commit 6b3eaddd270b8a87827d868b0f5bfbb9495bfde9
Author: miksa234 <milutin@popovic.xyz>
Date: Fri, 19 Mar 2021 11:56:11 +0100
initial commit
Diffstat:
19 files changed, 1670 insertions(+), 0 deletions(-)
diff --git a/AUTHORS b/AUTHORS
@@ -0,0 +1,5 @@
+Kraffert Jan <a11713707@unet.univie.ac.at>
+Avargues Noah <a11724509@unet.univie.ac.at>
+Sabo Filip <a11810435@unet.univie.ac.at>
+Scharinger Sophie <sophie.scharinger@univie.ac.at>
+Popovic Milutin <milutin@popovic.xyz>
diff --git a/LICENSE b/LICENSE
@@ -0,0 +1,674 @@
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+THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY
+GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE
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+DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD
+PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS),
+EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF
+SUCH DAMAGES.
+
+ 17. Interpretation of Sections 15 and 16.
+
+ If the disclaimer of warranty and limitation of liability provided
+above cannot be given local legal effect according to their terms,
+reviewing courts shall apply local law that most closely approximates
+an absolute waiver of all civil liability in connection with the
+Program, unless a warranty or assumption of liability accompanies a
+copy of the Program in return for a fee.
+
+ END OF TERMS AND CONDITIONS
+
+ How to Apply These Terms to Your New Programs
+
+ If you develop a new program, and you want it to be of the greatest
+possible use to the public, the best way to achieve this is to make it
+free software which everyone can redistribute and change under these terms.
+
+ To do so, attach the following notices to the program. It is safest
+to attach them to the start of each source file to most effectively
+state the exclusion of warranty; and each file should have at least
+the "copyright" line and a pointer to where the full notice is found.
+
+ <one line to give the program's name and a brief idea of what it does.>
+ Copyright (C) <year> <name of author>
+
+ This program is free software: you can redistribute it and/or modify
+ it under the terms of the GNU General Public License as published by
+ the Free Software Foundation, either version 3 of the License, or
+ (at your option) any later version.
+
+ This program is distributed in the hope that it will be useful,
+ but WITHOUT ANY WARRANTY; without even the implied warranty of
+ MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+ GNU General Public License for more details.
+
+ You should have received a copy of the GNU General Public License
+ along with this program. If not, see <https://www.gnu.org/licenses/>.
+
+Also add information on how to contact you by electronic and paper mail.
+
+ If the program does terminal interaction, make it output a short
+notice like this when it starts in an interactive mode:
+
+ <program> Copyright (C) <year> <name of author>
+ This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'.
+ This is free software, and you are welcome to redistribute it
+ under certain conditions; type `show c' for details.
+
+The hypothetical commands `show w' and `show c' should show the appropriate
+parts of the General Public License. Of course, your program's commands
+might be different; for a GUI interface, you would use an "about box".
+
+ You should also get your employer (if you work as a programmer) or school,
+if any, to sign a "copyright disclaimer" for the program, if necessary.
+For more information on this, and how to apply and follow the GNU GPL, see
+<https://www.gnu.org/licenses/>.
+
+ The GNU General Public License does not permit incorporating your program
+into proprietary programs. If your program is a subroutine library, you
+may consider it more useful to permit linking proprietary applications with
+the library. If this is what you want to do, use the GNU Lesser General
+Public License instead of this License. But first, please read
+<https://www.gnu.org/licenses/why-not-lgpl.html>.
diff --git a/README.md b/README.md
@@ -0,0 +1,8 @@
+Schrödinger Equation
+====================
+
+FEM, FD analysis and analytical comparison of the Schrödinger Equation:
+ * Simple 1D potentials
+ * H1
+ * H2
+ * Double Slit Simulation (time dependent SG)
diff --git a/h1/analytical/3d_analytical.py b/h1/analytical/3d_analytical.py
@@ -0,0 +1,52 @@
+import numpy as np
+from scipy.special import factorial, sph_harm, genlaguerre
+import matplotlib.pyplot as plt
+from dolfin import *
+import pandas as pd
+from mayavi import mlab
+from skimage import measure
+
+# build wave-function:
+def wave_function(n, l, m, x, y, z, a0):
+
+ r = np.sqrt(x**2+y**2+z**2)
+ theta = np.arccos(z/r)
+ phi = np.arctan(y/x)
+
+ # replace possible NaNs wirh 0
+ theta[np.argwhere(np.isnan(theta))] = 0
+ phi[np.argwhere(np.isnan(phi))] = 0
+
+ # Radial part
+ R = np.sqrt((2/n/a0)**3*factorial(n-l-1)/(2*n*(factorial(n+l))**3)) * (2*r/n/a0)**l * np.exp(-r/n/a0) * genlaguerre(n-l-1,2*l+1)(2*r/n/a0)
+ # spherical harmonics
+ Y = sph_harm(m, l, phi, theta)
+ # complete wavefunction
+ wf = R * Y
+ return wf
+
+
+def main():
+
+ d=0.1
+ min=-10
+ max=10
+ X = np.arange(min,max,d)
+ Y = np.arange(min,max,d)
+ Z = np.arange(min,max,d)
+ x, y, z = np.meshgrid(X, Y, Z)
+ a0 = 1
+
+# mlab.figure()
+
+ wf = np.abs(wave_function(2, 1, 0, x, y, z, a0))**2
+ mlab.contour3d(wf, transparent=True)
+
+ mlab.savefig('s-orbital.png')
+ mlab.colorbar()
+ mlab.outline()
+ mlab.show()
+
+
+if __name__ == "__main__":
+ main()
diff --git a/h1/analytical/h_orbit_analytisch.py b/h1/analytical/h_orbit_analytisch.py
@@ -0,0 +1,224 @@
+from scipy.constants import physical_constants
+import numpy as np
+from scipy.special import assoc_laguerre as lag #zugehörige Laguerre Polynome
+from scipy.special import sph_harm as kugelfl #Kugelflächenfunktionen
+import matplotlib.pyplot as plt
+
+(a0,blub,bla)=physical_constants["Bohr radius"]
+a0=0.529177210903 #in Angsröm
+
+def fak(n):
+ #Beschreibung: Diese Funktion ermittelt zur natürliche Zahl n die zugehörige Fakultät (n!)
