Reference.Variables.time.dof

Variable "time.dof" in MATLAB/Octave version of WavePacket

This structured(!) variable contains information mainly on the choice and parametrization of the initial wavefunction in the case where the initial wavefunction is to be constructed as a direct (outer) product of one-dimensional functions, for example

time.dof{1} = init.harmonic;
time.dof{1}.m_r = ... 
...
time.dof{2} = init.morse;
time.dof{2}.m_r = ... 
...

for a direct product of a harmonic oscillator (along the first coordinate) and a Morse oscillator eigen state (along the second coordinate). Below is a description of classdefs along with their properties; you can also find the complete list of model wavefunctions in the corresponding package folder.

init.gauss	Superposition of (one or more) Gaussian wave packets	default
pos_0	Vector (length N) of mean positions Xg	0
mom_0	Vector (length N)of mean momenta Kg	0
width	Vector (length N)of width parameters Wg ≡ position uncertainty	1
factor	Vector (length N) of weights Cg	1

Note that this Gaussian wave packets correspond to coherent (Glauber) states of harmonic oscillators
V(R) = k/2 (x-X_g)²=1/2 m ω² (x-X_g)²
with mass m, frequency ω=(k/m)^(1/2), and width parameter
W_g = (4 k m)^(-1/4) = (2 m ω)^(-1/2)

init.harmonic	Eigenstate of a harmonic oscillator	default
m_r	Optional: The mass of the particle (or reduced mass)	taken from the grid setting along the respective dof
r_e	Optional: Equilibrium distance, i.e., position of the minimum of the harmonic oscillator	0
omega	Optional: The angular frequency ω of the oscillator. Either this or ''v_2'' must be set. Throws an error if both are set.
v_2	Optional: The force constant v2 = mr ω². Either this or ''omega'' must be set. Throws an error if both are set.
n_q	Requested quantum number. Start counting from 0 (ground state)	0

init.morse	Eigenstate of a Morse oscillator	default
m_r	Optional: The mass of the particle (or reduced mass)	taken from the grid setting along the respective dof
r_e	Optional: The equilibrium distance of the oscillator	0
d_e	Dissociation energy of the oscillator
alf	Range parameter of the oscillator
n_q	Requested quantum number. Start counting from 0 (ground state)	0

init.pendulum1	Mathieu eigenfunctions of the simple planar pendulum V(θ) = ½ V0 (1 + cos (m(θ-θ0))) where θ is a 2π/m-periodic coordinate.
barrier	Height V0 of the barrier.
shift	Shift θ0 of the potential.
multiple	Multiplicity m of the potential. Only values of 1 and 2 are supported right now.
parity	Parity of the eigenstate. Valid choices are: ''c'' for cosine elliptic (ce) function ''s'' for sine elliptic (se) function ''l'' for localized superpositions of ce and se.
order	Order/quantum number of the eigenstate. Start counting from 1.

init.pendulum2	Eigenfunctions of the generalized planar pendulum V(θ) = - η cos(θ) - ζ cos^2(θ) for which the TISE is quasi-exactly solvable and conditionally exactly solvable. With optional shifts in position and/or momentum space
eta	orienting interaction η
zeta	aligning interaction ζ
beta	width parameter β=√ζ
kappa	topological index κ=η/β
n_q	Quantum number. Start counting from 0
irrep	irreducible representation A1 A2 B1 B2, specifying symmetry
pos_0	Mean initial position
mom_0	Mean initial momentum

Note that either η, ζ or β,κ have to be specified.

init.spherical	Associated Legendre polynomials in cos Θ. Apart from normalization and missing azimuthal functions, these functions are identical to spherical harmonics	default
l	Quantum number l ≥ 0	0
m	Quantum number m with 0 ≤ m ≤ l	0
sqst	Dividing wavefunction by sqrt(sin(Theta)) See e.g. Eq. (4) in this paper	false

init.fbr	Eigenstate of the FBR expansion underlying the grid along the respective dof e.g., a plane wave, a spherical harmonic, or a harmonic oscillator eigenstate. Using this is a bit tricky and is discouraged unless explicitly required.
state	Optional: The index of the eigenstate to use as initial state. Does not need to be set if ''coeffs'' is set. Note that this is an index, not a quantum number (i.e., it starts from 1)! Also, the mapping from index to actual eigenstate is not always straight-forward. For Legendre grids, the major quantum number l at fixed ml is given by l = state-1 + ml. For equally-spaced grids, the momentum starts at -kmax + Δk.
coeffs	Optional: An array of coefficients that gives the coefficients for the basis expansion, with the ''n''-th element giving the coefficient for the state with index ''n''. See ''state'' for an explanation of indices. This is overridden by the value of ''state'' if the latter is set.