Quantum Mechanics
Tirtho Biswas
Cal Poly Pomona
10th February
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’sexclusion principle
Excited states Thermodynamics (later)
Free Electron Loosely bound
How does the spectrum of a free particle in a boxlook like?
Almost continuous band of states
How do you think the spectrum will change if weadd a potential to the system?
A)No change
B)The spectrum will still be almost continuous, but thespacing will decrease
C)The spectrum will still be almost continuous, but thespacing will decrease
D)The spectrum will separate into different “bands”separated by “gaps”.
Kronig-Penney Model
How to model an electron free to move inside alattice?
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 adelta 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
Energy Gap
Depending upon the value of , there are valuesof 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 toshrink
Energy Bands
No object is really infinite…we can connect the twoends 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 everyvalue of .
Let’s not forget the spin => 2N states
https://phet.colorado.edu/en/simulation/band-structure
If each atom has q valence electrons, Nqelectrons 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 tocontrol the flow of current…semiconductors
Applications of semiconductors
Integrated circuits (electronics)
Photo cells
Diodes
Light emitting diodes (LED)
Solar cell…