+ # Variablenname: Datentyp: Beschreibung:
+ #Input: n integer Zu Berechnende Fakultät. ACHTUNG: es gilt n>=0 und n element N
+ #Output: - integer n!
+ #Fehleroutput integer=-1 Falsche Eingabe für n
+
+ if (n<0) or (n-int(n)>0):
+ print("Upps, das hätte leider nicht passieren dürfen.\nn muss eine natürliche Zahl sein.")
+ return -1
+ elif (n==0) or (n==1):
+ return 1
+ else:
+ return n*fak(n-1)
+
+def transform(x,y,z):
+ # Beschreibung: Diese Funktion Transfromiert Kartesische Koordinaten in Kugelkoordinaten
+ # Variablenname: Datentyp: Beschreibung:
+ # Input: x real kartesische x Koordinate
+ # y real kratesische y Koordinate
+ # z real kartesische z koordinate
+ # Output: r real Radius
+ # theta real Polarwinkel mit theta\in[0,pi]
+ # phi real Azimutwinkel mit phi\in[0,2*pi]
+
+ # Berechnung von r
+ r=np.sqrt(x**2+y**2+z**2)
+ #Problembehandlung: Koordinatenursprung
+ if(r==0):
+ phi=0.
+ theta=0.
+ else:
+ #Berechnung von theta
+ theta=np.arccos(z/r)
+
+ # Berechnung von phi
+ if (x>0)and(y>=0):
+ phi=np.arctan(y/x)
+ elif(x==0)and(y>0):
+ phi=np.pi/2
+ elif(x<0):
+ phi=np.arctan(y/x)+np.pi
+ elif(x==0)and(y<0):
+ phi=(3/2)*np.pi
+ elif(x>0)and(y<0):
+ phi=2*np.pi+np.arctan(y/x)
+ else:#if(x==0)and(y==0)
+ phi=0.
+ return r,theta,phi
+
+def norm(Z,n,l):
+ #Beschreibung: Diese Funktion ermittelt den Normierungsfaktor der Wellenfunktion für ein Elektron um einen Atomkeern der Ladung Z*e
+ # Variablenname: Datentyp: Beschreibung:
+ #Input: Z integer Anzahl der Protonen im Atomkern
+ # n integer Hauptquantenzahl
+ # l integer Nebenquantenzahl
+ #Output: - real Normierungsfaktor der Wellenfunktion
+
+ return np.sqrt((((2*Z)/(n*a0))**3)*(fak(n-l-1)/(2*n*fak(n+l))))
+
+def radial(Z,n,l,r,normierung):
+ #Beschreibung: Diese Funktion ermittelt den Radialen Anteil R_nl(r) der Wellenfunktion für das Wasserstoffatom
+ # Variablenname: Datentyp: Beschreibung:
+ #Input Z integer Anzahl der Protonen im Atomkern
+ # n integer Hauptquantenzahl
+ # l integer Nebenquantenzahl
+ # r real Abstand des Elektrons zum Atomkern
+ # normierung real Normierungsfaktor der Wellenfunktion
+ #Output: - real Radialer Anteil R_nl(r) der Wellenfunktion für das Wasserstoffatom
+
+ rho=(2*Z*r)/(n*a0)
+ return normierung*np.exp(-rho/2)*(rho**l)*lag(rho,n-l-1,2*l+1)
+
+def aufenthalt_radial(Z,n,l,r,normierung):
+ #Beschreibung: Diese Funktion ermittelt die Radiale Aufenthaltswahrscheinlichkeitsdichte des Wasserstoffatoms
+ # Variablenname: Datentyp: Beschreibung:
+ #Input Z integer Anzahl der Protonen im Atomkern
+ # n integer Hauptquantenzahl
+ # l integer Nebenquantenzahl
+ # r real Abstand des Elektrons zum Atomkern
+ # normierung real Normierungsfaktor der Wellenfunktion
+ #Output: - real Radiale Aufenthaltswahrscheinlichkeitsdichte des Wasserstoffatoms
+
+ return np.sqrt(radial(Z,n,l,r,normierung)**2)**2
+
+def aufenthalt_welle(Z,n,l,m,r,normierung,phi,theta):
+ #Beschreibung: Diese Funktion ermittelt die Radiale Aufenthaltswahrscheinlichkeitsdichte des Wasserstoffatoms
+ # Variablenname: Datentyp: Beschreibung:
+ #Input Z integer Anzahl der Protonen im Atomkern
+ # n integer Hauptquantenzahl
+ # l integer Nebenquantenzahl
+ # m integer magnetische Quantenzahl des Drehimpulses
+ # r real Abstand des Elektrons zum Atomkern
+ # normierung real Normierungsfaktor der Wellenfunktion
+ # phi real Azimutwinkel mit phi\in[0,2*pi]
+ # theta real Polarwinkel mit theta\in[0,pi]
+ #Output: result real Aufenthaltswahrscheinlichkeitsdichte des Wasserstoffatoms
+
+ result=radial(Z,n,l,r,normierung)*kugelfl(m,l,phi,theta)
+ return np.sqrt(result.conjugate()*result)**2
+
+
+def plotaufenthalt_radial(Z,n,l,aufl,name_nl,xmax):
