Simple Nature - Light and Matter

traveling waves going in opposite directions. The equation p = h/λreally only gives the magnitude of the momentum vector, not itsdirection, so we should really interpret the wave as a 50/50 mixtureof a right-going wave with momentum p = h/λ and a left-going onewith momentum p = −h/λ. The uncertainty in our knowledge ofthe electron’s momentum is ∆p = 2h/λ, covering the range betweenthese two values. Even if we make sure the electron is in the groundstate, whose wavelength λ = 2L is the longest possible, we have anuncertainty in momentum of ∆p = h/L. In general, we find∆p h/L ,with equality for the ground state and inequality for the higherenergystates. Thus if we reduce L to improve our knowledge of theelectron’s position, we do so at the cost of knowing less about itsmomentum. This trade-off is neatly summarized by multiplying thetwo equations to give∆p∆x h .Although we have derived this in the special case of a particle in abox, it is an example of a principle of more general validity:The Heisenberg uncertainty principleIt is not possible, even in principle, to know the momentum and theposition of a particle simultaneously and with perfect accuracy. Theuncertainties in these two quantities are always such that ∆p∆x h.(This approximation can be made into a strict inequality, ∆p∆x >h/4π, but only with more careful definitions, which we will notbother with.)Note that although I encouraged you to think of this derivationin terms of a specific real-world system, the quantum dot, noreference was ever made to any specific laboratory equipment or procedures.The argument is simply that we cannot know the particle’sposition very accurately unless it has a very well defined position,it cannot have a very well defined position unless its wave-patterncovers only a very small amount of space, and its wave-pattern cannotbe thus compressed without giving it a short wavelength anda correspondingly uncertain momentum. The uncertainty principleis therefore a restriction on how much there is to know abouta particle, not just on what we can know about it with a certaintechnique.An estimate for electrons in atoms example 17⊲ A typical energy for an electron in an atom is on the order of(1 volt)·e, which corresponds to a speed of about 1% of the speedof light. If a typical atom has a size on the order of 0.1 nm, howSection 13.3 Matter As a Wave 863