Physics 256F
Introduction to Quantum Physics: Problem Set 1.

• Problems from Gasiorowicz chapter 1:
  
  • 1.3
    
  • 1.4
    
  • 1.7
    
  • 1.12
    
• Additional problems:
  
  
  • Consider a Mach-Zehnder interferometer as discussed in class. There we assumed that each beam-splitter had transmission and reflection probabilities of 50%. Suppose instead that the reflection probability is 25%.
    We may assume, without loss of generality, that the transmission amplitude is real; however, the reflection amplitude will in general be a complex number in this case.
    
    • For a probability of 25%, what general form must the reflection amplitude take, including any hypothetical complex phase?
      What is the (real) transmission amplitude?
    • Consider light entering the interferometer from the left. What two paths can lead to light exiting the interferometer to the right? Individually, what are the probabilities for each of these paths? In the absence of interference, what would the total probability be for a photon to exit to the detector on the right?
      Taking interference into account, and including a phase-difference Q between the upper and lower paths, calculate the probability for a photon to reach this detector. What is its minimum value? What is its maximum value? What is its average value?
    • Now calculate the probability for a photon to take the other exit port, leaving the interferometer downwards instead of to the right. What are this probability's minimum, maximum, and average values?
    • What constraint does energy (or probability) conservation place on the reflection amplitude in light of the previous two parts?
    • Suppose the light exiting the bottom port is at its minimum intensity. If you block the light reflected from the first beam-splitter, what is the probability of a photon hitting your beam-block?
      What happens to the probability of light exiting the bottom port when you do this? Explain physically why this makes sense (or physically why it doesn't seem to make sense, as the case may be).
  • This problem relates to superposition and interference as discussed in class and tutorial, in particular, the generalisation of two-slit interference to the case of a large number of slits.
    • Consider a transmission diffraction grating with an infinite series of narrow slits separated by distance 3L. A stripe pattern will appear on a distant screen (assume the distance R is much greater than the separation between any of the slits in the problem, and sweep the "infinite" number of slits under the rug for now), indicating the points where the light from all the slits interferes constructively. Calculate the positions of the bright spots, using the fact that the different paths leading to a given bright spot must all differ in length by an integral number of wavelengths. (Make sure to use the approximation that R is much, much greater than L, to simplify the trigonometry-- i.e., the rays leaving two different slits for the same point on the screen can be treated as parallel.)
    • Now consider a similar grating, but with slit separation of only L. What happens to the diffraction (interference) pattern?
    • Finally, consider a diffraction grating that has a slit at 0, a slit at L, no slit at 2L; a slit at 3L, a slit at 4L, and no slit at 5L; et cetera. Use the superposition principle to deduce what the field at the slit must look like. Sketch the three diffraction patterns you have calculated.
    
    
    
    
    
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