Quantum Mechanics for Engineers 3 beta 4.2
© Leon van Dommelen
To:
Copyright
Fundamental Quantum Mechanics for Engineers
Leon van Dommelen
07/14/08 Version 3 beta 4.2
0
Copyright
Dedication
Preface
Book Structure
To the Student
Why Another Book on Quantum Mechanics?
Acknowledgments
Comments and Feedback
History
Wish List
Contents
List of Figures
List of Tables
1
. Mathematical Prerequisites
1
.
1
Complex Numbers
1
.
2
Functions as Vectors
1
.
3
The Dot, oops, INNER Product
1
.
4
Operators
1
.
5
Eigenvalue Problems
1
.
6
Hermitian Operators
1
.
7
Additional Points
1
.
7
.
1
Dirac notation
1
.
7
.
2
Additional independent variables
2
. Basic Ideas of Quantum Mechanics
2
.
1
The Revised Picture of Nature
2
.
2
The Heisenberg Uncertainty Principle
2
.
3
The Operators of Quantum Mechanics
2
.
4
The Orthodox Statistical Interpretation
2
.
4
.
1
Only eigenvalues
2
.
4
.
2
Statistical selection
2
.
5
Schrödinger's Cat [Background]
2
.
6
A Particle Confined Inside a Pipe
2
.
6
.
1
The physical system
2
.
6
.
2
Mathematical notations
2
.
6
.
3
The Hamiltonian
2
.
6
.
4
The Hamiltonian eigenvalue problem
2
.
6
.
5
All solutions of the eigenvalue problem
2
.
6
.
6
Discussion of the energy values
2
.
6
.
7
Discussion of the eigenfunctions
2
.
6
.
8
Three-dimensional solution
2
.
6
.
9
Quantum confinement
2
.
7
The Harmonic Oscillator
2
.
7
.
1
The Hamiltonian
2
.
7
.
2
Solution using separation of variables
2
.
7
.
3
Discussion of the eigenvalues
2
.
7
.
4
Discussion of the eigenfunctions
2
.
7
.
5
Degeneracy
2
.
7
.
6
Non-eigenstates
3
. Single-Particle Systems
3
.
1
Angular Momentum
3
.
1
.
1
Definition of angular momentum
3
.
1
.
2
Angular momentum in an arbitrary direction
3
.
1
.
3
Square angular momentum
3
.
1
.
4
Angular momentum uncertainty
3
.
2
The Hydrogen Atom
3
.
2
.
1
The Hamiltonian
3
.
2
.
2
Solution using separation of variables
3
.
2
.
3
Discussion of the eigenvalues
3
.
2
.
4
Discussion of the eigenfunctions
3
.
3
Expectation Value and Standard Deviation
3
.
3
.
1
Statistics of a die
3
.
3
.
2
Statistics of quantum operators
3
.
3
.
3
Simplified expressions
3
.
3
.
4
Some examples
3
.
4
The Commutator
3
.
4
.
1
Commuting operators
3
.
4
.
2
Noncommuting operators and their commutator
3
.
4
.
3
The Heisenberg uncertainty relationship
3
.
4
.
4
Commutator reference [Reference]
3
.
5
The Hydrogen Molecular Ion
3
.
5
.
1
The Hamiltonian
3
.
5
.
2
Energy when fully dissociated
3
.
5
.
3
Energy when closer together
3
.
5
.
4
States that share the electron
3
.
5
.
5
Comparative energies of the states
3
.
5
.
6
Variational approximation of the ground state
3
.
5
.
7
Comparison with the exact ground state
4
. Multiple-Particle Systems
4
.
1
Wave Function for Multiple Particles
4
.
2
The Hydrogen Molecule
4
.
2
.
1
The Hamiltonian
4
.
2
.
2
Initial approximation to the lowest energy state
4
.
2
.
3
The probability density
4
.
2
.
4
States that share the electrons
4
.
2
.
5
Variational approximation of the ground state
4
.
2
.
6
Comparison with the exact ground state
4
.
3
Two-State Systems
4
.
4
Spin
4
.
5
Instantaneous Interactions [Background]
4
.
6
Multiple-Particle Systems Including Spin
4
.
6
.
1
Wave function for a single particle with spin
4
.
6
.
2
Inner products including spin
4
.
6
.
3
Wave function for multiple particles with spin
4
.
6
.
4
Example: the hydrogen molecule
4
.
6
.
