If you asked members of the public to name a concept or idea from modern physics, the most popular answer would probably be E = mc2. However, the runner-up could well be Heisenberg's uncertainty principle. This principle states that the position and linear momentum of a particle cannot be known simultaneously with arbitrary precision. More precisely, it sets a lower bound for the product of the uncertainties of these quantities: ΔxΔp≥/2, where Δx is the uncertainty in the position, Δp is the uncertainty in the momentum and is Planck's constant divided by 2π.
If you asked members of the public to name a concept or idea from modern physics, the most popular answer would probably be E = mc2. However, the runner-up could well be Heisenberg's uncertainty principle. This principle states that the position and linear momentum of a particle cannot be known simultaneously with arbitrary precision. More precisely, it sets a lower bound for the product of the uncertainties of these quantities: ΔxΔp≥/2, where Δx is the uncertainty in the position, Δp is the uncertainty in the momentum and is Planck's constant divided by 2π.
Since the publication of Heisenberg’s paper1 on the “anschaulichen Inhalt” of quantum mechanics, discussions of the fundamental limitations on the accuracy of physical measurements have been much in the foreground. According to Heisenberg, the quantum mechanics implies that it is impossible to measure simultaneous values of a coordinate and its conjugate momentum with unlimited precision. Instead, if Δp be the estimated error or uncertainty in a momentum and Δq that in the associated coordinate one must have the inequality,
ΔpΔq>2πh
(1)
This inequality has come to be known quite generally as Heisenberg’s uncertainty relation.
From the outset, Heisenberg had resolved to eliminate classical space-time pictures involving particles and waves from the quantum mechanics of the atom. He had wanted to focus instead on the properties actually observed and recorded in laboratory experiments, such as the positions and intensities of spectral lines. Alone in Copenhagen in February 1927, he now pondered on the significance and meaning of such experimental observables. Feeling the need to introduce at least some form of ‘visualizability’, he asked himself some fundamental questions, such as: What do we actually mean when we talk about the position of an electron? He went on to discover the uncertainty principle: the product of the ‘uncertainties’ in certain pairs of variables—called complementary variables—such as position and momentum cannot be smaller than Planck’s constant h (now h / 4π).
The uncertainty principle places fundamental limits on the accuracy with which we are able to measure the values of different physical quantities (Heisenberg 1949 The Physical Principles of the Quantum Theory (New York: Dover); Robertson 1929 Phys. Rev. 34 127). This has profound effects not only on the microscopic but also on the macroscopic level of physical systems. The most familiar form of the uncertainty principle relates the uncertainties in position and linear momentum. Other manifestations include those relating uncertainty in energy to uncertainty in time duration, phase of an electromagnetic field to photon number and angular position to angular momentum (Vaccaro and Pegg 1990 J. Mod. Opt. 37 17; Barnett and Pegg 1990 Phys. Rev. A 41 3427). In this paper, we report the first observation of the last of these uncertainty relations and derive the associated states that satisfy the equality in the uncertainty relation. We confirm the form of these states by detailed measurement of the angular momentum of a light beam after passage through an appropriate angular aperture. The angular uncertainty principle applies to all physical systems and is particularly important for systems with cylindrical symmetry.
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.
Background: There has been altercation for and against Heisenberg uncertainty principles, whose earlier mathematical form has assumed other forms. This is despite the fact that the original form ought to be for the hydrogen atom only. Methods: the method is purely theoretical and computational, involving the use of spectroscopic data in the literature. Aim and objectives: The aim of the research was to show that half of reduced Planck’s constant (ℏ/2) cannot be restricted to a lower limit to the product of the uncertainty in the momentum and the uncertainty in the position. The objectives were: 1) to undertake the derivation of an equation unifying the Heisenberg uncertainty principle and atomic principles; and 2) to give evidence that half of the reduced Planck’s constant (ℏ/2) is strictly for hydrogen. Results and discussion: With hydrogen apart from other elements, fractions and multiples of ℏ/2, 0.0053–0.0264 corresponding to the radii ranging between a0/100 and a0/20 (a0 is the Bohr’s radius), and 5-26 corresponding to the radii ranging between 4a0 and 49a0 were demonstrated; the probability of locating the position and momentum increases with decreasing radii. Conclusion: The equations for the determination of the fractions and multiples of ℏ/2 were derived. Computation gave values less than and higher than ℏ/2; the latter was found to be restricted to hydrogen only. The possibility of linking the product of uncertainties in position and momentum to Pa may be reserved for a future study.
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.