PHYS 424 Notes

1. Chapter 1 [continued]
   3. Double Slit [continued]
      2. Low Intensity
         1. Experimental result
            • Decrease intensity of incident light
            • Compensate with intensification of signal at screen
            • Eventually you see individual flashes on the screen
            • Flashes add up to the interference pattern
            • Works for light and for electrons and nucleons
            • Measurement gives lambda = h / p , where h is Planck's constant and p is the particle momentum
            • Photoelectric effect shows that there is a time oscillation with angular frequency given by E= hbar w , where hbar is Planck's constant / (2 p) and E is the (free) particle energy.
         2. Interpretation
            • No choice:
            • Pattern gives probability of seeing flashes
            • To get interference, the probability must be the square of something wave-like, P = |y|², with y = y₁ + y₂.
            • Time dependence of y
              • cos(wt) or sin(wt) fails, because total probability is time-dependent
              • The only other choice is e^±iwt, which is fine because | e^{±iw t} | ² = 1, which is time-independent
              • Must choose a single sign, since e^-iwt + e^+iwt is not time independent.
              • Convention is y ~ e^-iwt
            • Therefore, for a free particle (as in the experiment), we need to use y = A e^{2 p i x / l} e^{-i w t}
            • In this case
              • - i hbar (d/dx)y = [2 p hbar / l] y = [h / l] y = p y
              • i hbar (d/dt)y = hbar w y = E y
              • There is also the obvious xy = xy
            • We call
              • -i hbar (d/dx) the operator p
              • i hbar (d/dt) the operator E
              • x the operator x [trivial of course]
              • Note the sign convention. It is designed for later consistency with special relativity
   4. Probability
      1. Discrete possibilities, probability P_i
         • S(P_i) = 1
         • <f> = S [f_i P_i]
         • s_f = Ö S { [f_i - <f>]² P_i}
      2. Continuous possibilities, probability P(x)dx
         • ò[P(x) dx] = 1, so the number of events between x and x + dx is dN = P(x) dx. Emphasize
         • <f(x)> = ò [f(x) P(x) dx]
         • s_f = Ö ò { [f(x) - <f(x)>]² P(x) dx}
         • Draw picture of a Gaussian distribution and interpret pictorially
      3. Some subtleties
         Card decks

Last revised 2002/09/03

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