The Applications of Nano Materials. Department of Chemical and Materials Engineering San Jose State University. Zhen Guo, Ph. D. Fundamentals of Nano Material Science Session II: Atomic Structure/Quantum Mechanics Session III: Bonding / Band Structures
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Department of Chemical and Materials Engineering
San Jose State University
Zhen Guo, Ph. D.
and Atomic Structure
-- Charge, Mass, Velocity, Energy
-- Follow Newton’s classic mechanics and Maxwell’s eletro-magnetic theory.
-- Diffraction: Young’s Double split experiment
-- De Broglie Relations: =h/P
The infringe pattern follows Bragg’s law if electron’s wave length obey De Broglie relations
Note: Wave Function itself does not have physical meaning while 2 has (probability of finding electron per unit volume)
where k is the wave number per unit length or k=2p/l
1-D Schrödinger Equation
3-D Schrödinger Equation
Free Electrons: Total Energy is equal to kinetic energy (V=0)
No energy quantizing needed. Electrons can occupy any energy
Inside Well (0<x<a), V=0
Outside Well (x<0 or x>a),
a=5cm, DE=4.53X10-16ev, no Quantum effect;
a=0.5nm, DE=4.53ev, (274nm light) Quantum Well
-- Incoming photons absorpted and excited electron from lower quantum state to higher state. Has to be exact wavelength / frequency.
-- Electron jump back from higher quantum state to lower one. Photons emitted are exact wavelength.
Coulomb Potential Energy
Principle Quantum Number: n=1, 2, 3, 4..... (or n=K, L, M, N...)
Orbital Angular Momentum Quantum Number l=0, 1, 2, ..(n-1), (or l=S, P, D, F...)
Magnetic Quantum Number ml=-l, -(l-1)...0, ...(l-1), l or |ml|<=l
Spin Angular Momentum Quantum Numberms=+1/2, -1/2 or |ms|=1/2
n=1, l=0, ml=0, => 1S state
n=2, l=0, ml=0, => 2S state
l=1, ml=-1, 0, 1 => 2Px, 2Py, 2Pz state.
Coulomb Potential Energy
=> Energy is a function of both n and l
-- Getting same ms number will allow electron take different ml, and thus different orbital (space) which can increase R12 and decrease Coulomb repulsive energy
vx or px
Easy to define wave length or p but not for x
Y(x) in X Space
g(K) in k Space (Reciprocal Space)
y(x)=eik0x, spread in x space
g(k)=k0 is a definite value
y(x)=x0, a delta equation
localized in the space
g(k)= e-ix0k, spread in k space
Dp x Dx > h / (2 p) = Planck's constant / (2 p)