As a result, one introduces a parameter called time to distinguish between the two systems (E) and (E1,E2). For a certain time range one has a single object, while for a later time interval, one has two objects. Each of the two objects in the second system, however, needs to be distinguished as well and so another parameter, namely space x is used for this. If the two objects existed at the same x and t, how would one know there were two unless another parameter were introduced? We use t and x as the distinguishing parameters and furthermore give these a measure i.e. dt and dx. We then argue that from these ideas one may show the notion of speed and direction (because there are two objects) as well as conservation of momentum arises. Given the notion of an interval, we argue it must be described by a probability distribution and we link this to constant/p and exp(ipx). We finally note that since spatial density is linked to dynamical probability, one must consider motion associated with a density which may lead even to collective motion.
This paper is a small discussion about the uncertainty principle. With the great discoveries and researches made by predecessors, I try to answer some questions within the uncertainty principle. Using the definition of momentum, the momentum in de Broglie wave, the property of cosine function, the Euler’s formula, the basic property of plane wave, the law of derivative, the energy of quantum discovered by Max Planck, the photon energy discovered by Albert Einstein, the energy equation of a system and the Hamiltonian operator, I figure out the Schrodinger equation successfully. With the help of Schrodinger equation, the possibility density of a wave function, the concept of normalization, the average value of position and momentum, the Fourier transform, the integration by parts and the limit of normalized wave functions, I find the position operator and the momentum operator. Then using some knowledge of the standard deviation and the variance, the concept of inner product of vectors, some property of Dirac symbol, the law of calculation of complex number and the law of operator calculation, I obtain the general uncertainty relationship of abstract mathematical quantities. According to the general relationship, I get the inequality of uncertainty principle, and discuss the specific relationship between the position and the momentum. After getting the inequality, I use Heisenberg’s experiment of measuring an electron by using the γ ray to hit the electron to explain why we cannot measure the position and the momentum simultaneously. Based on Heisenberg’s experiment, I give my own explanation by discussing the earth and a spacecraft. Transforming the perspective, we can easily see how the accuracies of position and momentum change. In addition, I attempt to figure out why the uncertainty exists. However, I can only explain it by discussing the time and frequency in mathematical method and have no idea to work it out in a pure physics approach.
Heisenberg guessed, after he established the matrix quantum mechanics, that the non-commutativity of the matrices of position and momentum implied that the position and momentum of a particle could not be precisely simultaneously determined. He consequently conjectured that time and energy should also have a similar relationship. Soon after, Robertson derived an inequality concerning the space coordinate and momentum, which was thought to be the mathematical expression of the uncertainty relation guessed by Heisenberg. Since then, people have tried various devices to prove the correctness of these two relations. However, no one conducted a careful analysis of Heisenberg’s primary paper. In this work, we point out some serious problems in Heisenberg’s paper and the literature talking about the uncertainty relationships: the physical concepts involved in the uncertainty relations are not clear; one physical concept had more than one explanation, i.e., switching concepts; there has never been measurement experiment to support the relations. The conclusions are that the so-called coordinate–momentum uncertainty relation has never been related to actual measurement and there does not exist a time–energy uncertainty relation.