/// <summary>
/// Creates a Gaussian wavepacket with given properties.
/// </summary>
public static WaveFunction CreateGaussianWavePacket(
int gridSizeX, int gridSizeY, float latticeSpacing, bool originAtLatticeCenter, float mass,
Vec2 packetCenter, Vec2 packetWidth, Vec2 avgMomentum, bool multiThread = true)
{
WaveFunction wf = new WaveFunction(gridSizeX, gridSizeY, latticeSpacing);
Complex I = Complex.I;
float rootPi = (float)Math.Sqrt(Math.PI);
float sigmaXSq = packetWidth.X * packetWidth.X;
float sigmaYSq = packetWidth.Y * packetWidth.Y;
float norm = (float)Math.Sqrt((packetWidth.X / (rootPi * sigmaXSq)) * (packetWidth.Y / (rootPi * sigmaYSq)));
int halfGridSizeX = (gridSizeX - 1) / 2;
int halfGridSizeY = (gridSizeY - 1) / 2;
TdseUtils.Misc.ForLoop(0, gridSizeY, (y) =>
{
float yf = (originAtLatticeCenter) ? (y - halfGridSizeY) * latticeSpacing : (y * latticeSpacing);
Complex expArgY = I * yf * avgMomentum.Y - (yf - packetCenter.Y) * (yf - packetCenter.Y) / (2 * sigmaYSq);
float[] wfDataY = wf.Data[y];
for (int x = 0; x < gridSizeX; x++)
{
float xf = (originAtLatticeCenter) ? (x - halfGridSizeX) * latticeSpacing : (x * latticeSpacing);
Complex expArgYX = expArgY + I * xf * avgMomentum.X - (xf - packetCenter.X) * (xf - packetCenter.X) / (2 * sigmaXSq);
Complex wfVal = norm * Complex.Exp(expArgYX);
wfDataY[2 * x] = wfVal.Re;
wfDataY[2 * x + 1] = wfVal.Im;
}
}, multiThread);
wf.Normalize();
return(wf);
}
/// <summary>
/// Calculates the value of a (freely evolving) Gaussian wavepacket at a given location and time.
/// </summary>
public static Complex FreeGaussianWavePacketValue(float x, float y, float z, float t,
Vec3 initialCenter, Vec3 initialWidth, Vec3 avgMomentum, float mass)
{
Complex I = Complex.I;
Complex effSigmaXSq = initialWidth.X * initialWidth.X + I * (t / mass);
Complex effSigmaYSq = initialWidth.Y * initialWidth.Y + I * (t / mass);
Complex effSigmaZSq = initialWidth.Z * initialWidth.Z + I * (t / mass);
float xRel = x - initialCenter.X - avgMomentum.X * t / mass;
float yRel = y - initialCenter.Y - avgMomentum.Y * t / mass;
float zRel = z - initialCenter.Z - avgMomentum.Z * t / mass;
float avgMomentumSq = avgMomentum.NormSq();
Complex expArg = I * (x * avgMomentum.X + y * avgMomentum.Y + z * avgMomentum.Z) - I * t * avgMomentumSq / (2 * mass) -
(xRel * xRel) / (2 * effSigmaXSq) - (yRel * yRel) / (2 * effSigmaYSq) - (zRel * zRel) / (2 * effSigmaZSq);
float rootPi = (float)Math.Sqrt(Math.PI);
Complex normX = Complex.Sqrt(initialWidth.X / (rootPi * effSigmaXSq));
Complex normY = Complex.Sqrt(initialWidth.Y / (rootPi * effSigmaYSq));
Complex normZ = Complex.Sqrt(initialWidth.Z / (rootPi * effSigmaZSq));
Complex wfVal = normX * normY * normZ * Complex.Exp(expArg);
return(wfVal);
}
/// <summary>
/// Creates a Gaussian wavepacket with given properties.
