Absolutely Small - Michael D. Fayer [21]
HOW FAST IS AN EJECTED ELECTRON
It is interesting to get a feel for how fast an electron moves when it is ejected from a piece of metal. Different metals have different binding energies called work functions. A binding energy for a metal can be determined by tuning the color redder and redder and seeing the wavelength of light at which photons cannot eject electrons. For a metal with a small binding energy, a typical cutoff wavelength for electron ejection is 800 nm. For λ = 800 nm, ν = 3.75 × 1014 Hz, and Eb = hν = 2.48 × 10-19 J. If we shine green light on the metal with a wavelength of 525 nm, the energy of the photon is 3.77 × 10 - 19 J. The kinetic energy of the electron that will be ejected from the metal is Ek = hν-Eb = 1.30 × 10 - 19 J. We can find out how fast the electron is moving using , where me is the electron mass, me = 9.11 × 10-31 kg (kg is kilograms, that is, 1000 grams). Multiplying the equation for Ek by 2 and dividing by me gives V2 = 2(1.30 × 10-19 J)/me = (2.60 × 10-19 J)/(9.11 × 10-31 kg) = 2.85 × 1011 m2/s2. This value is the square of the velocity. Taking the square root, V = 5.34 × 105 m/ s, which is about one million miles per hour. In this example of the photoelectric effect, the ejected electrons are really moving.
Classical electromagnetic theory describing light as waves seems to work perfectly in describing a vast array of phenomena including interference, but it can’t come close to explaining the photoelectric effect. Einstein explains the photoelectric effect, but now light can’t be waves, so what happens to the classical description of interference? Reconciling the photoelectric effect and interference brings us to the cusp of quantum theory and back to Schrödinger’s Cats.
5
Light: Waves or Particles?
THE EXPLANATION OF THE photoelectric effect discussed in Chapter 4 required a new theoretical description of the interferometer experiment discussed in connection with Figure 3.4. Understanding the interferometer experiment in a manner that does not contradict the description of the photoelectric effect requires the big leap into thinking quantum mechanically rather than thinking classically. In discussing absolute size in Chapter 2, the idea was introduced that for a system that is small in an absolute sense, a measurement will make an unavoidable nonnegligible disturbance. However, we did not discuss the nature or consequences of such a disturbance. Now, we need to come to grips with the true character of matter and what happens when we make measurements.
The problem we have is that light waves were used to explain the interference phenomenon in Figure 3.4, but “particles of light,” quanta called photons, were used to explain the photoelectric effect in connection with Figures 4.3 and 4.4. The classical mathematical description of light waves employed Maxwell’s equations to quantitatively describe interference. The mathematical entity that represented a light wave in the theory is called a wavefunction. A function gives a mathematical description of something, in this case a light wave. It describes the amplitude, frequency, and the spatial location of a light wave. The incoming light wave is described by a single wavefunction. In the classical description, after the light wave hits the 50% beam splitter, half of the wave goes into each leg of the interferometer (see Figure 3.4). There are now two waves and two wavefunctions, one for each wave. These wavefunctions describe waves that are each half of the intensity of the original incoming wave and are in different locations, the two legs of the interferometer. When these two wavefunctions are combined mathematically to describe the nature of the overlap region inside the circle in Figure 3.4, the interference pattern can be calculated. All of this worked so well that it was thought that the same math must apply to photons.
CLASSICAL DESCRIPTION OF INTERFERENCE DOESN’T WORK FOR PHOTONS
Figure 5.1 shows the interferometer again with everything exactly the same as in Figure 3.4 except that the incoming light beam