The Heisenberg uncertainty principle is used for understanding time-frequency distributions (TFDs) or density functions. The relationship between the uncertainty principle and proper TFDs is explored. Specifically, the authors address negative values in a TFD, the interpretation of TFDs as energy density functions, and local time-frequency resolution. This is done in light of having previously devised a method for constructing proper distributions.>
The Heisenberg uncertainty principle is used for understanding time-frequency distributions (TFDs) or density functions. The relationship between the uncertainty principle and proper TFDs is explored. Specifically, the authors address negative values in a TFD, the interpretation of TFDs as energy density functions, and local time-frequency resolution. This is done in light of having previously devised a method for constructing proper distributions.>
This contribution addresses the implications of Heisenberg's uncertainty relation on time-frequency energy distribution functions depending bilinearly on the signal. It is indicated that all these distribution functions must have a certain minimum spread in the time-frequency plane. Further-more such distribution functions can either have correct marginal distributions or be positive, but not both. Therefore a local interpretation of these energy distributions must be taken with care. Some facts determining the choice of an appropriate distribution function will be discussed. A comparison will be made between the spectrogram and the Wigner distribution.
This article establishes the proof of the Schrödinger equation for numerous quantum systems, utilizing Heisenberg's uncertainty principle. The Fourier transform connects functions in the time and frequency domains, resulting in the mathematical inequality that is the foundation of the uncertainty principle. In the part of Methods and Theory, the article derives the uncertainty principle through Fourier transforms by defining the mean and variance of angular frequency and time, and subsequently expanding the integral. This establishes the fundamental connection between time and frequency domains, illustrating the constraints imposed by quantum mechanics. In the part of Results and Application, the article applies the uncertainty principle to derive the Schrödinger equation under different conditions: free particle, particle in a box, harmonic oscillator, and hydrogen atom. For each case, the article assumes wave function solutions, uses the uncertainty in position and momentum to estimate kinetic and potential energies, and shows that the total energy matches the ground state energy derived from the Schrödinger equation. The results highlight the critical role of Heisenberg's uncertainty principle in understanding key aspects of quantum mechanics, providing a unified framework for these diverse systems.
This article establishes the proof of the Schrdinger equation for numerous quantum systems, utilizing Heisenberg's uncertainty principle. The Fourier transform connects functions in the time and frequency domains, resulting in the mathematical inequality that is the foundation of the uncertainty principle. In the part of Methods and Theory, the article derives the uncertainty principle through Fourier transforms by defining the mean and variance of angular frequency and time, and subsequently expanding the integral. This establishes the fundamental connection between time and frequency domains, illustrating the constraints imposed by quantum mechanics. In the part of Results and Application, the article applies the uncertainty principle to derive the Schrdinger equation under different conditions: free particle, particle in a box, harmonic oscillator, and hydrogen atom. For each case, the article assumes wave function solutions, uses the uncertainty in position and momentum to estimate kinetic and potential energies, and shows that the total energy matches the ground state energy derived from the Schrdinger equation. The results highlight the critical role of Heisenberg's uncertainty principle in understanding key aspects of quantum mechanics, providing a unified framework for these diverse systems.
It is sometimes said that one can't localize a signal simultaneously in time and frequency. This can be misleading. It's not that we can't localize the signal in time and frequency. The signal itself can't be concentrated simultaneously in time and in frequency. The Heisenberg uncertainty principle does not limit what we can know about reality; it describes that reality. The shorter-lived a function, the wider the band of frequencies given by its Fourier transform; the narrower the band of frequencies of its Fourier transform, the more the function is spread out in time.
Textbooks often emphasize the philosophical aspects of Heisenberg's Principle calling it “The Uncertainty Principle”. However, as a physical principle there is nothing uncertain about it. In its “exact” form it is a useful relationship between physical parameters. It is an expression of the fact that particles of matter behave simultaneously in two modes: one that is particle like, and the other that is wave like. These two modes are intimately connected, and it is this connection that is expressed in the Heisenberg Principle, and more generally in quantum mechanics including Schrödinger's equation (Atkins and Friedman, 1997). From this viewpoint, the principle can be used to deduce some of the properties of atoms and molecules in a simple way. This method of analysis is quantitative, but it is limited to cases that involve only a single coordinate that is given in advance (Born, 1989). The method has been extended somewhat by Simons and Bloch (1973) to cover some cases involving the angular momentum quantum number. It is a rudimentary form of density functional theory. In a somewhat different form than the one used here, it has been applied to molecules by Parr and his collaborators (Borkman and Parr, 1968). Physical properties can never be measured exactly. Errors, however small, are always present in a measurement procedure. Therefore, measured values of variables are average values derived from a distribution of individual measurements.
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.