# uncomment ONE line to choose matplotlib backend # if using Jupyter Notebook, use interactive "notebook" backend for best results # if using Jupyter Lab, use interactive "widget" backend for best results # if both fail, use static "inline" backend %matplotlib notebook #%matplotlib widget %matplotlib inline import numpy as np import matplotlib.pyplot as plt import scipy.linalg as linalg

h_bar = 1.05e-34 m = 9.11e-31 #elektronmasse [kg]

start = -1 stop = 1 L = stop - start N = 1000 x, dx = np.linspace(start, stop, N+1, retstep = True) n = [1, 2, 3] '''FUNKSJON SOM RETURNERER EGENVERDIER E OG TILHØRENDE, NORMERTE EGENVERDIFUNKSJONER PSI''' def calc_E_and_Psi(V): N_tot = len(V) d = [v + h_bar**2/(m*dx**2) for v in V] e = [-h_bar**2/(2*m*dx**2) for n in range(N_tot-1)] E,Psi = linalg.eigh_tridiagonal(d,e) return E, Psi

'''FUNKSJON SOM FRAMSTILLER POTENSIALET V, ENERGIEGENVERDIER E OG ENERGIEGENFUNKSJONER PSI''' def plot_psi(analytic, numeric, analytic_2, numeric_2, x_list, start, stop, xlim): ''' Denne funksjonen tar inn liste av lister med analytiske og numeriske verdier. Hvert element i listene er en liste av psi for en bestemt tilstand nr. n. ''' fig = plt.figure(figsize=(15,10)) for n in range(len(analytic)): ax1 = plt.subplot("221") ax1.plot(x_list, analytic[n][0], label="Ψ "+str(n+1)) ax1.set_xlabel("L", fontsize=13) ax1.set_ylabel('Ψ', fontsize=13) ax1.set_title("Analytisk", fontsize=20) ax1.grid() ax1.set_xlim(xlim) plt.legend() ax2 = plt.subplot("222") ax2.plot(x_list, numeric[n][0], label="Ψ" +str(n+1)) ax2.set_xlabel("L", fontsize=13) ax2.set_ylabel('Ψ', fontsize=13) ax2.set_title("Numerisk", fontsize=20) ax2.grid() ax2.set_xlim(xlim) plt.legend() ax3 = plt.subplot("223") ax3.plot(x_list, analytic_2[n][0], label="Ψ^2 "+str(n+1)) ax3.set_xlabel("L", fontsize=13) ax3.set_ylabel('Ψ^2', fontsize=13) ax3.set_title("Analytisk", fontsize=20) ax3.grid() ax3.set_xlim(xlim) plt.legend() ax4 = plt.subplot("224") ax4.plot(x_list, numeric_2[n][0], label="Ψ^2 "+str(n+1)) ax4.set_xlabel("L", fontsize=13) ax4.set_ylabel('Ψ^2', fontsize=13) ax4.set_title("Numerisk", fontsize=20) ax4.grid() ax4.set_xlim(xlim) plt.legend() plt.tight_layout(rect=[0, 0.03, 1, 0.95]) def plot_V_E(V_list, E_list_analytic, E_list_numeric, x_list, start, stop): ''' Denne funksjonen tar inn liste for potensialet og energiegenverdiene ''' #potensial plt.figure(1,figsize=(10,5)) plt.plot(x_list, V_list, label = 'Potential' ) plt.xlabel("L", fontsize=13) plt.ylabel('V', fontsize=13) plt.xticks(np.arange(start, stop, step=stop/10)) plt.title("Potensial", fontsize=16) plt.grid() # analytic solutions plt.figure(2, figsize=(10,5)) for i in range(len(n)): plt.plot(x_list, E_list_analytic[i][0], label = 'Eigenvalue ' + str(i+1)) plt.xlabel("L", fontsize=13) plt.ylabel('V, E', fontsize=13) plt.xticks(np.arange(start, stop, step=stop/10)) plt.title("Analytisk løsning", fontsize=16) plt.legend() plt.grid() plt.tight_layout(rect=[0, 0.03, 1, 0.95]) #numeric solutions plt.figure(3, figsize=(10,5)) for i in range(len(n)): plt.plot(x_list, E_list_numeric[i][0], label = 'Eigenvalue ' + str(i+1)) plt.xlabel("L", fontsize=13) plt.ylabel('V, E', fontsize=13) plt.xticks(np.arange(start, stop, step=stop/10)) plt.title("Numerisk løsning", fontsize=16) plt.legend() plt.grid() plt.tight_layout(rect=[0, 0.03, 1, 0.95])

