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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

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quantum mechanics
Quantum Mechanics

Tirtho Biswas

Cal Poly Pomona

10th February

review
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)
free electron loosely bound
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”.
kronig penney model
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
dirac kronig penney model
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?
matching boundary conditions
Matching Boundary conditions
  • Wavefunction is coninuous
  • The derivatives are discontinuous if there is a delta function:
  • Condition from wavefunction continuity
slide7

Lets calculate the

derivatives

  • What about region II?
slide8

Discontinuity of derivatives gives is

  • Eventually one finds
  • depends on the property of the

material

energy gap
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
energy bands
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

slide11

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…
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