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Quantum Mechanics

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Tirtho Biswas

Cal Poly Pomona

10th February

- 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)

- 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”.

- 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

- 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?

- Wavefunction is coninuous
- The derivatives are discontinuous if there is a delta function:
- Condition from wavefunction continuity

- Lets calculate the
derivatives

- What about region II?

- Discontinuity of derivatives gives is
- Eventually one finds
- depends on the property of the
material

- 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

- 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

- 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…