1 / 11

Quantum Mechanics

Quantum Mechanics. Tirtho Biswas Cal Poly Pomona 10 th February. Review. From one to many electron system Non-interacting electrons (first approximation) Solve Schroidinger equation With subject to Boundary conditions Obtain Energy eigenstates

lenora
Download Presentation

Quantum Mechanics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Quantum Mechanics Tirtho Biswas Cal Poly Pomona 10th February

  2. Review • From one to many electron system • Non-interacting electrons (first approximation) • Solve Schroidinger equation • With subject to Boundary conditions • Obtain Energy eigenstates • Include degeneracy (density of states) • Obtain ground state configuration according to Pauli’s exclusion principle • Excited states  Thermodynamics (later)

  3. Free Electron Loosely bound • How does the spectrum of a free particle in a box look like? Almost continuous band of states • How do you think the spectrum will change if we add a potential to the system? • No change • The spectrum will still be almost continuous, but the spacing will decrease • The spectrum will still be almost continuous, but the spacing will decrease • The spectrum will separate into different “bands” separated by “gaps”.

  4. Kronig-Penney Model • How to model an electron free to move inside a lattice? Periodic potential wells controlled by three important parameters: • Height of the potential barrier • Width of the potential barrier • Inter-atomic distance • Is there a clever way of solving this problem? Symmetry • Bloch’s theorem: If V(x+a) = V(x) then

  5. Dirac-Kronig-Penney Model • Simplify life to get a basic qualitative picture • What strategy to adopt in solving SE? Solve it separately in different regions and then match • What is the wave function in Region II?

  6. Matching Boundary conditions • Wavefunction is coninuous • The derivatives are discontinuous if there is a delta function: • Condition from wavefunction continuity

  7. Lets calculate the derivatives • What about region II?

  8. Discontinuity of derivatives gives is • Eventually one finds • depends on the property of the material

  9. Energy Gap • Depending upon the value of , there are values of k for which the |RHS|>1 => no solutions • There are ranges in energy which are forbidden! • Larger the , the bigger the band gaps • With increasing energy the band gaps start to shrink

  10. Energy Bands • No object is really infinite…we can connect the two ends to form a wire, for instance. •  = a can then only take certain discrete values LHS = cos  • N states in a given band, one solution of z, for every value of . • Let’s not forget the spin => 2N states https://phet.colorado.edu/en/simulation/band-structure

  11. If each atom has q valence electrons, Nq electrons around • q = 1 is a conductor…little energy to excite • q =2 is an insulator…have to cross the band gap • Doping (a few extra holes or electrons) allows to control the flow of current…semiconductors • Applications of semiconductors • Integrated circuits (electronics) • Photo cells • Diodes • Light emitting diodes (LED) • Solar cell…

More Related