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Understanding Quantum Tunneling and Electron Behavior in Potential Barriers

This lecture delves into the phenomenon of quantum tunneling, explaining how particles with kinetic energy can interact with potential barriers. It contrasts classical and quantum mechanical views, illustrating that classical mechanics would prevent particles with energy less than the barrier height from crossing. The lecture also discusses applications such as tunneling in nuclear fusion and scanning tunneling microscopy. Detailed calculations for electron probabilities and diffraction patterns are presented, enhancing the understanding of electron behavior at the quantum level.

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Understanding Quantum Tunneling and Electron Behavior in Potential Barriers

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  1. Lecture 5

  2. Tunneling classically An electron of such an energy will never appear here! Ekin= 1 eV 0 V -2 V x

  3. Potential barriers and tunneling According to Newtonian mechanics, if the total energy is E, a particle that is on the left side of the barrier can go no farther than x=0. If the total energy is greater than U0, the particle can pass the barrier.

  4. Tunneling – quantum approach Schroedinger eq. for region x>L Solution:

  5. Potential barriers and tunneling Two solutions: or Normalization condition: Solution: The probability to find a particle in the region II within

  6. Potential barriers and tunneling

  7. Potential barriers and tunneling A metal semiconductor example insulator Let electrons of kinetic energy E=2 eV hit the barrier height of energy U0= 5 eV and the width of L=1.0 nm. Find the percent of electrons passing through the barrier? T=7.1·10-8 If L=0.5 nm.then T=5.2 ·10-4!

  8. Scanning tunneling electron miscroscope

  9. Scanning tunneling electron miscroscope

  10. Scanning tunneling electron miscroscope

  11. Scanning tunneling electron miscroscope

  12. Scanning tunneling electron miscroscope Image downloaded from IBM, Almaden, Calif. It shows 48 Fe atoms arranged on a Cu (111) surface

  13. a particle decay Approximate potential - energy function for an a particle in a nucleus.

  14. Tunneling Nuclear fusion ( synteza ) is another example of tunneling effect E.g. The proton – proton cycle

  15. d   Young’s double slit experiment a) constructive interference For constructive interference along a chosen direction, the phase difference must be an even multiple of  m = 0, 1, 2, … b) destructive interference For destructive interference along a chosen direction, the phase difference must be an odd multiple of  m = 0, 1, 2, …

  16. Electron interference a, b, c – computer simulation d - experiment

  17. Im Re Franhofer Diffraction a dy R   R E 

  18. Electron Waves • Electrons with 20eV energy, have a wavelength of about 0.27 nm • This is around the same size as the average spacing of atoms in a crystal lattice • These atoms will therefore form a diffraction grating for electron “waves”

  19. C.J.Davisson and L.G.Germer dNi=0.215nm diffraction de Broglie

  20. Resolution Rayleigh’s criterion: When the location of the central maximum of one image coincides with the the location of the first minimum of the second image, the images are resolved. For a circular aperture:

  21. Electron Microscope

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