wavepacket.expression

Classes that wrap operators into expressions for use in partial differential equations.

Classes

`ExpressionBase`	Base class for expressions.
`CommutatorLiouvillian`	Represents a commutator expression in a Liouville von-Neumann equation.
`SchroedingerEquation`	Expression wrapper for a Schrödinger equation.

Package Contents

class wavepacket.expression.ExpressionBase

Bases: abc.ABC

Base class for expressions.

By deriving from this class and implementing the method ExpressionBase.apply(), you can add custom expressions.

Notes

All differential equations have the form $\dot \rho = \mathcal{L}(\rho)$ (or equivalently $\dot \psi = \hat H \psi$ ), that is, the left-hand side is just the time derivative. This matches the common convention for the Liouville von-Neumann equation, but differs from the usual form of the Schrödinger equation, where the imaginary factor is on the left-hand side of the equation.

abstractmethod apply(state: wavepacket.grid.State, t: float | None = None) → wavepacket.grid.State

Applies the expression to the input state and returns the result.

Parameters:

statewp.grid.State: The state on which the expression is applied.
tfloat, optional: The time at which the expression is evaluated. Default is None, which will raise an exception if the contained expression is time-dependent.

Raises:

wp.BadStateError: If the state is invalid or has the wrong time. For example, a Schroedinger equaiton makes little sense for a density operator.

class wavepacket.expression.CommutatorLiouvillian(op: wavepacket.operator.OperatorBase)

Bases: wavepacket.expression.expressionbase.ExpressionBase

Represents a commutator expression in a Liouville von-Neumann equation.

Given an operator H, this commutator expression is given by $\mathcal{L}(\hat \rho) = -\imath (\hat H \hat \rho - \hat \rho \hat H)$ .

Parameters:

opwp.operator.OperatorBase: The operator to commute with the density operator.

Notes

The extra factor of -i is added to ensure that the commutator can be directly plugged into a Liouville von-Neumann equation. defined as $\frac{\partial \hat \rho}{\partial t} = \mathcal{L}(\hat \rho)$ . If you need the raw commutator, you have to multiply the result with the imaginary number.

apply(rho: wavepacket.grid.State, t: float | None = None) → wavepacket.grid.State

Evaluates the commutator for the given density operator and time.

Parameters:

rhowp.grid.State: The density operator to commute with
tfloat, optional: The time at which the operator is evaluated. By default it is None, which will throw if the contained operator is time-dependent.

Returns:

wp.grid.State: The result of the commutator.

Raises:

wp.BadGridError: If the grids of the density operator and the wrapped operator do not match.
wp.BadStateError: If the input state is not a valid density operator.

class wavepacket.expression.SchroedingerEquation(hamiltonian: wavepacket.operator.OperatorBase)

Bases: wavepacket.expression.expressionbase.ExpressionBase

Expression wrapper for a Schrödinger equation.

You should wrap a Hamiltonian in an object of this type so that the solvers can subsequently solve the resulting equation.

Parameters:

hamiltonianwp.operator.OperatorBase: The Hamiltonian that is wrapped by this class.

Notes

The Schrödinger equation is given by $\dot \psi = -\imath \hat H \psi$ , so this class only multiplies the wave function with the negative imaginary number, and wraps the input Hamiltonian into an expression so that solvers can work with it.

apply(psi: wavepacket.grid.State, t: float | None = None) → wavepacket.grid.State

Evaluates the right-hand side of the Schrödinger equation for the given input state and time.

Parameters:

psiwp.grid.State: The input state that is evaluated.
tfloat, optional: The time at which the Schrödinger equation is evaluated. The default None throws if the wrapped operator is time-dependent.

Returns:

wp.grid.State: The result of the evaluation.

Raises:

wp.grid.BadGridError: If the state’s grid differs from that of the Hamiltonian.
wp.grid.BadStateError: If the input state is not a wave function.