5
Triplet and singlet states
4
.
7
Identical Particles
4
.
8
Global Symmetrization [Background]
4
.
9
Ways to Symmetrize the Wave Function
4
.
10
Matrix Formulation
4
.
11
Atoms Heavier Than Hydrogen [Descriptive]
4
.
11
.
1
The Hamiltonian eigenvalue problem
4
.
11
.
2
Approximate solution using separation of variables
4
.
11
.
3
Hydrogen and helium
4
.
11
.
4
Lithium to neon
4
.
11
.
5
Sodium to argon
4
.
11
.
6
Potassium to krypton
4
.
12
Exclusion-Principle Repulsion [Descriptive]
4
.
13
Chemical Bonds [Descriptive]
4
.
13
.
1
Covalent sigma bonds
4
.
13
.
2
Covalent pi bonds
4
.
13
.
3
Polar covalent bonds and hydrogen bonds
4
.
13
.
4
Promotion and hybridization
4
.
13
.
5
Ionic bonds
4
.
13
.
6
Limitations of valence bond theory
5
. Solids
5
.
1
Molecular solids [Descriptive]
5
.
2
Ionic solids [Descriptive]
5
.
3
Introduction to band theory [Descriptive]
5
.
4
Metals [Descriptive]
5
.
4
.
1
Lithium
5
.
4
.
2
One-dimensional crystals
5
.
4
.
3
Wave functions of one-dimensional crystals
5
.
4
.
4
Analysis of the wave functions
5
.
4
.
5
Floquet (Bloch) theory
5
.
4
.
6
Fourier analysis
5
.
4
.
7
The reciprocal lattice
5
.
4
.
8
The energy levels
5
.
4
.
9
Electrical conduction
5
.
4
.
10
Merging and splitting bands
5
.
4
.
11
Three-dimensional metals
5
.
5
Covalent Materials [Descriptive]
5
.
6
Confined Free Electrons
5
.
6
.
1
The Hamiltonian eigenvalue problem
5
.
6
.
2
Solution by separation of variables
5
.
6
.
3
Discussion of the solution
5
.
6
.
4
A numerical example
5
.
6
.
5
The density of states and confinement [Advanced]
5
.
6
.
6
Relation to Bloch functions
5
.
7
Nearly-Free Electrons [Advanced]
5
.
7
.
1
The lattice structure
5
.
7
.
2
The small perturbation approach
5
.
7
.
3
Zeroth order solution
5
.
7
.
4
First order solution
5
.
7
.
5
Second order solution
5
.
7
.
6
Discussion of the energy changes
5
.
8
Quantum Statistical Mechanics
5
.
9
Additional Points [Descriptive]
5
.
9
.
1
Thermal properties
5
.
9
.
2
Ferromagnetism
5
.
9
.
3
X-ray diffraction
6
. Basic and Quantum Thermodynamics [Advanced]
6
.
1
Temperature
6
.
2
Single-particle versus system energy eigenfunctions
6
.
3
How many system energy eigenfunctions are there?
6
.
4
The particle-energy distribution functions
6
.
5
The canonical probability distribution
6
.
6
Illustration of the low temperature behavior
6
.
7
The basic thermodynamic variables
6
.
8
Introduction to the second law
6
.
9
The reversible ideal
6
.
10
Entropy
6
.
11
The big lie of distinguishable particles
6
.
12
The new variables
6
.
13
Microscopic meaning of the new variables
6
.
14
Application to particles in a box
6
.
14
.
1
Bose-Einstein condensation
6
.
14
.
2
Fermions at low temperatures
6
.
14
.
3
A generalized ideal gas law
6
.
14
.
4
The ideal gas
6
.
14
.
5
Blackbody radiation
6
.
14
.
6
The Debye model
7
. Time Evolution
7
.
1
The Schrödinger Equation
7
.
1
.
1
Energy conservation
7
.
1
.
2
Stationary states
7
.
1
.
3
Time variations of symmetric two-state systems
7
.
1
.
4
Time variation of expectation values
7
.
1
.
5
Newtonian motion
7
.
2
Unsteady perturbations of two-state systems
7
.
2
.
1
Schrödinger equation for a two-state system
7
.
2
.
2
Stimulated and spontaneous emission
7
.
2
.
3
Effect of a single wave
7
.
2
.
4
Absorption of a single weak wave
7
.
2
.
5
Absorption of incoherent radiation
7
.
