- 79 Views
- Uploaded on

Download Presentation
## PowerPoint Slideshow about ' Quantum Mechanics' - lenora

**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.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

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

- 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

- 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

- 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

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

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

- 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…

Download Presentation

Connecting to Server..