[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
  • ${\psi }_s\left(x\right)\propto \exp \left[ik\sqrt{{\left(\frac{x-s}{2}\right)}^2+z_0^2}\right]+\exp \left[ik\sqrt{{\left(\frac{x+s}{2}\right)}^2+z_0^2}\right]$
• approximate formulas for the distance gives
  • ${\psi }_s\left(x\right)\propto \exp \left(i\alpha \right)(\exp \left[ik\left(\frac{sx}{2z_0}\right)\right]+\exp [-ik\left(\frac{sx}{2z_0}\right)])$
  • $\alpha =k(z_0+\frac{x^2}{2z_0}+\frac{s^2}{8z_0})$
• applying $\exp -\cos$ relationship
  • ${\psi }_s\left(x\right)\propto \exp \left(i\alpha \right)\cos \left(\frac{ksx}{2z_0}\right)$
• intensity of the beam
  • $\left|{\psi }_s\right(x){|}^2\propto {\cos }^2=\frac{1}{2}[1+\cos \left(\frac{2\pi sx}{\alpha z_0}\right)]$

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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