The Uncertainty Principle. (a) Imaging an atom: Suppose that you want to measure the position of a single H atom with an accuracy of 1 fm (i.e., about the size of the nucleus) by "shining'T electromagnetic waves on it. What would be the resulting uncertainty in the momentum Ap of the atom after the measurement? What is the wavelength of a photon whose energy is ~(Ap)2/2m? (b) A laser pulse: A laser emits a pulse of duration 10 fs having a nominal wavelength of 500 nm. Using the uncertainty relation AE - At 2 h/Z, what must be the fractional spread in wavelength AMA of the resulting pulse? (c) The energytime uncertainty principle indicates that if you examine a region of space for a nite period of time, your measurement of the energy will be uncertain. Since Einstein's Relativity tells us that mass and energy are equivalent, the uncertainty in energy can correspond to creation and annihilation of particles on short time scales. The energy to create an electron positron pair is 1 MeV. How long can this amount of energy be "borrowed" from the vacuum before the pair must annihilate? (In fact, the vacuum is not nothing as we assume classically but on short time scales is a quantum froth in which particles appear and disappear. Even more amazing, these virtual particles have measurable consequences for, e.g., the emission spectra of hydrogen atoms!)