+ #Beschreibung: Diese Funktion plottet die Radialenaufenthaltswahrscheinlichkeitsdichte*r**2, wobei nach Funktionsaufruf plt.show() ausgeführt werden muss
+ # Variablenname: Datentyp: Beschreibung:
+ #Input Z integer Anzahl der Protonen im Atomkern
+ # n integer,array Hauptquantenzahl, das Array muss dieselbe Größe wie l und name_nl haben
+ # l integer,array Nebenquantenzahl, das Array muss dieselbe Größe wie n und name_nl haben
+ # aufl integer Auflösung des Graphen
+ # name_nl characther,array Legende des Plots, das Array muss dieseleb Größe wie n und l haben
+ # xmax real maximaler Abstand des Elektrons zum Atomkern, wobei innerhald der Funktion xmax in xmax*a0 umgerechnet wird
+ #Output: - - Plotvorbereitung für die Radialenaufenthaltswahrscheinlichkeitsdichte*r**2
+
+ x=np.linspace(0,xmax,aufl)
+ x=x*a0
+ anz=np.shape(n)[0] #Anzahl der zu plottenden Orbitale
+ y=np.zeros((anz,aufl))
+ r=x
+ for i in range(0,anz):
+ normi=norm(Z,n[i],l[i])
+ y[i,:]=aufenthalt_radial(Z,n[i],l[i],r[:],normi)*r**2
+
+ # Plot
+ plt.figure()
+ for i in range(0,anz):
+ plt.plot(x,y[i,:],label=name_nl[i])
+ plt.title("Radiale Aufenthaltswahrscheinlichkeitsdichte *r²")
+ plt.ylabel("|R_nl(r)|*r²")
+ plt.xlabel("r*a0 [Angström]")
+ plt.legend()
+
+def plot2d_aufenthalt(Z,n,l,m,aufl,xmin,xmax,ymin,ymax,name):
+ #Beschreibung: Diese Funktion plottet die x-z Ebene der Aufenthaltswahrscheinlichkeitsdichte und die Aufenthaltswahrscheinlichkeitsdichte*r**2,
+ # wobei nach Funktionsaufruf plt.show() ausgeführt werden muss
+ # Variablenname: Datentyp: Beschreibung:
+ #Input Z integer Anzahl der Protonen im Atomkern
+ # n integer,array Hauptquantenzahl, das Array muss dieselbe Größe wie l und name haben
+ # l integer,array Nebenquantenzahl, das Array muss dieselbe Größe wie n und namehaben
+ # aufl integer Auflösung
+ # xmin,xmax real Gittergrenzen in x Richtung
+ # ymin,ymax real Gittergrenzen in y Richtung
+ # name character,array enthält die Spezifikation von n,l und m die geplottet werden und so in der Überschrift angezeigt werden,
+ # das Array muss dieselbe Größe wie n und l haben
+ #Output: - - Plotvorbereitung für die Aufenthaltswahrscheinlichkeitsdichte und die Aufenthaltswahrscheinlichkeitsdichte*r**2
+
+ #Definiere Gitter, Achtung y entspricht hier den Werten auf der z Achse
+ x=np.linspace(xmin,xmax,aufl)*a0
+ y=np.linspace(ymin,ymax,aufl)*a0
+ X,Y=np.meshgrid(x,y)
+
+ blub=np.shape(X)#Form von np.shape=(Anzahl y Werte, Anzahl x Werte, Anzahl z Werte)
+
+ #Bereite Transformation in Kugelkoordinaten vor
+ r=np.zeros(blub);theta=np.zeros(blub);phi=np.zeros(blub)
+ #Koordinatentransformation in Kugelkoordinaten
+ for i in range(0,blub[0]): #Schleife für z Werte
+ for j in range(0,blub[1]): #Schleife für x Werte
+ (r[i,j],theta[i,j],phi[i,j])=transform(X[i,j],0,Y[i,j])
+
+ #Plottvorbereitungen
+ anz=np.shape(n)[0] #Anazhl der zu Plottenden Orbitale
+
+ for number in range(0,anz):
+ #Berechne Normaisierungsfaktor
+ normi=norm(Z,n[number],l[number])
+ #Definiere Plot Array
+ orbit=np.zeros(np.shape(X))
+ #Berechne Raumpunkte
+ orbit=np.real(aufenthalt_welle(Z,n[number],l[number],m[number],r,normi,phi,theta))
+
+ #Plot
+ #Plot Aufenthaltswahrscheinlichkeitsdichte
+ plt.figure()
+ plt.imshow(orbit,cmap="gnuplot",extent=[x.min(),x.max(),y.min(),y.max()])
+ plt.title("Aufenthaltswahrscheinlichkeitsdichte\nfür "+name[number])
+ plt.xlabel("[Angström]")
+ plt.ylabel("[Angström]")
+ plt.colorbar()
+ #Plot Aufenthaltswahrscheinlichkeitsdichte*r²
+ plt.figure()
+ plt.imshow(orbit*r**2,cmap="gnuplot",extent=[x.min(),x.max(),y.min(),y.max()])
+ plt.title("Aufenthaltswahrscheinlichkeitsdichte*r²\n für "+name[number])
+ plt.xlabel("[Angström]")
