Uncertainty principle

In quantum physics, the Heisenberg uncertainty principle expresses a limitation on accuracy of (nearly) simultaneous measurement of observables such as the position and the momentum of a particle. It furthermore precisely quantifies the imprecision by providing a lower bound (greater than zero) for the product of the dispersions of the measurements. For instance, consider repeated trials of the following experiment: By an operational process, a particle is prepared in a definite state and two successive measurements are performed on the particle. The first one measures its position and the second immediately after measures its momentum. Suppose furthermore that the operational process of preparing the state is such that on every trial the first measurement yields the same value, or at least a distibution of values with a very small dispersion d_p around a value p. Then the second measurement will have a distribution of values whose dispersion d_q is at least inversely proportional to d_p. In quantum mechanical terminology, the operational process has produced a particle in a pure state with definite position. Any momentum measurement on the particle will necessarilly yield a dispersion of values on repeated trials. Moreover, if we follow the momentum measurement by a measurement of position, we will get a dispersion of position values. More generally, an uncertainty relation arises between any two observable quantities defined by non-commuting operators.\nIt is one of the cornerstones of quantum mechanics and was discovered by Werner Heisenberg in 1927.

Table of contents

1 Overview 2 Definition 2.1 Generalized uncertainty principle 2.2 Generalizations 3 Interpretations 4 References 5 External link

Overview

The uncertainty principle in quantum mechanics is sometimes erroneously explained by claiming that the measurement of position necessarily disturbs a particle's momentum. Heisenberg himself may have offered explanations which suggest this view, at least initially. That the role of disturbance is not essential can be seen as follows: Consider an ensemble of (non-interacting) particles all prepared in the same state for which we measure the momentum or the position (but not both). From the measurement results, we will obtain distributions of values for both these quantities and the uncertainty relations still hold for the dispersions of values. Correspondingly, any one particle (in the general sense, e. g. carrying discrete electric charge) cannot be described simultaneously as a "classic point particle" and as a wave. (The fact itself that either one of these descriptions can be appropriate at least in separate cases is called wave-particle duality; a change of appropriate descriptions according to measured values is known as wavefunction collapse.)\nThe uncertainty principle (as initially considered by Heisenberg) is concerned with cases in which neither of these two descriptions is fully and exclusively appropriate, such as a particle in a box with a particular energy value; i. e. systems which are characterized neither by one unique "position" (one particular value of distance form a potential wall) nor by one unique value of momentum (incl. its direction). Consider the following analogy: suppose you have a time-varying signal such as a sound wave, and you want to know the exact frequencies in your signal at an exact moment in time. \nThis is impossible: in order to determine the frequencies accurately, you need to sample the signal for some time and you thereby lose time precision. (In other words, a sound cannot have both a precise time, as in a short pulse, and a precise frequency, as in a continuous pure tone.) \nThe time and frequency of a wave in time are analogous to the position and momentum of a wave in space.

Definition

The statement is as follows. \nIf several identical copies of a system in a given state are prepared, measurements of position and momentum will vary according to known probability distributions; this is the fundamental postulate of quantum mechanics. \nWe could measure the standard deviation Δx of the position measurements and the standard deviation Δp of the momentum measurements. \nThen we will find that

where h is Planck's constant and π is Archimedes' constant.\n(In some treatments, the "uncertainty" of a variable is taken to be the smallest width of a range which contains 50% of the values, which, in the case of normally distributed variables, leads to a lower bound of h/2π for the product of the uncertainties.) \nNote that this inequality allows for several possibilities: the state could be such that x can be measured with high precision, but then p will only approximately be known, or conversely p could be sharply defined while x cannot be precisely determined. \nIn yet other states, both x and p can be measured with "reasonable" (but not arbitrarily high) precision. In everyday life, we don't observe these uncertainties because the value of h is extremely small.

Generalized uncertainty principle

The uncertainty principle does not just apply to position and momentum. In its general form, it applies to every pair of conjugate variables. An example of a pair of conjugate variables is the x-component of angular momentum (spin) vs. the y-component of angular momentum. \nIn general, and unlike the case of position versus momentum discussed above, the lower bound for the product of the uncertainties of two conjugate variables depends on the system state. The uncertainty principle becomes then a theorem in the theory of operators which we now state Theorem. For arbitrary symmetric operatorss A: H → H and B: H → H, and any element x of H such that A B x and B A x are both defined (so that in particular, A x and B x are also defined), then

This is an immediate consequence of the Cauchy-Bunyakovski-Schwarz inequality. Consequently, the following general form of the uncertainty principle, first pointed out in 1930 by Howard Percy Robertson and (independently) by Erwin Schrödinger, holds:

