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Dynamics of quantum systems, controlled by external fields

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Reference.Variables.time.dof

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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 (sourceforge.net)	Superposition of (one or more) Gaussian wave packets	default
pos_0	Vector of mean positions X_g
mom_0	Vector of mean momenta K_g	0
width	Vector of width parameters W_g ≡ position uncertainty
factor	Vector (length N) of weights C_g	1

$\psi(x) \propto \sum_{g=1}^N C_g \exp\left(\imath K_g (x - X_g) - \left(\frac{x - X_g}{2W_g}\right)^2\right)$

Note that this Gaussian wavepacket corresponds to a coherent (Glauber) state of a harmonic oscillator
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 (sourceforge.net)	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 = m_r ω². Either this or ''omega'' must be set. Throws an error if both are set.
n_q	The requested quantum number. Start counting from 0 (ground state)	0

init.morse (sourceforge.net)	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 (sourceforge.net)	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 ''s'' for sine elliptic ''l'' for localized superpositions.
order	Order/quantum number of the eigenstate. Start counting from 1.

init.pendulum2 (sourceforge.net)	Eigenfunctions of the generalized planar pendulum for which the TISE is quasi-exactly solvable and conditionally exactly solvable
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

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

init.spherical (sourceforge.net)	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 Eq. (4) in our paper	false

init.fbr (sourceforge.net)	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 m_l is given by l = state-1 + m_l. For equally-spaced grids, the momentum starts at -k_max + 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.

init.fbr (sourceforge.net)

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 m_l is given by l = state-1 + m_l. For equally-spaced grids, the momentum starts at -k_max + 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.