+ plt.ylabel("[Angström]")
+ plt.colorbar()
+
+
+#Plot der Radialenaufenthaltswahrscheinlichkeitsdichte*r**2
+Z=1 #Anzahl der Protonen im Kern
+n=np.array([1,2,2,])
+l=np.array([0,0,1])
+aufl=200 #Auflösung für den Plot der Radialenaufenthaltswahrscheinlichkeitsdichte*r**2
+name_nl=np.array(["n=1,l=0","n=2,l=0","n=2,l=1"]) #Legende des Plots
+xmax=20
+
+plotaufenthalt_radial(Z,n,l,aufl,name_nl,xmax)
+
+
+#Plot x-z Ebene
+Z=1
+n=np.array([1,2,2])
+l=np.array([0,0,1])
+m=np.array([0,0,0])
+#Definiere Gitterauflösung
+aufl=403
+#Definiere Gittergrenzen
+xmin=-20
+xmax=20
+ymin=-20
+ymax=20
+name=np.array(["n=1, l=0, m=0","n=2, l=0, m=0","n=2, l=1, m=0"]) #Quantenzahlen, die in der Überschrift angezeigt werden
+
+plot2d_aufenthalt(Z,n,l,m,aufl,xmin,xmax,ymin,ymax,name)
+
+plt.show()
diff --git a/h1/numerical/main.py b/h1/numerical/main.py
@@ -0,0 +1,72 @@
+#!/usr/bin/python3.9
+
+from dolfin import *
+import numpy as np
+import os
+
+def main():
+ mesh = UnitCubeMesh(25, 25, 25)
+ V = FunctionSpace(mesh, 'CG', 1)
+
+ m_e = 9.10e-31; m_k = 1.67e-27
+ hbar = 1.05e-34; k = 8.98e9
+ mu = m_e*m_k/(m_e + m_k)
+ e = 1.60e-19
+
+ # Potential for the Hydrogen Atom
+ # { 1/r if r > 0
+ # V = {
+ # { 0 else
+ pot_ = Expression('sqrt(x[0]*x[0]+x[1]*x[1]+x[2]*x[2])>0 ?\
+ -1/sqrt(x[0]*x[0]+x[1]*x[1]+x[2]*x[2]) : 0',\
+ degree=4)
+ pot = interpolate(pot_, V)
+
+
+ def boundary(x, on_boundary):
+ return on_boundary
+
+ bc = DirichletBC(V, Constant(0.), boundary)
+
+ u = TrialFunction(V)
+ v = TestFunction(V)
+
+ # Assemble system
+ a = hbar/(2*mu)*inner(grad(u), grad(v))*dx + e**2/k*pot*u*v*dx
+ m = u*v*dx
+
+ # Define Matrices
+ A = PETScMatrix()
+ M = PETScMatrix()
+
+ assemble(a, tensor=A)
+ assemble(m, tensor=M)
+
+ # Apply boundary on the system
+ bc.apply(A)
+ bc.apply(M)
+
+ # Create eigensolver
+ eigensolver = SLEPcEigenSolver(A, M)
+ eigensolver.parameters["spectrum"] = "smallest magnitude"
+
+ values = 30
+ eigensolver.solve(values)
+
+ u = Function(V)
+
+
+ if os.path.exists('./meshes') != True:
+ os.mkdir('./meshes')
+
+ f = File('./meshes/orbitals.pvd')
+
+ for i in range(values):
+ E, E_c, R, R_c = eigensolver.get_eigenpair(i)
+
+ u.vector()[:] = np.sqrt(R*R + R_c*R_c)
+
+ f << (u, i)
+
+if __name__ == "__main__":
+ main()
diff --git a/h2/h2.py b/h2/h2.py
@@ -0,0 +1,79 @@
+#!/usr/bin/python3.9
+
+
+from dolfin import *
+
+import numpy as np
+import os
+
+
+#verwenden atomare einheiten. hier werden alle in der schrodingergleicchung notigen naturkonstannten 1 und die langeneinheit wird der bohrradius
+
+def system(i):
+
+ d = i/100.0 #d in mesh units
+ dh = d/2.0
+
+ #define mesh and function space
+ mesh = UnitCubeMesh(20, 20, 20) #1 mesh unit = bohrradius = 0.529 Angstrom
+ V = FunctionSpace(mesh, 'CG', 1)
+
+ # Potential for the Hydrogen Atom
+ # { 1/r if r > 0
+ # V = {
+ # { 0 else
+
+ formula = 'sqrt(x[0]*x[0]+x[1]*x[1]+x[2]*x[2])>0 ?\
+ -(1/sqrt((x[0]-' + str(dh) + ')*(x[0]-' + str(dh) + ')+x[1]*x[1]+x[2]*x[2]) + 1/sqrt((x[0]+' + str(dh) + ')*(x[0]+' + str(dh) + ')+x[1]*x[1]+x[2]*x[2]) - 1.0/' + str(d) + ') : 0'
+
+
+
+ pot = Expression(formula,degree=3)
+
+
+ # Boundary 0 everywhere
+ def boundary(x, on_boundary):
+ return on_boundary
+ bc = DirichletBC(V, 0, boundary)
+
+ u = TrialFunction(V)
+ v = TestFunction(V)
+
+ # Assemble system
+ a = (1/2*inner(grad(u), grad(v)) + pot*u*v)*dx
+
+ # Define Matrices
+ A = PETScMatrix()
+ assemble(a, tensor=A)
+
+ # Apply boundary on the system
+ bc.apply(A)
+
+ # Create eigensolver
+ eigensolver = SLEPcEigenSolver(A)
+ eigensolver.parameters["spectrum"] = "smallest magnitude"
+
+ values = 1
+ eigensolver.solve(values)
+
+ u = Function(V)
