Review Questions
© Leon van Dommelen
To:
Abstract
Fundamental Quantum Mechanics for Engineers
Review Questions
Leon van Dommelen
Abstract
1
. Mathematical Prerequisites
1
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1
Complex Numbers
1
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2
Functions as Vectors
1
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3
The Dot, oops, INNER Product
1
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4
Operators
1
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5
Eigenvalue Problems
1
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6
Hermitian Operators
1
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7
Additional Points
1
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7
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1
Dirac notation
1
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7
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2
Additional independent variables
1
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7
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3
Self assessment
2
. Basic Ideas of Quantum Mechanics
2
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1
The Revised Picture of Nature
2
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2
The Heisenberg Uncertainty Principle
2
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3
The Operators of Quantum Mechanics
2
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4
The Orthodox Statistical Interpretation
2
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4
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1
Only eigenvalues
2
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4
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2
Statistical selection
2
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5
A Particle Confined Inside a Pipe
2
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5
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1
The physical system
2
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5
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2
Mathematical notations
2
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5
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3
The Hamiltonian
2
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5
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4
The Hamiltonian eigenvalue problem
2
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5
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5
All solutions of the eigenvalue problem
2
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5
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6
Discussion of the energy values
2
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5
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7
Discussion of the eigenfunctions
2
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5
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8
Three-dimensional solution
2
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5
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9
Quantum confinement
2
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6
The Harmonic Oscillator
2
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6
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1
The Hamiltonian
2
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6
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2
Solution using separation of variables
2
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6
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3
Discussion of the eigenvalues
2
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6
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4
Discussion of the eigenfunctions
2
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6
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5
Degeneracy
2
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6
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6
Non-eigenstates
3
. Single-Particle Systems
3
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1
Angular Momentum
3
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1
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1
Definition of angular momentum
3
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1
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2
Angular momentum in an arbitrary direction
3
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1
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3
Square angular momentum
3
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1
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4
Angular momentum uncertainty
3
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2
The Hydrogen Atom
3
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2
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1
The Hamiltonian
3
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2
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2
Solution using separation of variables
3
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2
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3
Discussion of the eigenvalues
3
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2
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4
Discussion of the eigenfunctions
3
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3
Expectation Value and Standard Deviation
3
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3
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1
Statistics of a die
3
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3
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2
Statistics of quantum operators
3
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3
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3
Simplified expressions
3
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3
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4
Some examples
3
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4
The Commutator
3
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4
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1
Commuting operators
3
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4
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2
Noncommuting operators and their commutator
3
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4
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3
The Heisenberg uncertainty relationship
3
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4
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4
Commutator reference [Reference]
3
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5
The Hydrogen Molecular Ion
3
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5
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1
The Hamiltonian
3
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5
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2
Energy when fully dissociated
3
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5
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3
Energy when closer together
3
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5
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4
States that share the electron
3
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5
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5
Comparative energies of the states
3
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5
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6
Variational approximation of the ground state
3
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5
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7
Comparison with the exact ground state
4
. Multiple-Particle Systems
4
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1
Wave Function for Multiple Particles
4
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2
The Hydrogen Molecule
4
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2
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1
The Hamiltonian
4
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2
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2
Initial approximation to the lowest energy state
4
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2
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3
The probability density
4
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2
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4
States that share the electrons
4
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2
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5
Variational approximation of the ground state
4
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2
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6
Comparison with the exact ground state
4
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3
Two-State Systems
4
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4
Spin
4
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5
Multiple-Particle Systems Including Spin
4
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5
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1
Wave function for a single particle with spin
4
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5
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2
Inner products including spin
4
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5
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3
Commutators including spin
4
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5
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4
Wave function for multiple particles with spin
4
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5
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5
Example: the hydrogen molecule
4
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5
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6
Triplet and singlet states
4
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6
Identical Particles
4
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7
Ways to Symmetrize the Wave Function
4
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8
Matrix Formulation
4
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9
Heavier Atoms [Descriptive]
4
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9
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1
The Hamiltonian eigenvalue problem
4
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9
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2
Approximate solution using separation of variables
4
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9
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3
Hydrogen and helium
4
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9
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4
Lithium to neon
4
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9
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5
Sodium to argon
4
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9
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6
Potassium to krypton
4
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10
Pauli Repulsion [Descriptive]
4
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11
Chemical Bonds [Descriptive]
4
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11
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1
Covalent sigma bonds
4
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11
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2
Covalent pi bonds
4
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11
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3
Polar covalent bonds and hydrogen bonds
4
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11
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4
Promotion and hybridization
4
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11
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5
Ionic bonds
4
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11
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6
Limitations of valence bond theory
5
. Time Evolution
5
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1
The Schrödinger Equation
5
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1
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1
Introduction to the equation
5
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1
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2
Some examples
5
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1
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3
Energy conservation [Descriptive]
5
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1
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4
Stationary states [Descriptive]
5
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1
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5
Particle exchange [Descriptive]
5
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1
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6
Energy-time uncertainty relation [Descriptive]
5
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1
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7
Time variation of expectation values [Descriptive]
5
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1
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8
Newtonian motion [Descriptive]
5
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1
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9
The adiabatic approximation [Descriptive]
5
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1
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10
Heisenberg picture [Descriptive]
5
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2
Conservation Laws and Symmetries
5
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3
Unsteady Perturbations of Systems
5
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3
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1
Schrödinger equation for a two-state system
5
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3
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2
Spontaneous and stimulated emission
5
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3
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3
Effect of a single wave
5
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3
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3
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1
The wave
5
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3
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3
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2
The Hamiltonian coefficients
5
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3
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4
Forbidden transitions
5
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3
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5
Selection rules
5
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3
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6
Angular momentum conservation
5
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3
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7
Parity
5
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3
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8
Absorption of a single weak wave
5
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3
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9
Absorption of incoherent radiation
5
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3
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10
Spontaneous emission of radiation
5
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4
Position and Linear Momentum
5
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4
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1
The position eigenfunction
5
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4
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2
The linear momentum eigenfunction
5
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5
Wave Packets in Free Space
5
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5
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1
Solution of the Schrödinger equation.
5
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5
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2
Component wave solutions
5
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5
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3
Wave packets
5
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5
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4
Group velocity
5
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6
Almost Classical Motion [Descriptive]
5
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6
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1
Motion through free space
5
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6
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2
Accelerated motion
5
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6
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3
Decelerated motion
5
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6
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4
The harmonic oscillator
5
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7
WKB Theory of Nearly Classical Motion
5
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8
Scattering
5
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8
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1
Partial reflection
5
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8
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2
Tunneling
5
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9
Reflection and Transmission Coefficients
To:
Abstract
FAMU-FSU College of Engineering
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