/// <summary>
/// <c>getGroundState</c> deteremines the Groundstate of a the wavefunction saved in <c>this.psi</c>
/// with the imaginary time method
/// </summary>
public void getGroundState()
{
//ComplexNumber[] psi_0 = psi;
double mu = 1;
double mu_old;
double mu_error = 1;
ComplexNumber[] psi_old = new ComplexNumber[psi.Length];
double NormOfPsi = 0;
for (int i = 0; i < psi.Length; i++)
{
NormOfPsi += Math.Pow(psi[i].Norm(), 2);
}
NormOfPsi = Math.Sqrt(NormOfPsi) * deltaX;
int j = 0;
while (mu_error > Math.Pow(10, -8))
{
// creation of new wave function
psi_old = (ComplexNumber[])psi.Clone();
for (int i = 0; i < psi.Length; i++)
{
psi[i] = psi[i] * Math.Exp(-0.5 * deltaT * (V[i] + g1D / PhysConst.hbar * Math.Pow(psi[i].Norm(), 2)));
}
//ComplexNumber[] psi_k = new ComplexNumber[psi.Length];
psi = FFT.BR(psi, reversedBits); // Fourier transformation of the wave function with the Cooley-Tukey algorithm
psi = FFT.Shift(psi); //
for (int i = 0; i < psi.Length; i++)
{
psi[i] = psi[i] / xSteps;
}
// psi = psi*exp((-0.5*deltaT*hbar*|k|^2)/m)
for (int i = 0; i < psi.Length; i++)
{
psi[i] = psi[i] * Math.Exp(-0.5 * deltaT * (PhysConst.hbar / this.mass * Math.Pow(K[i], 2)));
}
psi = FFT.Shift(psi);
psi = FFT.IBR(psi, reversedBits); //Inverse fourier transformation of the wave function with the bit reverse algorithm
for (int i = 0; i < psi.Length; i++)
{
psi[i] = psi[i] * kSteps;
}
// psi = psi*exp(-0.5*deltaT*V(x)+g1D/hbar * |psi|^2)
for (int i = 0; i < psi.Length; i++)
{
psi[i] = psi[i] * Math.Exp(-0.5 * deltaT * (V[i] + g1D / PhysConst.hbar * Math.Pow(psi[i].Norm(), 2)));
}
mu_old = mu;
mu = Math.Log((psi_old[psi_old.Length / 2] / psi[psi.Length / 2]).Norm()) / deltaT;
mu_error = Math.Abs(mu - mu_old) / mu;
double currentNormOfPsi = 0;
for (int i = 0; i < psi.Length; i++)
{
currentNormOfPsi += Math.Pow(psi[i].Norm(), 2);
}
currentNormOfPsi = Math.Sqrt(currentNormOfPsi) * deltaX;
for (int i = 0; i < psi.Length; i++)
{
psi[i] = psi[i] * Math.Sqrt(NormOfPsi) / Math.Sqrt(currentNormOfPsi);
}
if (j > Math.Pow(10, 8))
{
break;
}
j++;
}
}
/// <returns> /// The quotient of the current and another complex number. /// </returns> /// <param name="c">Complex number to be devided by.</param> public ComplexNumber Divide(ComplexNumber c) => new ComplexNumber((this.rr * c.rr + this.ii * c.ii) / (c.rr * c.rr + c.ii * c.ii), (this.ii * c.rr - this.rr * c.ii) / (c.rr * c.rr + c.ii * c.ii));
/// <summary>
/// Simulates on timestep of the time evolution by performing the splitstep fourier method only once.