+
+
+ for j in range(values):
+ E, E_c, R, R_c = eigensolver.get_eigenpair(j)
+
+ print('E_0 = ', E)
+
+ file = open("energy.dat","a")
+ file.write(str(d))
+ file.write(" ")
+ file.write(str(E))
+ file.write("\n")
+ file.close
+
+def main():
+ for i in range(100, 300, 2):
+ system(i)
+
+if __name__ == "__main__":
+ main()
diff --git a/potentials_1d/main.py b/potentials_1d/main.py
@@ -0,0 +1,362 @@
+#!/usr/bin/python3.9
+
+from dolfin import *
+import numpy as np
+import matplotlib.pyplot as plt
+from scipy.special import factorial, hermite, eval_laguerre, genlaguerre
+import matplotlib.patches as mpatches
+
+def harmonic_oscillator(xmin, xmax, mesh, V):
+
+ xmin=xmin
+ xmax=xmax
+ mesh=mesh
+ V=V
+
+ def boundary(x, on_bnd):
+ return on_bnd
+
+ bc = DirichletBC(V, Constant(0.), boundary)
+
+ u = TrialFunction(V)
+ v = TestFunction(V)
+
+ pot_ = Expression('0.5*pow(x[0], 2)', degree=2)
+ pot = interpolate(pot_, V)
+
+ a = 1/2*inner(grad(u), grad(v))*dx + pot*u*v*dx
+ m = u*v*dx
+
+ A = PETScMatrix()
+ M = PETScMatrix()
+
+ assemble(a, tensor=A)
+ assemble(m, tensor=M)
+
+ bc.apply(A)
+
+ # create eigensolver
+ eigensolver = SLEPcEigenSolver(A, M)
+ eigensolver.parameters['spectrum'] = 'smallest magnitude'
+
+ # solve for eigenvalues
+ values = 10
+ eigensolver.solve(values)
+
+ u = Function(V)
+
+ # plot
+ fig = plt.figure(figsize=[10, 10])
+ En = []
+ for i in range(0, values+1):
+ E, E_c, R, R_c = eigensolver.get_eigenpair(i)
+ En.append(E)
+ x = np.linspace(xmin, xmax, len(R))
+
+ I = np.trapz(R*R, x)
+ # print(I)
+
+ #plot eigenfunction
+ plt.plot(x, 20*R*R+E, c='blue')
+ plt.plot(x, R_c+E, lw=0.5, c='black')
+ plt.annotate(f'$E_{ { i } } = {round(En[i], 3)}$', (xmax-0.1, 0.05+E), fontsize=14)
+ plt.annotate(f'$|\Psi _{ { i } }|²$', (xmin, 0.1+E), fontsize=14)
+
+ plt.plot(x, 1/2*x**2, c='red')
+ plt.ylim(0, max(En)+0.5)
+ plt.title('Harmonic Oscillator $V(x) =\\frac{1}{2}x^2 $', fontsize=20)
+ plt.xticks([])
+ plt.yticks([])
+
+
+
+ # plot analytical solutions
+
+ def harm_osc(n, x):
+ return 1/sqrt(sqrt(np.pi)*(2**n)*factorial(n)) * hermite(n)(x) * np.exp(-1/2*x**2)
+
+ x = np.linspace(xmin, xmax, 70)
+
+ # print(En)
+
+ E = []
+ for n in range(0, values+1):
+
+ E.append(n+1/2)
+ psi = harm_osc(n, x)
+ I = np.trapz(psi*psi, x)
+ # print(I)
+ plt.scatter(x, psi*psi+En[n], s=20, c='green', marker="x")
+
+ numerical = mpatches.Patch(color='blue', label='numerical solution', linestyle='-')
+ analytical = mpatches.Patch(color='green', label='analytical solution', linestyle='-')
+ plt.legend(handles = [numerical, analytical], loc='upper left')
+ # plt.show()
+ # plt.savefig('./plots/harmonic_oscillator.png')
+
+ return E, En
+
+def box(xmin, xmax, mesh, V):
+ xmin=xmin
+ xmax=xmax
+ mesh=mesh
+ V=V
+
+ def boundary(x, on_bnd):
+ return on_bnd
+
+ bc = DirichletBC(V, Constant(0.), boundary)
+
+ u = TrialFunction(V)
+ v = TestFunction(V)
+
+ pot_ = Expression('x[0] == xmin || x[0] == xmax ? 1000 : 0', xmin=xmin, xmax=xmax, degree=2)
+
+ pot = interpolate(pot_, V)
+
+ a = 1/2*inner(grad(u), grad(v))*dx + pot*u*v*dx
+ m = u*v*dx
+
+ A = PETScMatrix()
+ M = PETScMatrix()
+
+ assemble(a, tensor=A)
+ assemble(m, tensor=M)
+
+ bc.apply(A)
+
+ # create eigensolver
+ eigensolver = SLEPcEigenSolver(A, M)
+ eigensolver.parameters['spectrum'] = 'smallest magnitude'
+
+ # solve for eigenvalues
+ values = 6
+ eigensolver.solve(values)
+
+ u = Function(V)
+
+ # plot
+ fig = plt.figure(figsize=[10, 10])
+ En = []
+ for i in range(0, values+1):
+ E, E_c, R, R_c = eigensolver.get_eigenpair(i)
+ En.append(E)
+ x = np.linspace(xmin, xmax, len(R))
+
+ I = np.trapz(R*R, x)
+ # print(I)
+
+ #plot eigenfunction