3
Conservation Laws and Symmetries [Background]
7
.
4
Position and Linear Momentum Eigenfunctions
7
.
4
.
1
The position eigenfunction
7
.
4
.
2
The linear momentum eigenfunction
7
.
5
Wave Packets in Free Space
7
.
5
.
1
Solution of the Schrödinger equation.
7
.
5
.
2
Component wave solutions
7
.
5
.
3
Wave packets
7
.
5
.
4
The group velocity
7
.
6
Motion near the Classical Limit
7
.
6
.
1
General procedures
7
.
6
.
2
Motion through free space
7
.
6
.
3
Accelerated motion
7
.
6
.
4
Decelerated motion
7
.
6
.
5
The harmonic oscillator
7
.
7
Scattering
7
.
7
.
1
Partial reflection
7
.
7
.
2
Tunneling
8
. Numerical Procedures
8
.
1
The Variational Method [Advanced]
8
.
1
.
1
Basic variational statement
8
.
1
.
2
Differential form of the statement
8
.
1
.
3
Example application using Lagrangian multipliers
8
.
2
The Born-Oppenheimer Approximation [Advanced]
8
.
2
.
1
The Hamiltonian
8
.
2
.
2
The basic Born-Oppenheimer approximation
8
.
2
.
3
Going one better
8
.
3
The Hartree-Fock Approximation [Advanced]
8
.
3
.
1
Wave function approximation
8
.
3
.
2
The Hamiltonian
8
.
3
.
3
The expectation value of energy
8
.
3
.
4
The canonical Hartree-Fock equations
8
.
3
.
5
Additional points
8
.
3
.
5
.
1
Meaning of the orbital energies
8
.
3
.
5
.
2
Asymptotic behavior
8
.
3
.
5
.
3
Hartree-Fock limit
8
.
3
.
5
.
4
Configuration interaction
9
. Some Additional Topics
9
.
1
All About Angular Momentum [Advanced]
9
.
1
.
1
The fundamental commutation relations
9
.
1
.
2
Ladders
9
.
1
.
3
Possible values of angular momentum
9
.
1
.
4
A warning about angular momentum
9
.
1
.
5
Triplet and singlet states
9
.
1
.
6
Clebsch-Gordan coefficients
9
.
1
.
7
Pauli spin matrices
9
.
2
The Relativistic Dirac Equation [Advanced]
9
.
2
.
1
The Dirac idea
9
.
2
.
2
Emergence of spin from relativity
9
.
3
The Electromagnetic Field [Advanced]
9
.
3
.
1
The Hamiltonian
9
.
3
.
2
Maxwell's equations
9
.
3
.
3
Electrons in magnetic fields
9
.
4
Nuclear Magnetic Resonance [Advanced]
9
.
4
.
1
Description of the method
9
.
4
.
2
The Hamiltonian
9
.
4
.
3
The unperturbed system
9
.
4
.
4
Effect of the perturbation
9
.
5
Quantum Field Theory in a Nanoshell [Advanced]
9
.
5
.
1
Occupation numbers
9
.
5
.
2
Annihilation and creation operators
9
.
5
.
3
Field operators
9
.
5
.
4
An example using field operators
9
.
6
Some Topics Not Covered [Advanced]
9
.
7
The Meaning of Quantum Mechanics [Background]
9
.
7
.
1
Failure of the Schrödinger Equation?
9
.
7
.
2
The Many-Worlds Interpretation
A. Notes
A.
1
Notes on the Mathematical Prerequisites
A.
1
.
1
Derivation of the Euler identity
A.
1
.
2
Nature and real eigenvalues
A.
1
.
3
Completeness of Fourier modes
A.
1
.
4
Are Hermitian operators really like that?
A.
2
Notes on the Basic Ideas of Quantum Mechanics
A.
2
.
1
Why the linear momentum operators are Hermitian
A.
2
.
2
Why boundary conditions are tricky
A.
2
.
3
Three-dimensional solutions from one-dimensional ones
A.
2
.
4
Derivation of the harmonic oscillator solution
A.
2
.
5
More on the harmonic oscillator and uncertainty
A.
3
Notes on the Single-Particle Systems
A.
3
.
1
Derivation of a vector identity
A.
3
.
2
Derivation of the spherical harmonics
A.
3
.
3
Derivation of the hydrogen radial wave functions
A.
3
.
4
Definitions of the Laguerre polynomials
A.
