PHYS 424 Notes

1. Chapter 1.
   1. Administrative Matters
      1. Roster
      2. Syllabus
         1. Reasons for coverage
         2. Credit for assignments [existance only]
            • 110% if turned in by due date
            • 100% if turned in by (actual) discussion date
            • 75% if turned in within one week of due date
            • 50% if turned in before relevant exam
         3. Work in groups
         4. Exam dates
         5. Grades: 20% homework, 30% two higher exams, 20% lowest exam
      3. Text: Griffiths, Introduction to Quantum Mechanics
         1. Good text
         2. Better than Griffiths's E & M because there are few long derivations, which Griffiths is not very good at motivating
         3. Footnotes are substantive and interesting
         4. Will cover chapters 1-6 with a few omissions + a little on WKB approximation (8.1 and 8.2)
      4. Office hours
   2. Double slit
      1. Waves and double slits

         

         Our first step is to look in detail at the double-slit experiment in order to learn what atom-level physics looks like and to surmise what a quantum theory might look like. We will find that the world is inherantly probabilistic and that what we can calculate is a function whose square gives the probability of a given experimental result. We will see how to calculate probabilities given this function, and guess (subject to later experimental confirmation) how to calculate the function.

         We can evaluate the magnitude of a wave coming through two slits using the diagram on the right. We see that

         z_{1, 2}² = x² + (a-+y)²
                 ~ x² + y² -+ 2ay
         so
         z_1,2 ~ R +- ay/R
         

         Therefore
         

         (A/2)[e^ikz₁ + e^ikz₂]e^-iwt
                 = (A/2) e^ikR-iwt [e^-ikay/R + e^ikay/R]
                 = A e^ikR-iwt cos(kay/R)
         

         where R = Ö[x² + y²]

         The intensity of a wave is given by its absolute-magnitude squared, so

         I = A² cos²(kay/R)               [1-1]

         On the other hand, if we had a spray of particles coming through the slits, we would have a roughly constant distribution of particles hitting the screen from each slit, the number of particles hitting at any given point would be the sum of those hitting from each slit, and hence there would be a roughly constant number of particles hitting the screen at any point. Thus, oscillations in the intensity of collisions with the screen as a function of height is a clean signal of the presence of waves in the double-slit experiment.

         Eq. 1-1 gives a quantitative match to experiments with electromagnetic waves or electrons or nucleons coming through two slits and landing on a detecting screen. Therefore we are forced to conclude that all three systems involve the propagation of waves. In the case of EM waves, there is obviously no surprise. For electrons and nucleons the result is unexpected, to say the least.

         Notice that the positions of the minima of the cosine, which look like dark lines on the screen, are determined by the location of both slits. If information about the position of either slit were not carried to the screen, there would be no way to determine that a minimum should be present.

Summarize

Last revised 2002/09/03

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