+ plt.plot(x, 10*R*R+E, c='blue')
+ plt.plot(x, R_c+E, lw=0.5, c='black')
+ plt.annotate(f'$E_{ { i } } = {round(En[i], 3)}$', (xmax-0.1, 0.05+E), fontsize=14)
+ plt.annotate(f'$|\Psi _{ { i } }|²$', (xmin, 0.05+E), fontsize=14)
+
+ plt.axvline(x=xmin, c='red')
+ plt.axvline(x=xmax, c='red')
+ plt.ylim(0, max(En)+0.5)
+ plt.title('Particle in \'infinite\' well', fontsize=20)
+ plt.xticks([])
+ plt.yticks([])
+
+
+
+ # plot analytical solutions
+ x = np.linspace(xmin, xmax, 70)
+
+ # print(En)
+
+ for i in range(values+1, values+2):
+ E, E_c, R, R_c = eigensolver.get_eigenpair(i)
+ # En.append(E)
+
+ E = []
+ for n in range(1, values+2):
+
+ # particle in a box
+ E.append(n**2 * np.pi**2 / (2*100))
+ kn = n*np.pi/10
+ if n%2 == 0:
+ psi = sqrt(2/10) * np.sin(kn*x)
+ if n%2 == 1:
+ psi = sqrt(2/10) * np.cos(kn*x)
+
+
+ # plot
+
+ I = np.trapz(psi*psi, x)
+ # print(I)
+ plt.scatter(x, 0.5*psi*psi+En[n-1], s=20, c='green', marker="x")
+
+
+
+ # plt.tight_layout()
+
+ numerical = mpatches.Patch(color='blue', label='numerical solution', linestyle='-')
+ analytical = mpatches.Patch(color='green', label='analytical solution', linestyle='-')
+ plt.legend(handles = [numerical, analytical], loc='upper left')
+ # plt.show()
+ # plt.savefig('./plots/box_potential.png')
+
+ return E, En
+
+def morse(xmin, xmax, mesh, V):
+ xmin=xmin
+ xmax=xmax
+ mesh=mesh
+ V=V
+ D = 12
+ a = 1
+
+ def boundary(x, on_bnd):
+ return on_bnd
+
+ bc = DirichletBC(V, Constant(0.), boundary)
+
+ u = TrialFunction(V)
+ v = TestFunction(V)
+
+ pot_ = Expression('D*pow((1-exp(a*(x[0]-x0))), 2)', D=D, a=a, x0=0, degree=2)
+ # pot_ = Expression('D*(exp(-2*a*(x[0]-x0))-2*exp(-a*(x[0]-x0)))', D=D, a=a, x0=0, degree=2)
+
+ pot = interpolate(pot_, V)
+
+ a = 1/2*inner(grad(u), grad(v))*dx + pot*u*v*dx
+ m = u*v*dx
+
+ A = PETScMatrix()
+ M = PETScMatrix()
+
+ assemble(a, tensor=A)
+ assemble(m, tensor=M)
+
+ bc.apply(A)
+
+ # create eigensolver
+ eigensolver = SLEPcEigenSolver(A, M)
+ eigensolver.parameters['spectrum'] = 'smallest magnitude'
+
+ # solve for eigenvalues
+ values = 3
+ eigensolver.solve(values)
+
+ u = Function(V)
+
+ # plot
+ fig = plt.figure(figsize=[10, 10])
+ En = []
+ for i in range(0, values+1):
+ E, E_c, R, R_c = eigensolver.get_eigenpair(i)
+ En.append(E)
+ x = np.linspace(xmin, xmax, len(R))
+
+ I = np.trapz(R*R, x)
+ # print(I)
+
+ #plot eigenfunction
+ plt.plot(x, 20*R*R+E, c='blue')
+ plt.plot(x, R_c+E, lw=0.5, c='black')
+ plt.annotate(f'$E_{ { i } } = {round(En[i], 3)}$', (xmax-0.1, 0.1+E), fontsize=14)
+ plt.annotate(f'$|\Psi _{ { i } }|²$', (xmin, 0.2+E), fontsize=14)
+
+ plt.plot(x, 12*(1-np.exp(-x))**2, c='red')
+ # plt.plot(x, 12*(np.exp(-2*x)-2*np.exp(-x)))
+ plt.ylim(0, max(En)+2)
+ # plt.title('Particle in \'infinite\' well', fontsize=20)
+ plt.title('Morse Potential $V(x) = D \cdot (1 - e^{- \\alpha (x - x0)})^2$', fontsize=20)
+ plt.xticks([])
+ plt.yticks([])
+
+
+
+
+
+ # plot analytical solutions
+ x = np.linspace(xmin, xmax, 70)
+
+ # print(En)
+
+ E = []
+
+ for n in range(0, values+1):
+ # morse
+
+ w = np.sqrt(2*12)/(2*np.pi)
+ l = np.sqrt(2*D)
+ z = 2*l*np.exp(-x)
+ N = np.sqrt(factorial(n)*(2*l-2*n-1)/factorial(2*l-n-1))
+ psi = N * z**(l-n-1/2) * np.exp(-z/2) * genlaguerre(n, 2*l-2*n-1)(z)
+
+ E.append(-1/2*(l-n-1/2)**2 + 12)
+ # print(E)
+ # print(En)
+
+ # plot
+
+ I = np.trapz(psi*psi, x)
+ # print(I)
+ plt.scatter(x, psi*psi+En[n], s=20, c='green', marker="x")
+
+
+
+ # plt.tight_layout()
+
+ numerical = mpatches.Patch(color='blue', label='numerical solution', linestyle='-')
+ analytical = mpatches.Patch(color='green', label='analytical solution', linestyle='-')