3
.
5
Justification of the expression for the expectation value
A.
3
.
6
Why commuting operators have a common set of eigenvectors
A.
3
.
7
Derivation of the generalized uncertainty relationship
A.
3
.
8
Derivation of the commutator rules
A.
3
.
9
How the hydrogen molecular ion integrals are done
A.
3
.
10
In what sense the variational approximation is best
A.
3
.
11
More on the accuracy of variational approximation
A.
3
.
12
Why the hydrogen molecular ion wave function is positive
A.
3
.
13
Symmetries of the hydrogen molecular ion wave function
A.
4
Notes on the Multiple-Particle Systems
A.
4
.
1
Additional hydrogen molecule model data
A.
4
.
2
Bell’s actual analysis
A.
4
.
3
Why spin does not change the hydrogen molecule ground state
A.
4
.
4
Limitations of the shielding approximation
A.
4
.
5
Why the s states have the least energy
A.
5
Notes on Solids
A.
5
.
1
Explanation of the London forces
A.
5
.
2
Ambiguities in the definition of electron affinity
A.
5
.
3
Why Floquet theory should be called so
A.
5
.
4
Superfluidity versus BEC
A.
5
.
5
Explanation of Hund’s first rule
A.
5
.
6
The qualitative mechanism of ferromagnetism
A.
6
Notes on Thermodynamics
A.
6
.
1
System eigenfunctions for given bucket numbers
A.
6
.
2
The fundamental assumption of quantum statistics
A.
6
.
3
A problem if the energy is given
A.
6
.
4
Derivation of the particle energy distributions
A.
6
.
5
Derivation of the canonical probability distribution
A.
6
.
6
Analysis of the ideal gas Carnot cycle
A.
6
.
7
Checks on the expression for entropy
A.
6
.
8
Chemical potential and the distribution functions
A.
6
.
9
The Fermi-Dirac integral for small temperature
A.
7
Notes on Time Evolution
A.
7
.
1
Why energy eigenstates are physically stationary
A.
7
.
2
More precise description of two-state systems
A.
7
.
3
Derivation of the evolution of expectation values
A.
7
.
4
The virial theorem
A.
7
.
5
The energy-time uncertainty relationship
A.
7
.
6
Justification of the two-state approximation in atom radiation
A.
7
.
7
About spectral broadening
A.
7
.
8
Derivation of the Einstein B coefficients
A.
7
.
9
Why symmetry eigenvalues are preserved
A.
7
.
10
Remark on the edges of the wave packet
A.
8
Notes on the Numerical Procedures
A.
8
.
1
A basic description of Lagrangian multipliers
A.
8
.
2
The generalized variational principle
A.
8
.
3
Born-Oppenheimer approximation and spin-degeneracy
A.
8
.
4
Derivation of the Born-Oppenheimer approximation
A.
8
.
5
Why a single Slater determinant is not exact
A.
8
.
6
Simplification of the Hartree-Fock energy
A.
8
.
7
Constraints on the Coulomb and exchange integrals
A.
8
.
8
Generalized orbitals
A.
8
.
9
Derivation of the Hartree-Fock equations
A.
8
.
10
Why the Fock operator is Hermitian
A.
8
.
11
Basic science (BS) behind “correlation energy”
A.
9
Notes on the Additional Topics
A.
9
.
1
Physical justification of the fundamental commutation relations
A.
9
.
2
Angular momentum components have only zero in common
A.
9
.
3
Components of vectors are less than the total vector
A.
9
.
4
Finding the spherical harmonics using ladder operators
A.
9
.
5
Why angular momenta components can be added
A.
9
.
6
Why the Clebsch-Gordan tables can be read either way
A.
9
.
7
How to make your very own Clebsch-Gordan tables
A.
9
.
8
Machine language version of the Clebsch-Gordan tables
A.
9
.
9
The triangle inequality in quantum mechanics
A.
9
.
10
Awkward questions about spin
A.
9
.
11
More awkwardness about spin
A.
9
.
12
Derivation of a vectorial triple product property
A.
9
.
13
More on Maxwell’s third law
A.
9
.
14
Setting the record straight on alignment
A.
9
.
15
Solving the NMR equations
A.
9
.
16
Everett’s theory and vacuum energy
A.
9
.
17
A tenth of a googol in universes
Bibliography
Web Pages
Notations
Index
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Copyright
FAMU-FSU College of Engineering
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