/// </summary>
public static WaveFunction CreateGaussianWavePacket(
GridSpec gridSpec, float latticeSpacing, bool originAtLatticeCenter, float mass,
Vec3 packetCenter, Vec3 packetWidth, Vec3 avgMomentum, bool multiThread = true)
{
WaveFunction wf = new WaveFunction(gridSpec, latticeSpacing);
int sx = gridSpec.SizeX;
int sy = gridSpec.SizeY;
int sz = gridSpec.SizeZ;
float[][][] wfData = wf.Data;
Complex I = Complex.I;
float rootPi = (float)Math.Sqrt(Math.PI);
float sigmaXSq = packetWidth.X * packetWidth.X;
float sigmaYSq = packetWidth.Y * packetWidth.Y;
float sigmaZSq = packetWidth.Z * packetWidth.Z;
float norm = (float)Math.Sqrt(1.0 / (rootPi * packetWidth.X * rootPi * packetWidth.Y * rootPi * packetWidth.Z));
int halfGridSizeX = (sx - 1) / 2;
int halfGridSizeY = (sy - 1) / 2;
int halfGridSizeZ = (sz - 1) / 2;
TdseUtils.Misc.ForLoop(0, sz, (z) =>
{
float zf = (originAtLatticeCenter) ? (z - halfGridSizeZ) * latticeSpacing : (z * latticeSpacing);
Complex expArgZ = I * zf * avgMomentum.Z - (zf - packetCenter.Z) * (zf - packetCenter.Z) / (2 * sigmaZSq);
for (int y = 0; y < sy; y++)
{
float yf = (originAtLatticeCenter) ? (y - halfGridSizeY) * latticeSpacing : (y * latticeSpacing);
Complex expArgZY = expArgZ + I * yf * avgMomentum.Y - (yf - packetCenter.Y) * (yf - packetCenter.Y) / (2 * sigmaYSq);
float[] wfDataZY = wf.Data[z][y];
for (int x = 0; x < sx; x++)
{
float xf = (originAtLatticeCenter) ? (x - halfGridSizeX) * latticeSpacing : (x * latticeSpacing);
Complex expArgZYX = expArgZY + I * xf * avgMomentum.X - (xf - packetCenter.X) * (xf - packetCenter.X) / (2 * sigmaXSq);
Complex wfVal = norm * Complex.Exp(expArgZYX);
wfDataZY[2 * x] = wfVal.Re;
wfDataZY[2 * x + 1] = wfVal.Im;
}
}
}, multiThread);
wf.Normalize();
return(wf);
}
/// <summary>
/// Calculates the value of a (freely evolving) Gaussian wavepacket at a given location and time.
/// </summary>
public static Complex FreeGaussianWavePacketValue(float x, float y, float t,
PointF initialCenter, PointF initialWidth, PointF avgMomentum, float mass)
{
Complex I = Complex.I;
Complex effSigmaXSq = initialWidth.X * initialWidth.X + I * (t / mass);
Complex effSigmaYSq = initialWidth.Y * initialWidth.Y + I * (t / mass);
float xRel = x - initialCenter.X - avgMomentum.X * t / mass;
float yRel = y - initialCenter.Y - avgMomentum.Y * t / mass;
float avgMomentumSq = avgMomentum.X * avgMomentum.X + avgMomentum.Y * avgMomentum.Y;
Complex expArg = I * (x * avgMomentum.X + y * avgMomentum.Y) - I * t * avgMomentumSq / (2 * mass) - (xRel * xRel) / (2 * effSigmaXSq) - (yRel * yRel) / (2 * effSigmaYSq);
float rootPi = (float)Math.Sqrt(Math.PI);
Complex normX = Complex.Sqrt(initialWidth.X / (rootPi * effSigmaXSq));
Complex normY = Complex.Sqrt(initialWidth.Y / (rootPi * effSigmaYSq));
Complex wfVal = normX * normY * Complex.Exp(expArg);
return(wfVal);
}