+ plt.legend(handles = [numerical, analytical], loc='upper left')
+ # plt.show()
+# plt.savefig('./plots/morse_potential.png')
+
+ return E, En
+
+def main():
+
+ xmin = -5; xmax = 5
+ s_h = []
+ s_b = []
+ s_m = []
+ meshfinesse = []
+
+ for n in range(20, 300, 20):
+ meshfinesse.append(n)
+ mesh = IntervalMesh(n, xmin, xmax)
+ V = FunctionSpace(mesh, 'CG', 1)
+ E_h_an, E_h_num = harmonic_oscillator(xmin, xmax, mesh, V)
+ E_b_an, E_b_num = box(xmin, xmax, mesh, V)
+ E_m_an, E_m_num = morse(xmin, xmax, mesh, V)
+
+ d_h = np.subtract(E_h_an, E_h_num)/E_h_an
+ s_h.append(np.average(np.abs(d_h)))
+
+ d_b = np.subtract(E_b_an, E_b_num)/E_b_an
+ s_b.append(np.average(np.abs(d_b)))
+
+ # print(E_m_an, E_m_num)
+ d_m = np.subtract(E_m_an, E_m_num)/E_m_an
+ s_m.append(np.average(np.abs(d_m)))
+
+ s_h = np.array(s_h)
+ s_b = np.array(s_b)
+ s_m = np.array(s_m)
+ print(s_h)
+ print(s_b)
+ print(s_m)
+
+ fig = plt.figure()
+ plt.plot(meshfinesse, 100*s_h, marker='x', c='green', label='harmonic oscillator')
+ plt.plot(meshfinesse, 100*s_b, marker='x', c='red', label='box potential')
+ plt.plot(meshfinesse, 100*s_m, marker='x', c='blue', label='morse potential')
+ plt.title('Mittlerer prozentualer Fehler der Eigenwerte', fontsize=20)
+ plt.ylabel('MAPE (%)')
+ plt.xlabel('number of intervalls (meshsize = 10)')
+# plt.xticks(np.arange(0, 1, 0.1))
+# plt.gca().invert_xaxis()
+# plt.xscale('log')
+
+ plt.legend()
+ plt.show()
+
+
+
+if __name__ == "__main__":
+ main()
diff --git a/presentation.pptx b/presentation.pptx
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diff --git a/slit_simulation/main.py b/slit_simulation/main.py
@@ -0,0 +1,75 @@
+#!/usr/bin/python3.9
+
+import numpy as np
+from scipy import sparse
+from scipy.sparse.linalg import spsolve
+
+import matplotlib.pyplot as plt
+import os
+
+from system import *
+from plotter import *
+
+import time
+
+"""
+ Solves the 2D time dependent Schrodinger Eq. on a square mesh.
+ Initial condition is a Gaussian wave packet.
+ The Potential is in form of the double/single slit
+
+ H * Psi = Psi_t
+
+ FD approximation of the Laplace Operator,
+ Crank Nicolson time discretization and
+ hbar = m = 1 makes the system equal to the following
+
+ (i - dt/2 * H) * Psi_{n+1} = (dt/2 * H + i) * Psi_n
+"""
+
+def main():
+ dx = 0.7; dt = 0.2
+ n = 100; steps = 100
+ xmin = 0; xmax = 10
+
+ x = np.linspace(xmin, xmax, n)
+ X, Y = np.meshgrid(x, x)
+
+ x0 = 8; y0 = 5
+ k0 = -50; V0 = 100
+
+ # Initial Condition
+ psi0 = wave_packet(X, Y, dx, x0, y0, k0)
+ psi0 = (psi0/norm(psi0, x)).T.reshape(n*n)
+
+ # Potential
+ pos = 5; thickness = 0.07
+ xs1 = 4.6; xs2 = 5.4;
+
+ h0 = indicator(V0, x, x, xmin, xs1, pos, pos+thickness)
+ h1 = indicator(V0, x, x, xs2, xmax, pos, pos+thickness)
+ h2 = indicator(V0, x, x, 4.8, 5.2, pos, pos+thickness)
+ V = h0 + h1 + h2
+
+ #spalt = 0.2
+ #for i in range(1, int((xs2-xs1)/spalt)):
+ # if i%3 != 0:
+ # V += indicator(V0, x, x, xs1+spalt*i, xs1+spalt*(i+1), pos, pos+thickness)
+
+
+ # Assemble system
+ A = left_side(n, V, dx, dt)
+ M = right_side(n, V, dx, dt)
+
+ start = time.time()
+ Psi = mysolver(A, M, psi0, dt, steps)
+ timing = time.time() - start
+
+ print(f'Time for the calculation: {round(timing, 2)}')
+
+ U = convert(Psi, x, x)
+
+ plotter(U, V, dt, X, Y, 'single_slit')
+
+if __name__ == "__main__":
+ main()
+
diff --git a/slit_simulation/plotter.py b/slit_simulation/plotter.py
@@ -0,0 +1,39 @@
+#!/usr/bin/python3.9
+
+import numpy as np
+import matplotlib.pyplot as plt
+import os
+
+
+def plotter(U, V, dt, xx, yy, name:str):
+
+ n = len(xx)
+ path = './.plotcache'
+ if os.path.exists(path) != True:
+ os.mkdir(path)
+
+ print('Making gif...')
