One of the fundamental questions resolved in the early development of quantum mechanics was whether the electron motion was wavelike or particle-like. The answer was provided by De Broglie who showed that the particle linearmomentum, p, was related to an electromagnetic wavelength, l, as follows: l ¼ h p= , where h¼Planck’s constant ¼ 6:65 1034 J sec. This relationship was important, because it says that the motion of a

particle can be described in terms of wave propagation. Thus, for example, a particle enclosed in a microwave cavity can be described in terms of an electromagnetic standing mode or as a particle in a box. De Broglie’s equation proved to be a great triumph in the early development of quantum physics, especially when applied to atomic physics. Let’s apply the De Broglie relationship to a ring resonator. The ring resonator consists of a simple wire loop and is coupled to a microwave source. Further, let’s imagine a hypothetical situation in which the loop is so small that very few, perhaps one single electron, is flowing around the wire loop. Thus, the motion is constrained to be circular. Remarkably, it radiates maximum energy at resonance, as determined by nl ¼ 2pr, where n ¼ 1, 2, 3, . . . , l is the electrical wavelength of the radiated energy or the wavelength of the particle within the wire loop, and r is the radius of the loop. If we assume that De Broglie’s equation is applicable, l is related to the linear momentum, p, of the few particles ‘‘entrapped’’ in the resonator. Since the motion is circular, the angular momentum, G, is simply G ¼ pr. This implies that (applying the De Broglie relationship)

2pr ¼ h G rn


G ¼ nh:

G is in units of h and discrete. This implies that the magnetic moment of the ring resonator must also be quantized, since G is discrete.