Elements of Photoionization Quantum Dynamics Methods

I Basics on ab-initio quantum dynamics

 

1 Introduction

1.1 The theoretical minimum

1.2 Laser-atom quantum dynamics methods

 

2 Hilbert-space dynamics

2.1 Time-development

2.2 TDSE on the position representation

2.3 TDSE in the energy representation

 

3 Hilbert space rotations

3.1 Introduction

3.2 Central Hamiltonian

3.3 Angular momentum representation

3.3.1 Laplace operator

3.3.2 Dipole operators

3.3.3 Molecular diatomic potential

 

II One-electron atomic systems in laser fields

 

4 Radial SE of the atomic & molecular hydrogen

4.1 Atomic hydrogen radial SE

4.2 Molecular diatomic

4.2.1 Dipole matrix elements

 

5 SE on a finite basis

5.1 The TD wavefunction in a finite region

5.2 Finite basis expansion

5.2.1 Fixed boundary conditions

5.2.2 Free boundary conditions

5.3 Molecular hydrogen ion

5.4 Appendix: Normalization of the continuum states

 

6 Laser pulses

6.1 Radiation gauge

6.2 Laser beams in the paraxial approximation

6.2.1 Fourier-limited pulses

 

7 TDSE

7.1 Hamiltonian

7.2 TDSE of atomic hydrogen

7.2.1 Energy-angular momentum basis

7.2.2 TDSE on the angular momentum basis

7.3 TDSE of the molecular hydrogen ion

7.3.1 Energy-angular momentum basis

7.3.2 Angular momentum basis

7.4 TDSEs matrix notation

7.4.1 Initial state calculation-diffusion equation

 

8 TDSE post-processing

8.1 Preliminary

8.2 Bound state populations and ionization yield

8.2.1 Photoelectron Angular distribution (PAD)

8.2.2 High harmonic generation (HHG)

 

III The RMT division-of-space method

 

9 Division-of-space method

9.1 Introduction

9.2 RMT in 1-dimensional square well

9.3 TDSE in the inner region

9.3.1 Fixed boundary conditions (Green’s theorem)

9.3.2 Free-boundary conditions (Bloch-operator)

9.4 Outer-region, finite-difference approach

 

10 RMT in hydrogenic systems

10.1 Introduction

10.2 Inner and outer TDSE

10.3 The atomic case

10.4 Population and ionization yields

10.4.1 Bound state populations and ionization yield

 

Appendices

 

A Spherical harmonics and Glebsch-Gordan coefficients

A.1 Basic properties

A.2 Expansions on spherical harmonics

 

B Numerical approximations of operators

B.1 Spatial operators

B.1.1 Finite-differences

B.1.2 B-splines

B.2 Time-evolution operators

B.2.1 The Truncated Taylor Series Propagator

B.2.2 Runge-Kutta methods