+ for i in range(0, len(U)):
+ levels = np.linspace(0, max(U[i].reshape(n*n)), 200)
+ fig, ax = plt.subplots(figsize=[7,5])
+
+ c = ax.contourf(xx, yy, U[i], levels=levels, zorder=1,\
+ cmap=plt.cm.inferno)
+
+ ax.contour(xx, yy, V.reshape(n, n), extend='both',\
+ cmap=plt.cm.binary)
+
+ ax.set_title(f'Time {round(i*dt, 2)}')
+ fig.colorbar(c)
+
+ plt.savefig(f'{path}/img-{i}.png')
+ plt.close()
+
+ plt.plot(U[-1][15])
+ plt.savefig('./inteferenz.png')
+ plt.close()
+
+ os.system(f'ffmpeg -start_number 0 -i {path}/img-%d.png {name}.gif')
+ os.system(f'ffmpeg -start_number 0 -i {path}/img-%d.png {name}.mp4')
+ os.system('rm -rf ' + path)
+ os.system('rm -rf __pycache__')
diff --git a/slit_simulation/result/double_slit.gif b/slit_simulation/result/double_slit.gif
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diff --git a/slit_simulation/result/double_slit.mp4 b/slit_simulation/result/double_slit.mp4
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diff --git a/slit_simulation/result/gitter.gif b/slit_simulation/result/gitter.gif
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diff --git a/slit_simulation/result/gitter.mp4 b/slit_simulation/result/gitter.mp4
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diff --git a/slit_simulation/result/inteferenz_double.png b/slit_simulation/result/inteferenz_double.png
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diff --git a/slit_simulation/result/single_slit.gif b/slit_simulation/result/single_slit.gif
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diff --git a/slit_simulation/result/single_slit.mp4 b/slit_simulation/result/single_slit.mp4
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diff --git a/slit_simulation/system.py b/slit_simulation/system.py
@@ -0,0 +1,80 @@
+#!/usr/bin/python3.9
+
+import numpy as np
+from scipy import sparse
+from scipy.sparse.linalg import spsolve
+from scipy.sparse.linalg import inv
+
+import time
+
+def indicator(V0, x, y, x0, x1, y0, y1):
+ n = len(x)
+ h = np.zeros(n*n)
+ for i in range(n):
+ for j in range(n):
+ if x[i] >= x0 and x[i] <= x1:
+ if y[j] >= y0 and y[j] <= y1:
+ h[i + j*n] = 1
+
+ return V0*h
+
+def hamiltonian(n, V, dx):
+ dim, n = n, n*n
+
+ d0 = -2*(1/dx**2 + 1/dx**2) * np.ones(n) + V
+ d1 = np.array([1/dx**2 if i%dim != 0 else 0 for i in range(1, n)])
+ dn = 1/dx**2 * np.ones(n-dim)
+
+ return d0, d1, dn
+
+def left_side(n, V, dx, dt):
+ dim, n = n, n*n
+
+ d0, d1, dn = hamiltonian(dim, V, dx)
+
+ d_0 = -dt/2 * d0 + 1j
+ d_1 = -dt/2 * d1
+ d_n = -dt/2 * dn
+
+ return sparse.diags([d_n, d_1, d_0, d_1, d_n], [-dim, -1, 0, 1, dim], format='csc')
+
+
+def right_side(n, V, dx, dt):
+ dim, n = n, n*n
+
+ d0, d1, dn = hamiltonian(dim, V, dx)
+
+ d_0 = dt/2 * d0 + 1j
+ d_1 = dt/2 * d1
+ d_n = dt/2 * dn
+
+ return sparse.diags([d_n, d_1, d_0, d_1, d_n], [-dim, -1, 0, 1, dim], format='csc')
+
+
+def wave_packet(x, y, dx, x0, y0, k0):
+ return 1/(2*dx**2*np.pi)**(1/2) *\
+ np.exp(-((x-x0)/(2*dx)) ** 2) *\
+ np.exp(-((y-y0)/(2*dx)) ** 2) *\
+ np.exp(1.j * (k0*x))
+
+def mysolver(A, M, u0, dt, steps):
+ U = np.array([u0] + [np.zeros(u0.size) for i in range(steps)])
+ for i in range(steps):
+ U[i+1] = spsolve(A, M.dot(U[i]))
+ return U
+
+def norm(u, x):
+ n = len(x)
+ u = np.sqrt(np.real(u)**2 + np.imag(u)**2)**2
+ u = u.reshape(n, n)
+ return np.trapz(np.trapz(u, x), x)
+
+def convert(U, x, y):
+ """normalization of each solution"""
+ n = len(x)
+ Unew = []
+ for u in U:
+ unew = abs(u)**2/norm(u, x)
+ Unew.append(unew.reshape(n, n))
+ return Unew
+
