Heisenberg's uncertainty relation and a new measurement-axiom in quantum theory

In this note, we shall propose a new measurement-axiom in nonrelativistic quantum theory, which can be considered a kind of generalized Copenhagen interpretation.This interpretation can assert rather radical statements, which may irritate our minds familiar with Copenhagen interpretation.We shall, under this interpretation, assert the following two statements: (i) the uncertainties (i.e.Δq and Δp) in Heisenberg's uncertainty relation "Δq・Δp≥■/2" can be characterized as (average) errors in the approximates simultaneous measurement, and so, Heisenberg's uncertainty relation can be clarified as the statement in physics for the first time, (ii) the (discrete) trajectory of a particle is enough significant (though this, of course, includes errors).Furthermore, concerning (ii) we shall show the numerical results of the trajectory of a particle in two slit experiment.(Though almost all parts of this note are composed of summaries of references [11], [12], [13] and [14], we add some discussions from other angles.)In this note, we shall propose a new measurement-axiom in nonrelativistic quantum theory, which can be considered a kind of generalized Copenhagen interpretation.This interpretation can assert rather radical statements, which may irritate our minds familiar with Copenhagen interpretation.We shall, under this interpretation, assert the following two statements: (i) the uncertainties (i.e.Δq and Δp) in Heisenberg's uncertainty relation "Δq・Δp≥■/2" can be characterized as (average) errors in the approximates simultaneous measurement, and so, Heisenberg's uncertainty relation can be clarified as the statement in physics for the first time, (ii) the (discrete) trajectory of a particle is enough significant (though this, of course, includes errors).Furthermore, concerning (ii) we shall show the numerical results of the trajectory of a particle in two slit experiment.(Though almost all parts of this note are composed of summaries of references [11], [12], [13] and [14], we add some discussions from other angles.)

Jan 1, 19921 citation

Uncertainty principle for joint measurement of noncommuting variables

M. G. Raymer

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.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.

American Journal of PhysicsNov 1, 199470 citations

Remarks on Uncertainty Principles

E. U. Condon

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, $\Delta p\Delta q>\frac{h}{{2\pi}}$ (1) This inequality has come to be known quite generally as Heisenberg’s uncertainty relation.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, $$ \Delta p\Delta q>\frac{h}{{2\pi}} $$ (1) This inequality has come to be known quite generally as Heisenberg’s uncertainty relation.

ScienceMay 31, 192915 citations

Measurement uncertainty relations: characterising optimal error bounds for qubits : Topical Review

Paul Busch, Tom Bullock

In standard formulations of the uncertainty principle, two fundamental features are typically cast as impossibility statements: two noncommuting observables cannot in general both be sharply defined (for the same state), nor can they be measured jointly. The pioneers of quantum mechanics were acutely aware and puzzled by this fact, and it motivated Heisenberg to seek a mitigation, which he formulated in his seminal paper of 1927. He provided intuitive arguments to show that the values of, say, the position and momentum of a particle can at least be unsharply defined, and they can be measured together provided some approximation errors are allowed. Only now, nine decades later, a working theory of approximate joint measurements is taking shape, leading to rigorous and experimentally testable formulations of associated error tradeoff relations. Here we briefly review this new development, explaining the concepts and steps taken in the construction of optimal joint approximations of pairs of incompatible observables. As a case study, we deduce measurement uncertainty relations for qubit observables using two distinct error measures. We provide an operational interpretation of the error bounds and discuss some of the first experimental tests of such relations.In standard formulations of the uncertainty principle, two fundamental features are typically cast as impossibility statements: two noncommuting observables cannot in general both be sharply defined (for the same state), nor can they be measured jointly. The pioneers of quantum mechanics were acutely aware and puzzled by this fact, and it motivated Heisenberg to seek a mitigation, which he formulated in his seminal paper of 1927. He provided intuitive arguments to show that the values of, say, the position and momentum of a particle can at least be unsharply defined, and they can be measured together provided some approximation errors are allowed. Only now, nine decades later, a working theory of approximate joint measurements is taking shape, leading to rigorous and experimentally testable formulations of associated error tradeoff relations. Here we briefly review this new development, explaining the concepts and steps taken in the construction of optimal joint approximations of pairs of incompatible observables. As a case study, we deduce measurement uncertainty relations for qubit observables using two distinct error measures. We provide an operational interpretation of the error bounds and discuss some of the first experimental tests of such relations.

Jul 13, 20180 citations

Measurement uncertainty relations: characterising optimal error bounds for qubits

Tom Bullock, Paul Busch

Abstract In standard formulations of the uncertainty principle, two fundamental features are typically cast as impossibility statements: two noncommuting observables cannot in general both be sharply defined (for the same state), nor can they be measured jointly. The pioneers of quantum mechanics were acutely aware and puzzled by this fact, and it motivated Heisenberg to seek a mitigation, which he formulated in his seminal paper of 1927. He provided intuitive arguments to show that the values of, say, the position and momentum of a particle can at least be unsharply defined, and they can be measured together provided some approximation errors are allowed. Only now, nine decades later, a working theory of approximate joint measurements is taking shape, leading to rigorous and experimentally testable formulations of associated error tradeoff relations. Here we briefly review this new development, explaining the concepts and steps taken in the construction of optimal joint approximations of pairs of incompatible observables. As a case study, we deduce measurement uncertainty relations for qubit observables using two distinct error measures. We provide an operational interpretation of the error bounds and discuss some of the first experimental tests of such relations.Abstract In standard formulations of the uncertainty principle, two fundamental features are typically cast as impossibility statements: two noncommuting observables cannot in general both be sharply defined (for the same state), nor can they be measured jointly. The pioneers of quantum mechanics were acutely aware and puzzled by this fact, and it motivated Heisenberg to seek a mitigation, which he formulated in his seminal paper of 1927. He provided intuitive arguments to show that the values of, say, the position and momentum of a particle can at least be unsharply defined, and they can be measured together provided some approximation errors are allowed. Only now, nine decades later, a working theory of approximate joint measurements is taking shape, leading to rigorous and experimentally testable formulations of associated error tradeoff relations. Here we briefly review this new development, explaining the concepts and steps taken in the construction of optimal joint approximations of pairs of incompatible observables. As a case study, we deduce measurement uncertainty relations for qubit observables using two distinct error measures. We provide an operational interpretation of the error bounds and discuss some of the first experimental tests of such relations.

arXiv:1512.00104Journal of Physics A Mathematical and TheoreticalMay 23, 201813 citations

Measurement uncertainty: the problem of characterising optimal error bounds

Tom Bullock, Paul Busch

Dec 1, 20151 citation

\Delta p\Delta q>\frac{h}{{2\pi}}

(1) This inequality has come to be known quite generally as Heisenberg’s uncertainty relation.