Schrödinger's equations are the most basic equations of (non-relativistic) quantum mechanics of distinguishable particles. Formulated as partial differential equations, they describe the structure and evolution of complex-valued wave functions ψ(R,t), where R is an N-dimensional position vector and t is the time. The corresponding densities ρ(R,t)=|ψ(R,t)|^{2} are usually interpreted as probability densities to find the physical particles at position R.
Schrödinger's equations describe the structure and time evolution of wavefunctions with the help of a quantum-mechanical Hamiltonian operator H. There is a time-independent Schrödinger equation (TISE) and a time-dependent Schrödinger equation (TDSE) where the Hamiltonian in the latter one may or may not be time-dependent.
Here, E_{n} are the eigenvalues of the TISE and -i∇_{R} is the quantum-mechanical momentum operator. Note that the solutions of the TISE, multiplied with an overall phase factor, are an obvious solution to the TDSE if and only if the Hamiltonian operator H(R,-i∇_{R}) is time-independent
Depending on whether you want to solve a single ($M=1$) or several ($M>1$, i.e. coupled channels) Schrödinger equation(s), the wavefunction is either scalar or vector valued and the corresponding Hamiltonian operator is either scalar or matrix valued. The general structure of the Hamiltonians eligible for use in WavePacket is given by
with the following symbols:
T(R,−i∇_{R}) is the kinetic energy operator the implementation of which is highly entangled with the choice of the discrete variable representation scheme. This operator is identical for all M>1 coupled equations.
V(R) is the potential energy function which is matrix valued for M>1 (real symmetric; diabatic representation). In order to adapt to the different physical/chemical situations you want to simulate, WavePacket comes with a choice of different model functions for the potential energy functions as specified in the Reference Manual of our Matlab version.
W(R) is the negative imaginary potential, used to absorb the wavefunctions near the boundaries. This absorber is identical for all M>1 coupled equations. In order to use different absorbing boundary conditions in your simulations, WavePacket comes with a choice of different model functions for negative imaginary potentials as specified in the Reference Manual of our Matlab version.
F(t) is the external electric field, semiclassically interacting with the system's dipole moment. In order to adapt to the photophysical/photochemical situation you want to simulate, WavePacket comes with a choice of different model functions for the time-dependence of external electric fields. The choice of different laser pulse envelope shapes can also be found in the Reference Manual of our Matlab version. If you cannot find a model fitting your needs, you can use an interpolation of tabulated values or write your own model function. The former case may be used in connection with optimally taylored laser pulses.
μ(R) is the dipole moment function, which is matrix valued for M>1 (real symmetric). In that case, the diagonal and off-diagonal elements of that matrix are referred to as permanent, μ_{p}(t), or transition, μ_{t}(R), dipole moments, respectively. In order to adapt to the photophysical/photochemical situation you want to simulate, the WavePacket program package comes with different models for the coordinate-dependence of (permanent or transition) dipole moments A choice of functions is given in the Reference Manual of our Matlab version.
If you cannot find a model fitting your specific needs, you can alternatively use interpolation of tabulated values or even write your own model function. In the latter case, please contact the authors if you need support. In turn, it would be highly appreciated if you could send us such files so that they can be published with future versions of WavePacket.
To further enhance flexibility of the program package, each part of the Hamiltonian can be optionally omitted in simulations with WavePacket!
Future version might/should also contain vector potentials to treat, e. g., quantum dynamics of charged particles interacting with magnetic fields ...
WavePacket has a rather large choice of different initial wave functions for solving the TDSE already built in
These functions serve as initial conditions for the solution of the time-dependent Schrödinger equation or for the solution of the time-independent Schrödinger equation by imaginary time propagation. In order to be applied to different physical/chemical situations, the program package comes with a choice of different model functions for the coordinate-dependence of initial wave functions. For the Matlab version there is the choice of either using an outer product of one-dimensional wave functions or using a correlated wave function. If you cannot find a model fitting your specific needs, you can use an interpolation of tabulated values or even write your own model function.
The expectation values of an operator O with respect to stationary wavefunctions ψ_{n} or dynamical wavefunctions ψ(t) are defined in the following way
WavePacket calculates these mean values for a choice of different elementary operators, such as position and momentum, potential and kinetic energy, etc. If coupled Schrödinger equations are solved (M>1), those expectation values can be calculated either for every component of the wavefunction individually or for the total wavefunction. In the latter case, the resulting expectation values also account for the coupling parts of the Hamiltonian.
To enhance flexibility of the program package, WavePacket allows to use self-defined spatial projection operators in position space, for which stationary or dynamical expectation values are calculated, respectively. In order to adapt to different simulation scenarios, the WavePacket program package comes with a choice of different model functions for the coordinate-dependence of spatial projection functions as specified in the Reference Manual of the Matlab version. If you cannot find a model fitting your specific needs, you can write your own model function.
The potential energy in the above formulation of the Hamiltonian operator is represented in a diabatic representation. However, in certain cases it may be advantageous to transform to an adiabatic representation where the potential energy matrix becomes diagonal. This comes, however, at the expense of coupling caused by the kinetic operator. However, under certain circumstances these non-adiabatic coupling tensors (NACTs) can be neglected which offers new insights into the interpretation of quantum dynamics in coupled channel calculations within the adiabatic approximation. Read more ...
If the electric fields F(t) are periodically oscillating in time, it may be advantageous to transform to the Floquet picture of light-dressed states. Apart from numerical advantages, this offers new interpretations of quantum dynamics of externally driven systems. Read more ...
Currently, the WavePacket codes are using atomic units throughout. These units are adapted to the scales of the hydrogen atom, i. e. hbar, electronic mass and charge being scaled to one. Read more ...
Wiki: Numerics.Adiabatic
Wiki: Numerics.AtomicUnits
Wiki: Numerics.Control.Closed
Wiki: Numerics.Control
Wiki: Numerics.DVR
Wiki: Numerics.FloquetStates
Wiki: Numerics.Main
Wiki: Numerics.TDSE