/// </summary>
/// <param name="FT">Algorithm which will be used for the Fouriertransformation</param>
public void splitStepFourier(string FT)
{
int size = this.psi.Length;
// psi=psi.*exp(-0.5*1i*dt*(V+(g1d/hbar)*abs(psi).ˆ2));
for (int i = 0; i < size; i++)
{
psi[i] = psi[i] * ComplexNumber.Exp(-0.5 * ComplexNumber.ImaginaryOne * deltaT
* (V[i] + g1D / PhysConst.hbar
* Math.Pow(psi[i].Norm(), 2)));
}
// decides which algorithm will be used for the FT
switch (FT)
{
case "DFT":
psi = FFT.DFT(psi);
break;
case "CT":
psi = FFT.CT(psi);
break;
case "BR":
psi = FFT.BR(psi, reversedBits);
break;
default:
break;
}
psi = FFT.Shift(psi); // shift the lower half with the upper one to restore normal order
for (int i = 0; i < size; i++)
{
psi[i] = psi[i] / size;
}
// psi_k=psi_k*exp(-0.5*dt*1i*(hbar/m)*kˆ2)
for (int i = 0; i < size; i++)
{
psi[i] = psi[i] * ComplexNumber.Exp(-0.5 * ComplexNumber.ImaginaryOne * deltaT * PhysConst.hbar / mass * Math.Pow(K[i], 2));
}
psi = FFT.Shift(psi); // shifts again, so that the result of the IFT will be normal orderd
// decides which algorithm will be used for the IFT
switch (FT)
{
case "DFT":
psi = FFT.IDFT(psi);
break;
case "CT":
psi = FFT.ICT(psi);
break;
case "BR":
psi = FFT.IBR(psi, reversedBits);
break;
default:
break;
}
for (int i = 0; i < size; i++)
{
psi[i] = psi[i] * size;
}
//psi = psi.* exp(-0.5 * 1i * dt * (V + (g1d / hbar) * abs(psi).ˆ2));
for (int i = 0; i < size; i++)
{
psi[i] = psi[i] * ComplexNumber.Exp(-0.5 * ComplexNumber.ImaginaryOne * deltaT
* (V[i] + g1D / PhysConst.hbar
* Math.Pow(psi[i].Norm(), 2)));
}
}
/// <returns> /// The difference of the current and another complex number. /// </returns> /// <param name="c">Complex number to be substracted.</param> public ComplexNumber Substract(ComplexNumber c) => new ComplexNumber(this.rr - c.rr, this.ii - c.ii);
/// <returns> /// The product of the current and another complex number. /// </returns> /// <param name="c">Complex number to be multiplicated.</param> public ComplexNumber Multiply(ComplexNumber c) => new ComplexNumber(this.rr * c.rr - this.ii * c.ii, this.rr * c.ii + this.ii * c.rr);
/// <returns> /// The sum of the current and another complex number. /// </returns> /// <param name="c">Complex number to be added.</param> public ComplexNumber Add(ComplexNumber c) => new ComplexNumber(this.rr + c.rr, this.ii + c.ii);
/// <summary> /// Computes the exponantial of a complex number. /// </summary> /// <param name="c">Complex exponent.</param> /// <returns>e to the power of the complex parameter.</returns> public static ComplexNumber Exp(ComplexNumber c) => Math.Exp(c.rr) * (new ComplexNumber(Math.Cos(c.ii), Math.Sin(c.ii)));
/// <returns></returns> /// <param name="c">Complex number whose imaginary part is desired.</param> public static double imaginaryPart(ComplexNumber c) => c.imaginaryPart();
/// <returns></returns> /// <param name="c">Complex number whose real part is desired.</param> public static double realPart(ComplexNumber c) => c.realPart();
/// <summary>
/// Transformation algorithm to perform a FFT using the Bit reversal structure.
/// </summary>
/// <remarks>
/// The FT is performed by setting the <c>negativeTwidleFactor</c> to <c>true</c>, its inverse by setting it to <c>false</c>.
/// </remarks>
/// <returns>Transformed array</returns>
/// <param name="negativeTwidleFactor">If set to <c>true</c> FT is performed, if <c>false</c> its inverse.</param>
/// <param name="f">Array to be transformed</param>
/// <param name="reversedBits">Reversed bit array, needs to be precalculated by <c>BitReverse</c></param>
public static ComplexNumber[] Transform(bool negativeTwidleFactor, ComplexNumber[] f, uint[] reversedBits)
{
ComplexNumber[] workF = (ComplexNumber[])f.Clone();
int size = f.Length;
if (size != reversedBits.Length)
{
throw new Exception("The arrays dont have the same lenght");
}
int a = 0;
int findA = size;
while (findA > 1)
{
findA = findA >> 1;
a++;
}
// sets the prefactor for the twiddle factor to perform either FFT or IFFT
int preFactor;
if (negativeTwidleFactor)
{
preFactor = -1;
}
else
{
preFactor = 1;
}
// prepare necassary variables
ComplexNumber even, odd, wM, w;
int shift, m;
//orders the array according to the bit reversal
for (int i = 0; i < f.Length; i++)
{
workF[i] = f[reversedBits[i]];
}
// three for loops to process the full FFT/IFFT
// the result of the loops gives the FFT/IFFT in the normal ordered form
for (int s = 1; s <= a; s++)
{
shift = 1 << (s - 1); // = 2^(s-1) via bitshift
m = 1 << s; // = 2^s
wM = ComplexNumber.Exp(preFactor * ComplexNumber.ImaginaryOne * 2 * Math.PI / m); // twiddle factor, with prefactor for FFT/IFFT
for (int i = 0; i < size; i += m)
{
w = 1;
for (int j = 0; j < shift; j++)
{
// the call to the reverse bit structure accounts for the bitflipped order
even = workF[i + j];
odd = w * workF[i + j + shift];
// joins together the desired even and odd parts, with the twidle factor multiplied to the odd part
workF[i + j] = even + odd;
workF[i + j + shift] = even - odd;
// increase phase of twiddle factor
w = w * wM;
}
}
}
return(workF);
}