Dreams of a Final Theory: The Scientist's Search for the Ultimate Laws of Nature

by Steven Weinberg

E.g., the frequency with which the wave function of any system in a state of definite energy oscillates is given by the energy divided by a constant of nature known as Planck’s constant. This system appears much the same to two observerswho have set their watches differently by one second, but, if they both observe the system when the hands on their watches both point precisely to twelve noon, they observe that the oscillation is at a different phase; because their watches are set differently they are really observing the system at different times, so that one observer may, e.g., see a crest in the wave, while the other sees a trough. Specifically, the phase is different by the number of cycles (or parts of cycles) that occur in one second; in other words, by the frequency of the oscillation in cycles per second, and hence by the energy divided by Planck’s constant. In today’s quantum mechanics, we define the energy of any system as the change in phase (in cycles or parts of cycles) of the wave function of the system at a given clock time when we shift the way our clocks are set by one second. Planck’s constant gets into the act only because energy is historically measured in units like calories or kilowatt hours or electron volts that were adopted before the advent of quantum mechanics; Planck’s constant simply provides the conversion factor between these older systems of units and the natural quantum-mechanical unit of energy, which is cycles per second. It can be shown ...