This inequality is called the Robertson-Schrödinger relation. The operator A B - B A is called the commutator of A, B and is denoted [A, B]. It is defined on those x for which A B x and B A x are both defined. From the Robertson-Schrödinger relation, the following Heisenberg uncertainty relation is immediate: Suppose A and B are two observables which are identified to self-adjoint (and in particular symmetric) operators. If B A ψ and A B ψ are defined then\n:\nwhere

is the operator mean of observable X in the syatem state ψ and

is the operator standard deviation of observable X in the system state ψ The above definitions of mean and standard deviation are defined formally in purely operator-theoretic terms. The statement becomes more meaningful however, once we note that these actually are the mean and standard deviation for the measured distribution of values. See quantum statistical mechanics. It may be evaluated not only for pairs of conjugate operators (e.g. those defining measurements of distance and of momentum, or of duration and of energy) but generally for any pair of Hermitian operators. There is also an uncertainty relation between the field strength and the number of particles which is responsible for the phenomenon of virtual particles. Note that it is possible to have two non-commuting self-adjoint operators A and B which share an eigenvector ψ in this case ψ represents a pure state which is simultaneously measurable for A and B.

Generalizations

Other forms of the uncertainty principle can be formulated for the Fourier transform on general locally compact groups or for Fourier integral operators on manifolds. For example, Hirschman proved in 1957 a form of the uncertainty principle which is stronger than the Weyl form stated above.

Interpretations

Main article: Interpretation of quantum mechanics Albert Einstein was not happy with the uncertainty principle, and he challenged Niels Bohr with a famous thought experiment: we fill a box with a radioactive material which randomly emits radiation. \nThe box has a shutter, which is opened and immediately thereafter shut by a clock at a precise time, thereby allowing some radiation to escape. \nSo the time is already known with precision. \nWe still want to measure the conjugate variable energy precisely. \nEinstein proposed doing this by weighing the box before and after. The equivalence between mass and energy from special relativity will allow you to determine precisely how much energy left the box. \nBohr countered as follows: should energy leave, then the now lighter box will rise slightly on the scale. \nThat changes the position of the clock. \nThus the clock deviates from our stationary reference frame, and again by special relativity, its measurement of time will be different from ours, leading to some unavoidable margin of error. \nIn fact, a detailed analysis shows that the imprecision is correctly given by Heisenberg's relation. Within the widely but not universally accepted Copenhagen interpretation of quantum mechanics, the uncertainty principle is taken to mean that on an elementary level, the physical universe does not exist in a deterministic form—but rather as a collection of probabilities, or potentials. \nFor example, the pattern (probability distribution) produced by millions of photons passing through a diffraction slit can be calculated using quantum mechanics, but the exact path of each photon cannot be predicted by any known method. \nThe Copenhagen interpretation holds that it cannot be predicted by any method. It is this interpretation that Einstein was questioning when he said "I cannot believe that God would choose to play dice with the universe." \nBohr, who was one of the authors of the Copenhagen interpretation responded, "Einstein, don't tell God what to do." Einstein was convinced that this interpretation was in error.\nHis reasoning was that all previously known probability distributions arose from deterministic events. \nThe distribution of a flipped coin or a rolled dice can be described with a probability distribution (50% heads, 50% tails). \nBut this does not mean that their physical motions are unpredictable. \nOrdinary mechanics can be used to calculate exactly how each coin will land, if the forces acting on it are known. \nAnd the heads/tails distribution will still line up with the probability distribution (given random initial forces). Einstein assumed that there are similar hidden variabless in quantum mechanics which underlie the observed probabilities. Neither Einstein or anyone since has been able to construct a satisfying hidden variable theory, and the Bell inequality illustrates some very thorny issues in trying to do so. Although the behavior of an individual particle is random, they are also correlated with the behavior of other particles. Therefore, if the uncertainty principle is the result of some deterministic process, it must be the case that particles at great distances instantly transmit information to each other to ensure that the correlations in behavior between particles occur. In some situations the Heisenberg uncertainty principle is called the Heisenberg indeterminacy principle. \nSee: Quantum indeterminacy.

References

• G. Folland and A. Sitaram, The Uncertainty Principle: A Mathematical Survey, Journal of Fourier Analysis and Applications, 1997 pp 207-238.
• W. Heisenberg, Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Zeitschrift für Physik, 43 1927, pp 172-198. English translatation: J. A. Wheeler and H. Zurek, Quantum Theory and Measurement Princeton Univ. Press, 1983, pp. 62-84.
• I Hirschman, A Note on Entropy, American Journal of Mathematics, 1957. This paper formulates a general uncertainty principle using a measure of uncertainty based on entropy.
• J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeon University Press, 1955. This book has been reissued in paprpack and is widely available
• R. Omnès, Understanding Quantum Mechanics, Princeton University Press, 1999.
• H. Weyl, The Theory of Groups and Quantum Mechanics, Dover Publications 1950. Was originally published in German in 1928.

External link

• Critical Resources: Heisenberg Uncertainty Principle

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