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Physics A: Problem Set 23: The Wave Nature of Matter

recommended reading

Barron's Let's Review: n/a
physics.info: Matter Waves
Wikipedia: Matter wave, Wave-particle duality
HyperPhysics: Wave-Particle Duality, Wave Nature of Electron

writing

  1. Light and other forms of electromagnetic radiation have both particle and wave properties. Answer the following questions. Provide an example of a device, object, phenomena, or situation that can be used to demonstrate each of the two different models of light. Choose examples from common experience. No fancy laboratory experiments or abstract textbook situations.
    1. What evidence do we have that light is a wave?
    2. What evidence do we have that light is composed of particles (photons)?
    1. Diffraction and interference can be observed in the light reflected from the closely spaced grooves of a CD, DVD, Blu-Ray disc, or phonograph record. Diffraction and intereference are both properties of waves. The thin film intereference pattern that appears when gasoline is spilled on water is also evidence that light is a wave.
    2. The photoelectric effect of photons pusheing electrons around is a good example of a particle property. Photovoltaic cells like the kind found on some calculators, homes, and emergency call boxes all rely on the photoelectric effect to work. The charge couple device (CCD) sensor that is at the heart of digital cameras would not work if photons did not push electrons around like they were particles.
  2. Matter has both particle and wave properties.
    1. Why don't we notice the particle nature of matter in our everyday experience?
    2. What experimental evidence do we have to show that matter is composed of particles (atoms, molecules, ions)?
    3. Why don't we notice the wave nature of matter in our everyday experience?
    4. What experimental evidence do we have to show that elementary particles can behave like waves?
    1. We don't notice the particle nature of matter in our everyday experience because the particles that make up the world around us (atoms, ions, molecules) are very, very small.
    2. All of chemistry is based on the notion that matter can be decomposed into finitely small entities that combine chemically in definite proportions. This is known as the law of definite proportions. For example, when water decomposes into hydrogen and oxygen, there is always twice as much hydrogen gas by volume than there is oxygen gas (or 18 times more oxygen gas than hydrogen gas by mass).
    3. We notice the wave nature of matter in our everyday experience because the wavelengths of ordinary sized objects are ridiculously small — smaller than anything that has ever been measured. For an object to have a wavelength that is measurable, it would have to be small (have a small momentum). Electrons are probably the best example of a small material object with a measurable wavelength, but as was pointed out in part a, they are hard to notice since they are so very small themselves.
    4. The first experimental evidence of elementary particles behaving like waves was the electron diffraction experiment of Davisson and Germer in the 1920s. The results of this experiment lead to the development of the electron microscope, which by its very name is an analog to an ordinary light microscope. Light behaves like a wave and so do electrons.

calculation

  1. The Davisson-Germer experiment that first demonstrated the wave nature of matter used electrons accelerated to 54 V.
    1. Determine the energy of the electrons in…
      1. electron volts
      2. joules
    2. How fast were the electrons moving in this experiment?
    3. What momentum did these electrons have?
    4. Determine the wavelength of these electrons.
    5. What form of electromagnetic radiation has the same wavelength as the value you calculated in part d.?
    1. When electrons are accelerated through a potential difference of 54 V they acquire 54 eV of energy. To convert this to joules, multiply by the elementary charge.

      (54 eV)(1.60 × 10−19 C) = 8.64 × 10−18 J

    2. Rearrange the kinetic energy equation to compute the speed of the electrons.

      v = 
      2K
      m
       
       
      v =  2(8.64 × 10−18 J)  
      9.11 × 10−31 kg  
      v = 4,360,000 m/s  
       
    3. Use the definition of momentum to compute the momentum of an electron.

      p = mv
      p = (9.11 × 10−31 kg)(4,360,000 m/s)
      p = 3.97 × 10−24 kgm/s
    4. Use de Broglie's equation to find the wavelength of an electron.

      λ = 
      h
      p
       
       
      λ =  6.63 × 10−34 Js  
      3.97 × 10−24 kgm/s  
      λ = 1.67 × 10−10 m  
       
    5. This is comparable to an x-ray photon.