[QMSE] W02 - Two Slit Diffraction

young’s slits

• two slits at a distance \( s \) are made in a mask
  • then a wave is passed through a slit

• for large waves:
  • the waves are approximately uniformly “bright”
    • using exponential waves for convenience
  • \( \psi_s(x) \propto \exp{ \left[ ik \sqrt{ \left( \frac{x-s}{2} \right)^2 + z_0^2 } \right] } + \exp{ \left[ ik \sqrt{ \left( \frac{x+s}{2} \right)^2 + z_0^2 } \right] }\)
• approximate formulas for the distance gives
  • \( \psi_s(x) \propto \exp{(i\alpha)} \left( \exp\left[ ik \left( \frac{sx}{2z_0} \right) \right] + \exp\left[ -ik \left( \frac{sx}{2z_0} \right) \right] \right) \)
  • \( \alpha = k\left( z_0 + \frac{x^2}{2z_0} + \frac{s^2}{8z_0} \right) \)
• applying \( \exp{}-\cos{} \) relationship
  • \( \psi_s(x) \propto \exp{(i\alpha)} \cos\left( \frac{ksx}{2z_0} \right) \)
• intensity of the beam
  • \( \lvert \psi_s (x) \rvert^2 \propto \cos^2 = \frac{1}{2} \left[ 1 + \cos{\left( \frac{2\pi sx}{\alpha z_0} \right) } \right] \)

---

interpreting diffraction by two slits

• in the quantum mechanics paradigm, there is no specific size to the photon or electron

• the act of the electron or photon hitting the screen causes a measurement to be made
  • the wavefunction collapses into one with a definite position
    • whose probability is given by the born’s rule
• the establishment of which slit the electron goes through is meaningless
  • we can either have the interference pattern or know which slit the electron went through