3.024 Electrical, Optical, and Magnetic Properties of Materials
Recitation 7 Notes
Exam Major Concepts Review
Spring 2012
Classical Hamiltonian Mechanics
∑
2nd Order Ordinary Differential Equation General Solution
̈
Euler’s Formula
Classical Free Particle
Massless Wave
De Broglie Particle-Wave
I.
∫
{
{
=
=
II.
Probability Density
∫
∫
III.
⟨ | ̂| ⟩
̂
1
∫
̂
3.024 Electrical, Optical, and Magnetic Properties of Materials
Recitation 7 Notes
IV.
Total State
|⟨
V.
|
⟩|
Spring 2012
∑
|∫
| |
|
̂
VI.
̂
̂
Operators
1D Position ̂
3D Position ̂
⃗
̂
1D Momentum ̂
3D Momentum ̂
̂
̂
(̂
̂
̂
)
1D Hamiltonian ̂
3D Hamiltonian ̂
⃗ =
(
)
Systems Studied
Particle-In-A-Box
Quantum Simple Harmonic Oscillator
Hydrogen Atom
Free Electrons
Electron in a Crystal Periodic Potential
⟨ ̂⟩
Conservation
̂
Ehrenfest Theorem
̂
⟨ ̂ ⟩ ⟨[ ̂ ̂ ]⟩
⟨ ⟩
Real Material Effects of Element Choice (Periodic Table Trends)
N = Atomic Number
2
⃗
3.024 Electrical, Optical, and Magnetic Properties of Materials
Recitation 7 Notes
Spring 2012
Free Electrons
Eigenstates are given by energy E (magnitude) and momentum in k (direction)
√
If the total wave function of a system is a sum of different momentum eigenstates:
∑
The average value of momentum is then given as:
⟨ ⟩
∑
∑
The average value of energy is similarly given as:
⟨ ⟩
∑
∑
Bloch Theorem
Band Diagrams
Plots of E vs k for electrons
Plot below shows 1st 5 bands for
∑
to a 2nd order Central Matrix Equation approximation
(| |
| |)
3
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3.024 Electronic, Optical and Magnetic Properties of Materials
Spring 2013
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