Feynman units.”) The UP forces us to consider the position and momentum of the particle to be fundamentally imprecise or “fuzzy” in such a way that the particle occupies a region of at least ℏ in phase space ( x, p ). Thus we cannot speak of the trajectory of a quantum mechanical particle but must instead take account of the fuzziness of such a particle in our description, that is in terms of a wave function or probability amplitude. Most quantum mechanics textbooks also give a derivation of the UP from the commu- tation relation for the position and momentum operators, and it is also readily obtained from the fact that the wave functions in position and momentum space are Fourier trans- forms of each other. 16 However neither derivation is as conceptually simple as that using the Heisenberg microscope. There is an energy-time analog of the UP Eq.(6), but it has a somewhat different meaning. Consider a wave of frequency approximately ν and duration T , which thus consists of N ≈ νT cycles. The finite duration of the wave means that its leading and trailing edges will in general be distorted somewhat from sinusoidal, so that N will not be precisely well-defined and measurable, but will have an uncertainty of order Δ N ≈ 1. This uncertainty in N implies in turn an uncertainty in the frequency, given by Δ νT ≈ Δ N ≈ 1. (This relation is well-known in many fields, such as optics and electrical engineering, wherein it relates band-width and pulse-width; 17 the time T is subject to a number of somewhat different interpretations. The relation is also easy to derive more formally by calculating the Fourier transform of a finite nearly monochromatic wave train, which is its frequency spectrum.) Since the energy of a photon of light is given by E = hν = $_{ℏ}$ω we see that its energy can be measured only to an accuracy given by where Δ E is, of course, the absolute value of the energy uncertainty. The same relation holds by similar reasoning for most any quantum system. The expression Eq.(7) formally resembles the UP in Eq.(6), but unlike the position of a particle, time is not an observable in quantum mechanics, so Eq.(7) has a different meaning: T is the characteristic time of the system (eg. pulse width), and not an uncertainty in a time measurement. 18 Thus, for example the light emitted by an atom is only approximately monochromatic, with an energy uncertainty given by Δ E ≈ $_{ℏ}$/T where T is the lifetime of the excited atomic state. 5 Δ ET ≈ h ≈ ℏ (7)