WAVEPACKET
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Approximate the behaviour of a 1D Gaussian wavepacket in an infinite-well potential box. This example uses the Crank-Nicolson Method to solve the 1D Schrodinger equation.
Further reading:
http://staff.ustc.edu.cn/~zqj/posts/Numerical_TDSE/ Params: L_box : float The width of the box in Bohr lengths. center : float The center of the initial wavepacket. momentum : float The momentum of the initial wavepacket. sigma : float The width of the initial wavepacket. dt : float Time steps in atomic units, 1 a.u. = 24.188 as. Returns: out : OrderedPair The most recent wavepacket.
Python Code
import numpy as np
import scipy.sparse as spa
from scipy.sparse.linalg import splu
from flojoy import flojoy, OrderedPair, SmallMemory, DefaultParams
from typing import Optional
def gaussian_wavepacket(x, x0, k, sigma=0.1):
# One dimensional Gaussian wavepacket.
x = np.asarray(x)
g = np.sqrt(1 / np.sqrt(np.pi) / sigma) * np.exp(-((x - x0) ** 2) / 2 / sigma**2)
return np.exp(1j * k * (x - x0)) * g
def CrankNicolson(psi, V, x, dt):
# Crank-Nicolson method for the 1D Schrodinger equation.
# No. of spatial grid points
J = x.size - 1
dx = x[1] - x[0]
# the external potential
V_e = spa.diags(V)
# the kinetic operator
O_k = np.ones(J + 1)
T = (-1 / 2 / dx**2) * spa.spdiags([O_k, -2 * O_k, O_k], [-1, 0, 1], J + 1, J + 1)
# the two unitary matrices
U2 = spa.eye(J + 1) + (1j * 0.5 * dt) * (T + V_e)
U1 = spa.eye(J + 1) - (1j * 0.5 * dt) * (T + V_e)
# splu requires CSC matrix format for efficient decomposition
U2 = U2.tocsc()
LU = splu(U2)
b = U1.dot(psi)
PSI_t = LU.solve(b)
return PSI_t
memory_key = "WAVEPACKET"
@flojoy(inject_node_metadata=True)
def WAVEPACKET(
default_params: DefaultParams,
default: Optional[OrderedPair] = None,
L_box: float = 20,
center: float = 0,
momentum: float = 5,
sigma: float = 1,
dt: float = 0.2,
) -> OrderedPair:
"""Approximate the behaviour of a 1D Gaussian wavepacket in an infinite-well potential box.
This example uses the Crank-Nicolson Method to solve the 1D Schrodinger equation.
Further reading:
http://staff.ustc.edu.cn/~zqj/posts/Numerical_TDSE/
Parameters
----------
L_box : float
The width of the box in Bohr lengths.
center : float
The center of the initial wavepacket.
momentum : float
The momentum of the initial wavepacket.
sigma : float
The width of the initial wavepacket.
dt : float
Time steps in atomic units, 1 a.u. = 24.188 as.
Returns
-------
OrderedPair
The most recent wavepacket.
"""
node_id = default_params.node_id
psi_t = SmallMemory().read_memory(node_id, memory_key)
if psi_t is None:
initialize = True
elif isinstance(psi_t, np.ndarray):
initialize = False
else:
raise TypeError("Error loading object from SmallMemory.")
# Box parameters.
xmin = -L_box / 2.0 # left boundary
J = 999 # No. of spatial grid points
x = np.linspace(xmin, xmin + L_box, J + 1, endpoint=True)
# The gaussian wavepacket as the initial wavefunction.
psi0 = gaussian_wavepacket(x, x0=center, k=momentum, sigma=sigma)
V_e = np.zeros_like(x) # The externial potentials.
# The time evolution of the Schrodinger equation.
if initialize:
PSI = CrankNicolson(psi0, V_e, x, dt)
elif not initialize:
PSI = CrankNicolson(psi_t, V_e, x, dt)
SmallMemory().write_to_memory(node_id, memory_key, PSI)
y = np.abs(PSI)
return OrderedPair(x=x, y=y)
Example App
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In this example, the WAVEPACKET node simulates the behavior of a wavepacket in a box. The wavepacket is visualized by the LINE node, and the wavepacket is evolved through time with the use of the LOOP node.
The initial state of the WAVEPACKET node can be altered through the peak center (center), phase velocity (momentum), and width (sigma) parameters. The box can be altered with the box width (L_box) and time step (dt) parameters.
For further reading, see the Theory appendix below.