Wigner function in science and technology, The
Wigner distribution Mathematical physics Phase space (Statistical physics) Quantum theory
IOP Publishing
2018
EISBN 9780750316712
1. Introduction.
1.1. Classical mechanics.
1.2. Rise of quantum mechanics.
1.3. Eugene Wigner.
1.4. Modern devices and simulation.
1.5. Our approach
2. Approaches to quantum transport.
2.1. Modes and the Landauer formula.
2.2. The scattering matrix approach.
2.3. The density matrix.
2.4. Green's functions.
2.5. What are the relative advantages?
3. Wigner functions.
3.1. Preliminary considerations.
3.2. The equations of motion.
3.3. Generalizing the Wigner function.
3.4. Other phase space approaches.
3.5. Wigner-Weyl transforms.
3.6. The hydrodynamic equations
4. Effective potentials.
4.1. Size of the electron.
4.2. The Bohm potential.
4.3. Bohm and the two-slit experiment.
4.4. The Wigner potential.
4.5. Feynman and effective potentials
5. Numerical solutions.
5.1. The initial state.
5.2. Numerical techniques.
5.3. The resonant tunneling diode : Wigner function simulations.
5.4. Other devices
6. Particle methods.
6.1. The classical Monte Carlo technique.
6.2. Paths in quantum mechanics.
6.3. Using particles with the Wigner function
7. Collisions and the Wigner function.
7.1. The interaction representation.
7.2. The electron-phonon interaction.
7.3. The Wigner scattering integrals.
7.4. Collisions in the Monte Carlo approach
8. Entanglement.
8.1. An illustration of entanglement.
8.2. Entanglement in harmonic oscillators.
8.3. Measures of entanglement.
8.4. Some illustrative examples
9. Quantum chemistry.
9.1. Quantum statistics.
9.2. Reactions and rates.
9.3. Tunneling.
9.4. Spectroscopy
10. Signal processing.
10.1. Signal propagation.
10.2. Wavelets
11. Quantum optics.
11.1. Propagation.
11.2. The Jaynes-Cummings model.
11.3. Squeezed states.
11.4. Coherence I.
11.5. Coherence II.
11.6. Bell states
12. Quantum physics.
12.1. The harmonic oscillator.
12.2. Quantum physics.
12.3. Superconductivity.
12.4. Plasmas.
12.5. Relativistic systems.
12.6. Quantum cascade laser.
This book is designed to give a background on the origins and development of Wigner functions, as well as its mathematical underpinnings. Along the way the authors emphasise the connections, and differences, from the more popular non-equilibrium Green's function approaches. But, the importance of the text lies in the discussions of the applications of the Wigner function in various fields of science, including quantum information, coherent optics, and superconducting qubits. These disciplines approach it differently, and the goal here is to give a unified background and highlight how it is utilized in the different disciplines.
1.1. Classical mechanics.
1.2. Rise of quantum mechanics.
1.3. Eugene Wigner.
1.4. Modern devices and simulation.
1.5. Our approach
2. Approaches to quantum transport.
2.1. Modes and the Landauer formula.
2.2. The scattering matrix approach.
2.3. The density matrix.
2.4. Green's functions.
2.5. What are the relative advantages?
3. Wigner functions.
3.1. Preliminary considerations.
3.2. The equations of motion.
3.3. Generalizing the Wigner function.
3.4. Other phase space approaches.
3.5. Wigner-Weyl transforms.
3.6. The hydrodynamic equations
4. Effective potentials.
4.1. Size of the electron.
4.2. The Bohm potential.
4.3. Bohm and the two-slit experiment.
4.4. The Wigner potential.
4.5. Feynman and effective potentials
5. Numerical solutions.
5.1. The initial state.
5.2. Numerical techniques.
5.3. The resonant tunneling diode : Wigner function simulations.
5.4. Other devices
6. Particle methods.
6.1. The classical Monte Carlo technique.
6.2. Paths in quantum mechanics.
6.3. Using particles with the Wigner function
7. Collisions and the Wigner function.
7.1. The interaction representation.
7.2. The electron-phonon interaction.
7.3. The Wigner scattering integrals.
7.4. Collisions in the Monte Carlo approach
8. Entanglement.
8.1. An illustration of entanglement.
8.2. Entanglement in harmonic oscillators.
8.3. Measures of entanglement.
8.4. Some illustrative examples
9. Quantum chemistry.
9.1. Quantum statistics.
9.2. Reactions and rates.
9.3. Tunneling.
9.4. Spectroscopy
10. Signal processing.
10.1. Signal propagation.
10.2. Wavelets
11. Quantum optics.
11.1. Propagation.
11.2. The Jaynes-Cummings model.
11.3. Squeezed states.
11.4. Coherence I.
11.5. Coherence II.
11.6. Bell states
12. Quantum physics.
12.1. The harmonic oscillator.
12.2. Quantum physics.
12.3. Superconductivity.
12.4. Plasmas.
12.5. Relativistic systems.
12.6. Quantum cascade laser.
This book is designed to give a background on the origins and development of Wigner functions, as well as its mathematical underpinnings. Along the way the authors emphasise the connections, and differences, from the more popular non-equilibrium Green's function approaches. But, the importance of the text lies in the discussions of the applications of the Wigner function in various fields of science, including quantum information, coherent optics, and superconducting qubits. These disciplines approach it differently, and the goal here is to give a unified background and highlight how it is utilized in the different disciplines.
