We've considered that light, which is typically thought of as a wave, can also act as a particle. We'll now consider how electrons, which are typically thought of as particles, can also act as waves. If we run a diffraction experiment with a beam of particles, the particles will pass straight through our slits without diffracting and produce two bright spots on our film corresponding to the two slits. If we run our diffraction experiment with an electron source that sends electrons slowly enough so we essentially have one electron at a time, a single electron can simultaneously enter both slits and interfere with itself on the other side. The result is an interference pattern similar to light acting as a wave: with the most electrons detected in the center, and alternating constructive and destructive interference moving out from the center. de Broglie hypothesized that if light can act as a wave or a particle, perhaps particles can also act as waves. The de Broglie wavelength of a particle is equal to Planck's constant divided by the particle's mass times its speed. Notice that this is the speed of the particle, v, instead of frequency, nu. Also, notice that the mass of the particle must be in kilograms in this equation. This is because one joule equals 1 kg meter squared per second squared. If we calculated the de Broglie wavelength of an electron, it would be on the scale of an atom, so the de Broglie wavelength is significant for the electron compared to its size. To recap, if we run our diffraction experiment with a stream of electrons, we get an interference pattern as if it's functioning as a wave. What's even more interesting is if we add a laser on the other side of the slit, we get different results. The laser will allow us to observe the electron after it passes through the slit. As the electron crosses the laser, it scatters a photon from the laser and we see a flash. This flash allows us to tell something about the electron's position. When electrons are sent at a slow enough rate so that one electron passes through the slit at a time, the flash only occurs at one slit or the other; it never occurs at both simultaneously. The act of observing the electron's position forces the electron to act as a particle, and we lose our diffraction pattern. Instead, the typical pattern for a particle forms on our film. We cannot observe the electron simultaneously acting as both a wave and a particle. The wave nature and particle nature of an electron are complementary properties: we cannot observe both at once. In fact, the more we observe about the particle nature (or position) of the electron, the less we can simultaneously observe about the wave nature (or velocity or energy) of the electron. According to Heisenberg's uncertainty principle, if you simultaneously measure the position and velocity of an electron, the uncertainty in your position measurement times the mass of the electron times the uncertainty of your velocity measurement must be greater than or equal to Planck's constant divided by 4 pi. So the more exact you know the electron's position, the less exact you can tell its velocity, and vice versa. In classical physics, we can know the trajectory of a particle. For example, if the baseball is hit into the outfield, the outfielder is able to position him or herself to catch the baseball because the path of that baseball follows a classical trajectory. If we calculated the de Broglie wavelength of the baseball, it would be around 5 times 10 to the -35 meters. This is insignificant compared to the size of the baseball, so the baseball will act as a particle, not a wave. Not so with an electron; the future path of an electron is indeterminate. It is not possible to know both the position and velocity of an electron, but we can consider the probability of finding an electron in a particular area. We'll call this a probability distribution map. Continuing with our baseball analogy, let's say a pitcher throws 1000 pitches, and we track where those pitches cross the plate. We could come up with a graph that shows how likely a pitch is to cross the plate within a particular distance from the center of the strike zone. This is the idea of a probability distribution map. We could do the same with an electron, showing how likely it is that an electron will be found in a particular volume. This will be used in our quantum mechanical model of the atom.