Quantum Mechanics Tirtho Biswas Cal Poly Pomona 10 th February.
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Quantum Mechanics Tirtho Biswas Cal Poly Pomona 10 th February
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Transcript of Quantum Mechanics Tirtho Biswas Cal Poly Pomona 10 th February.
Quantum Mechanics
Tirtho Biswas
Cal Poly Pomona
10th February
ReviewFrom 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 boundHow does the spectrum of a free particle in a box
look like?
Almost continuous band of statesHow do you think the spectrum will change if we
add a potential to the system?A) No change
B) The spectrum will still be almost continuous, but the spacing will decrease
C) The spectrum will still be almost continuous, but the spacing will decrease
D) The spectrum will separate into different “bands” separated by “gaps”.
Kronig-Penney ModelHow 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
)()( xeax ai
Dirac-Kronig-Penney ModelSimplify life to get a basic qualitative picture
What strategy to adopt in solving SE?
Solve it separately in different regions and then matchWhat is the wave function in Region II?
n
naxxV )()(
)])(sin[)](cos[()()
)])(sin[)](cos[()()
)]sin()cos([)()
)sin()cos()()
axkBaxkAexD
axkBaxkAexC
kxBkxAexB
kxBkxAxA
ai
ai
ai
Matching Boundary conditionsWavefunction is coninuous
The derivatives are discontinuous if there is a delta function:
Condition from wavefunction continuity
)0()0( xx III
kae
ka
B
Aai cos
sin
2
2
m
x
Lets calculate the
derivatives
What about region II? kBxC
kBxD
kAxC
kAxB
BAkxA
I
I
I
I
I
)0(')
)0(')
)0(')
)0(')
][)0(')
)]sin()cos([)(')
)]cos()sin([)(')
)]cos()sin([)(')
)]cos()sin([)(')
kaBkaAkexD
kaBkaAkexC
kaBkaAkexB
kaBkaAkexA
aiII
aiII
aiII
aiII
Discontinuity of derivatives gives is
Eventually one finds
depends on the property of the material
z
zz
ka
kakaa
sincos
sincoscos
ai
ai
ekm
ka
kae
B
A
2
2sin
cos
2am
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
NN
NNNN
2)1(...,
24,
23,
22,
2,0
If each atom has q valence electrons, Nq electrons around
q = 1 is a conductor…little energy to exciteq =2 is an insulator…have to cross the band gapDoping (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…
