Numerics.Schroedinger

Schrödinger's equations and Hamiltonians

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.

Time-independent and time-dependent formulation

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

Structure of the Hamiltonian operator

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/Octave 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 control the absorption behavior 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/Octave 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/Octave 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, 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 model functions for the coordinate-dependence of (permanent or transition) dipole moments. A choice of functions is given in the Reference Manual of our Matlab/Octave version.

• α(R) is the polarizability, which is (so far) vector valued for M>1. In order to adapt to the photophysical/ photochemical situation you want to simulate, the WavePacket program package comes with different model functions for the coordinate-dependence of polarizabilities. A choice of functions is given in the Reference Manual of our Matlab/Octave version.

The dot product in the last two terms of the Hamiltonian given above stands for different field components F_k(t) interacting with the quantum system through corresponding dipole μ_p(R) or polarizability α_pp(R) components. For example, p=1 and p=2 stand for the x and y polarization directions, respectively, which allows to also simulate effects of differently polarized light using WavePacket software. In principle, the number of components is unlimited; it is deduced from the number of components of the field amplitude specified by the user. We note that the dipole coupling mechanism can also be used to specify time-dependent potentials as long as those can be expressed as sums of products of time- and coordinate-dependent functions.

If you cannot find suitable model functions for 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!

Initial conditions

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. The corresponding densities are used for sampling of trajectories in fully classical or quantum-classical simulations. 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.

Expectation values of observables

The expectation values of an observable (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 are calculated for every component of the wavefunction individually and also for 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 additional multiplicative operators (AMO) 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 AMO 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.

Diabatic vs. adiabatic formulation

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 ...

Floquet Theory: light-dressed states

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 ...

Atomic units

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 ...