In principle, it is possible to make simultaneous and precise measurements of a particle's velocity using two different methods, which should give identical results. It is shown that these results permit the calculation of the particle's position and momentum with a combined error [Formula: see text] at every instant of time in an arbitrarily large interval. It is also shown that, in the special case considered, the statistical interpretation of the wave function is invalid.
In principle, it is possible to make simultaneous and precise measurements of a particle's velocity using two different methods, which should give identical results. It is shown that these results permit the calculation of the particle's position and momentum with a combined error [Formula: see text] at every instant of time in an arbitrarily large interval. It is also shown that, in the special case considered, the statistical interpretation of the wave function is invalid.
Does the Heisenberg uncertainty principle refer to errors of measurement, or to the spread of values of the physical variables intrinsic to a particle’s state, or to some combination of these? The most commonly quoted form of the uncertainty principle relates the spread of an ensemble of separate measurements of some variable to the analogous spread of its conjugate variable. In contrast, Heisenberg’s original argument for the uncertainty principle involved the perturbation to a particle’s state by a measurement of one variable, which affects one’s ability to predict the outcome of a subsequent measurement of the conjugate variable. The relation between these two views of the uncertainty principle is discussed in this paper. A familiar example is considered: an ensemble of identically prepared particles passing through a slit, and after further propagation being detected. From this arrangement it is possible to infer joint (although necessarily imprecise) information on both transverse position and conjugate momentum for each member of the ensemble. It is shown that in this case of joint measurement the product of standard deviations for the measurement outcomes is at least twice as large as the lower bound implied by the usual uncertainty principle. The discussion is meant to help clarify the different roles played in the various statements of the uncertainty principle by the initial state and by measurement error.
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.
A simple ideal experiment to measure simultaneously the position and momentum of a free electron, also its kinetic energy and the instant when this kinetic energy is measured, taking into account the finite velocity of propagation of light as embodied in the Lorentz transformation, is proposed. This experiment is as follows: an observer sends out a light signal which strikes a given free electron and returns along its own track. From measurements made on the outgoing and returning signals the required quantities, and the uncertainties in their measurement, are calculated. In the absence of detailed knowledge concerning the nature of the interaction between the signal and the electron, it is found that for any electron velocity between 0 and c, $\ensuremath{\Delta}p\ensuremath{\Delta}x$ and $\ensuremath{\Delta}E\ensuremath{\Delta}t$ have an upper and a lower bound. The upper bound depends only on the initial electron velocity and the lower bound is a function of the electron velocity and the outgoing frequency. These bounds come close together for low electronic velocities and low frequencies, but their separation increases as the initial electron velocity increases becoming infinite in the limit v=c. In the limit $c\ensuremath{\rightarrow}\ensuremath{\infty}$ the upper and lower bounds coalesce for all frequencies and all electron velocities giving Heisenberg's results. The possibility, but not the necessity, of a relativistic quantum mechanics with $\ensuremath{\Delta}p\ensuremath{\Delta}x=h$ (and $\ensuremath{\Delta}E\ensuremath{\Delta}t=h$) is, we believe, established by the present results.
In two articles, the authors claim that the Heisenberg uncertainty principle limits the precision of simultaneous measurements of the position and velocity of a particle and refer to experimental evidence that supports their claim. It is true that ever since the inception of quantum mechanics, the uncertainty relation that corresponds to a pair of observables represented by non-commuting operators is interpreted by many scientists and engineers, including Heisenberg himself, as a limitation on the accuracy with which observables can be measured. However, such a limitation cannot be deduced from the postulates and theorems of quantum thermodynamics.
According to Heisenberg’s uncertainty principle [1], both, position and momentum of a particle, cannot be known with ultimate accuracy; the product: Δ x . Δ p > h . Whereas in the case of Laser-cooled atoms, we know that velocity of the particle is close to zero; and its position x is so perfectly known, that an atom is said to be held in Laser forceps [2]. These experiments show that both position and momentum of atoms are accurately knowable simultaneously! Similarly, when momentum of an atom is zero, the wavelength of its de Broglie wave is very very long; and according to quantum-mechanical interpretation, the atom is likely to be detected anywhere within its de Broglie wavelength, with higher probability at the peaks of the wave. Whereas the experiments with Laser-cooled atoms show that the atom is confined precisely within the Laser tweezers. Based on this observation a new experiment is proposed here in which, instead of quantum mechanical waves, which have wavelength h / m v , corresponding to phase velocity of the wave, we can carefully observe the waves corresponding to group-velocity, of the wavelength h v / m c^2. And we can expect that a particle may deterministically tunnel from one peak of the group-wave to next peak of the group wave; and can be detected at deterministically predictable points.