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Appendix 11. Heisenberg’s Uncertainty Principle
- University of Arkansas Press
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Appendix 11 HEISENBERG’S UNCERTAINTY PRINCIPLE transitory Species at the Foundation of Continuous Identity: Chaos at the Foundation of Order One of the characteristic aspects of the ordinary objects of our conscious experience is the fact that their physical attributes seem innate, inexorably a part of them. The table I write on exists, because it owns its position in space; it also owns its linear momentum or other dynamic (variable) attributes. In addition, it is a characteristic aspect that any combination of physical attributes can seemingly be measured with some precision, subject only to the limitations of the measuring device. In contrast, quantum particles are different. Their dynamic attributes are conjugated in pairs. If one of them is known exactly, the other one is unknowable. Consider a particle residing on an axis in space, say the x-axis. When its position is measured repeatedly, to some extent the results will vary. In general, all measurements are affected by experimental error, afflicted with experimental uncertainty. We characterize the differences that may occur between various measurements by the Greek letter delta, ∆. When measurements of position, x, are made, we say that the uncertainty in position is ∆x. For example, if the particle was found at x = . cm within an accuracy of plus or minus . cm, then ∆x = . cm. Similarly, when momentum, px, is measured along the same axis, it will be known with a degree of uncertainty designated by ∆px. Heisenberg’s uncertainty principle states that the dynamic variables of a particle come in pairs which are connected in such a way that the uncertainty in one multiplied by the uncertainty in the other cannot yield a value that is smaller than Planck’s constant, h, divided by π. Position and momentum are such a conjugate pair. When the uncertainty in position, ∆x, is multiplied by the uncertainty in momentum, ∆px, the result cannot be smaller than h/π, meaning that to the extent we are sure of the value of one we cannot know the value of the other. We write ∆x∆px≥h/4π. 1SCHÄFER_PAGES:SCHÄFER PAGES 4/29/10 11:14 AM Page 174 Heisenberg derived the uncertainty principle by a thought experiment that he performed using a microscope to determine position and momentum of a particle. We are not concerned here with the technicalities of the matter, but will consider some specific situations to illustrate its meaning. Say that, for example, the position of a particle is known exactly. This means that the uncertainty is zero; ∆x = . In that case, in order to satisfy the uncertainty principle, ∆px must be infinitely large, because only then the product with ∆x = can yield a value that is greater than h/π. When the uncertainty in momentum is infinitely large, nothing is known about the momentum. When position is known exactly, momentum is unknowable. And vice versa, when the momentum is known exactly, nothing is known about the particle’s whereabouts. The relationship is exactly what the name says; that is, a matter of principle. It has nothing to do with the precision of measuring devices or the skills of experimenters. If an instrument existed that could determine position without any uncertainty and if, further, a second instrument could determine momentum without any uncertainty, Heisenberg’s principle says that the two instruments could not do their job at the same time. The two variables can never be known exactly at the same time, ever. All kinds of situations were designed in attempts to falsify the uncertainty principle. None of them has succeeded, because the relation is fundamentally connected with the wave-particle duality. For example, the solution to Schrödinger’s equation for a free particle moving in some direction in space, say x, is a wave ψ(x) with an exactly defined wavelength, λ. When λ is known exactly, momentum is also known exactly, since the two are related by the equation p = h/λ. When momentum is defined exactly, its uncertainty is zero and we say that the particle is in a definite momentum state. To test the uncertainty principle we will now consider the position of the particle in a definite momentum state. As usual, information about position is given by squared quantum amplitude |ψ(x)|2. For a free particle the wave function ψ(x) is such that the probability to find it is the same everywhere in the universe. Thus, the position of a free particle in a definite momentum state...