Quantum mechanics
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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

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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


Quantum mechanics

  • Lets calculate the

    derivatives

  • What about region II?


Quantum mechanics

  • 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


Quantum mechanics

  • 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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