Quantum Mechanics for Engineers
© Leon van Dommelen
Next:
2. Mathematical Prerequisites
II
. Basic Quantum Mechanics
Subsections
2
. Mathematical Prerequisites
2
.
1
Complex Numbers
2
.
2
Functions as Vectors
2
.
3
The Dot, oops, INNER Product
2
.
4
Operators
2
.
5
Eigenvalue Problems
2
.
6
Hermitian Operators
2
.
7
Additional Points
2
.
7
.
1
Dirac notation
2
.
7
.
2
Additional independent variables
3
. Basic Ideas of Quantum Mechanics
3
.
1
The Revised Picture of Nature
3
.
2
The Heisenberg Uncertainty Principle
3
.
3
The Operators of Quantum Mechanics
3
.
4
The Orthodox Statistical Interpretation
3
.
4
.
1
Only eigenvalues
3
.
4
.
2
Statistical selection
3
.
5
A Particle Confined Inside a Pipe
3
.
5
.
1
The physical system
3
.
5
.
2
Mathematical notations
3
.
5
.
3
The Hamiltonian
3
.
5
.
4
The Hamiltonian eigenvalue problem
3
.
5
.
5
All solutions of the eigenvalue problem
3
.
5
.
6
Discussion of the energy values
3
.
5
.
7
Discussion of the eigenfunctions
3
.
5
.
8
Three-dimensional solution
3
.
5
.
9
Quantum confinement
4
. Single-Particle Systems
4
.
1
The Harmonic Oscillator
4
.
1
.
1
The Hamiltonian
4
.
1
.
2
Solution using separation of variables
4
.
1
.
3
Discussion of the eigenvalues
4
.
1
.
4
Discussion of the eigenfunctions
4
.
1
.
5
Degeneracy
4
.
1
.
6
Noneigenstates
4
.
2
Angular Momentum
4
.
2
.
1
Definition of angular momentum
4
.
2
.
2
Angular momentum in an arbitrary direction
4
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2
.
3
Square angular momentum
4
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2
.
4
Angular momentum uncertainty
4
.
3
The Hydrogen Atom
4
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3
.
1
The Hamiltonian
4
.
3
.
2
Solution using separation of variables
4
.
3
.
3
Discussion of the eigenvalues
4
.
3
.
4
Discussion of the eigenfunctions
4
.
4
Expectation Value and Standard Deviation
4
.
4
.
1
Statistics of a die
4
.
4
.
2
Statistics of quantum operators
4
.
4
.
3
Simplified expressions
4
.
4
.
4
Some examples
4
.
5
The Commutator
4
.
5
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1
Commuting operators
4
.
5
.
2
Noncommuting operators and their commutator
4
.
5
.
3
The Heisenberg uncertainty relationship
4
.
5
.
4
Commutator reference
4
.
6
The Hydrogen Molecular Ion
4
.
6
.
1
The Hamiltonian
4
.
6
.
2
Energy when fully dissociated
4
.
6
.
3
Energy when closer together
4
.
6
.
4
States that share the electron
4
.
6
.
5
Comparative energies of the states
4
.
6
.
6
Variational approximation of the ground state
4
.
6
.
7
Comparison with the exact ground state
5
. Multiple-Particle Systems
5
.
1
Wave Function for Multiple Particles
5
.
2
The Hydrogen Molecule
5
.
2
.
1
The Hamiltonian
5
.
2
.
2
Initial approximation to the lowest energy state
5
.
2
.
3
The probability density
5
.
2
.
4
States that share the electrons
5
.
2
.
5
Variational approximation of the ground state
5
.
2
.
6
Comparison with the exact ground state
5
.
3
Two-State Systems
5
.
4
Spin
5
.
5
Multiple-Particle Systems Including Spin
5
.
5
.
1
Wave function for a single particle with spin
5
.
5
.
2
Inner products including spin
5
.
5
.
3
Commutators including spin
5
.
5
.
4
Wave function for multiple particles with spin
5
.
5
.
5
Example: the hydrogen molecule
5
.
5
.
6
Triplet and singlet states
5
.
6
Identical Particles
5
.
7
Ways to Symmetrize the Wave Function
5
.
8
Matrix Formulation
5
.
9
Heavier Atoms
5
.
9
.
1
The Hamiltonian eigenvalue problem
5
.
9
.
2
Approximate solution using separation of variables
5
.
9
.
3
Hydrogen and helium
5
.
9
.
4
Lithium to neon
5
.
9
.
5
Sodium to argon
5
.
9
.
6
Potassium to krypton
5
.
9
.
7
Full periodic table
5
.
10
Pauli Repulsion
5
.
11
Chemical Bonds
5
.
11
.
1
Covalent sigma bonds
5
.
11
.
2
Covalent pi bonds
5
.
11
.
3
Polar covalent bonds and hydrogen bonds
5
.
11
.
4
Promotion and hybridization
5
.
11
.
5
Ionic bonds
5
.
11
.
6
Limitations of valence bond theory
6
. Macroscopic Systems
6
.
1
Intro to Particles in a Box
6
.
2
The Single-Particle States
6
.
3
Density of States
6
.
4
Ground State of a System of Bosons
6
.
5
About Temperature
6
.
6
Bose-Einstein Condensation
6
.
6
.
1
Rough explanation of the condensation
6
.
7
Bose-Einstein Distribution
6
.
8
Blackbody Radiation
6
.
9
Ground State of a System of Electrons
6
.
10
Fermi Energy of the Free-Electron Gas
6
.
11
Degeneracy Pressure
6
.
12
Confinement and the DOS
6
.
13
Fermi-Dirac Distribution
6
.
14
Maxwell-Boltzmann Distribution
6
.
15
Thermionic Emission
6
.
16
Chemical Potential and Diffusion
6
.
17
Intro to the Periodic Box
6
.
18
Periodic Single-Particle States
6
.
19
DOS for a Periodic Box
6
.
20
Intro to Electrical Conduction
6
.
21
Intro to Band Structure
6
.
21
.
1
Metals and insulators
6
.
21
.
2
Typical metals and insulators
6
.
21
.
3
Semiconductors
6
.
21
.
4
Semimetals
6
.
21
.
5
Electronic heat conduction
6
.
21
.
6
Ionic conductivity
6
.
22
Electrons in Crystals
6
.
22
.
1
Bloch waves
6
.
22
.
2
Example spectra
6
.
22
.
3
Effective mass
6
.
22
.
4
Crystal momentum
6
.
22
.
5
Three-dimensional crystals
6
.
23
Semiconductors
6
.
24
The
P-N
Junction
6
.
25
The Transistor
6
.
26
Zener and Avalanche Diodes
6
.
27
Optical Applications
6
.
27
.
1
Atomic spectra
6
.
27
.
2
Spectra of solids
6
.
27
.
3
Band gap effects
6
.
27
.
4
Effects of crystal imperfections
6
.
27
.
5
Photoconductivity
6
.
27
.
6
Photovoltaic cells
6
.
27
.
7
Light-emitting diodes
6
.
28
Thermoelectric Applications
6
.
28
.
1
Peltier effect
6
.
28
.
2
Seebeck effect
6
.
28
.
3
Thomson effect
7
. Time Evolution
7
.
1
The Schrödinger Equation
7
.
1
.
1
The equation
7
.
1
.
2
Solution of the equation
7
.
1
.
3
Energy conservation
7
.
1
.
4
Stationary states
7
.
1
.
5
The adiabatic approximation
7
.
2
Time Variation of Expectation Values
7
.
2
.
1
Newtonian motion
7
.
2
.
2
Energy-time uncertainty relation
7
.
3
Conservation Laws and Symmetries
7
.
4
Conservation Laws in Emission
7
.
4
.
1
Conservation of energy
7
.
4
.
2
Combining angular momenta and parities
7
.
4
.
3
Transition types and their photons
7
.
4
.
4
Selection rules
7
.
5
Symmetric Two-State Systems
7
.
5
.
1
A graphical example
7
.
5
.
2
Particle exchange and forces
7
.
5
.
3
Spontaneous emission
7
.
6
Asymmetric Two-State Systems
7
.
6
.
1
Spontaneous emission revisited
7
.
7
Absorption and Stimulated Emission
7
.
7
.
1
The Hamiltonian
7
.
7
.
2
The two-state model
7
.
8
General Interaction with Radiation
7
.
9
Position and Linear Momentum
7
.
9
.
1
The position eigenfunction
7
.
9
.
2
The linear momentum eigenfunction
7
.
10
Wave Packets
7
.
10
.
1
Solution of the Schrödinger equation.
7
.
10
.
2
Component wave solutions
7
.
10
.
3
Wave packets
7
.
10
.
4
Group velocity
7
.
10
.
5
Electron motion through crystals
7
.
11
Almost Classical Motion
7
.
11
.
1
Motion through free space
7
.
11
.
2
Accelerated motion
7
.
11
.
3
Decelerated motion
7
.
11
.
4
The harmonic oscillator
7
.
12
Scattering
7
.
12
.
1
Partial reflection
7
.
12
.
2
Tunneling
7
.
13
Reflection and Transmission Coefficients
8
. The Meaning of Quantum Mechanics
8
.
1
Schrödinger’s Cat
8
.
2
Instantaneous Interactions
8
.
3
Global Symmetrization
8
.
4
A story by Wheeler
8
.
5
Failure of the Schrödinger Equation?
8
.
6
The Many-Worlds Interpretation
8
.
7
The Arrow of Time
Next:
2. Mathematical Prerequisites
FAMU-FSU College of Engineering
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