'''POTENSIALET''' V_infwell = [0]*len(x) '''ANALYTISK LØSNING''' def psi_analytic(x,n): return np.sqrt(2/L)*np.sin(n*np.pi*(x+1)/L) def E_analytic(x_vals,n): E_an = np.zeros(len(x_vals)) for i in range(len(x_vals)): E_an[i] = n**2*np.pi**2*h_bar**2/(2*m*L**2) return E_an '''NUMERISK LØSNING''' E_num_infwell_calc, Psi_num_infwell_calc = calc_E_and_Psi(V_infwell) '''LISTENE SOM SKAL PLOTTES''' E_an_infwell = [[] for _ in range(len(n))] Psi_an_infwell = [[] for _ in range(len(n))] E_num_infwell = [[] for _ in range(len(n))] Psi_num_infwell = [[] for _ in range(len(n))] for i in range(len(n)): E_an_infwell[i].append(E_analytic(x, n[i])) Psi_an_infwell[i].append(psi_analytic(x, n[i])) Psi_num_infwell[i].append(np.array([0] + Psi_num_infwell_calc[:, n[i]-1] + [0])) E_num_infwell[i] = np.full((1, len(x)), E_num_infwell_calc[n[i]-1]) xlim_infwell = (-1,1) '''PLOTTER RESULTATENE''' print(plot_V_E(V_infwell, np.array(E_an_infwell), np.array(E_num_infwell), x, start, stop)) print(plot_psi(np.array(Psi_an_infwell), np.array(Psi_num_infwell), np.abs(np.array(Psi_an_infwell))**2, np.abs(np.array(Psi_num_infwell))**2, x, start, stop, xlim_infwell))

from scipy import special ''' gammel kode ''' #w = 1 #V0=1.6E-19 #N = 100 #start_h = -0.1 #stop_h = 0.1 ''' ny kode ''' w = 1 N = 1000 start_h = -1 stop_h = 1 '''Diskretiserer potensialet for harmonisk ocsillator, gammel kode''' #V_harmosc = [V0]*4*N + [V0*((n-N)/(N*1.0))**2 for n in range(2*N+1)] + [V0]*4*N #x_harmosc = np.linspace(start_h, stop_h, len(V_harmosc)) ''' Diskretiserer potensialet, ny kode ''' # definerer x-akse for hele plottet (konstant + varierende potensial) x_harmosc, dx = np.linspace(start_h, stop_h, N*3 , retstep = True) ''' Potensialet der det ikke er konstant''' def V_middle_calc(m, w, x_list): V = [0]*len(x_list) for i in range(len(x_list)): V[i] = 1/2 * m * w**2 * x_list[i]**2 return V # potensialet i midten skal variere etter HO formel V_middle = V_middle_calc(m, w, x_harmosc[N: 2* N]) # konstant potensial før og etter det varierende V_const = [V_middle[0]]*N # Samler de til en array V_harmosc = np.array(V_const + V_middle + V_const) '''Analytisk løsning''' def psi_harmosc_an(x,n): '''x: en array, n: et heltall. returnerer en array med psi-verdier for denne n-verdien''' psi = np.array(1/(np.sqrt((2**n)*np.math.factorial(n)))*(m*w/(np.pi*h_bar))**(1/4)*np.exp(-m*w*x**2/(2*h_bar))*special.hermite(n)((x*(m*w/h_bar)**(1/2)))) return psi def E_harmosc_an(x, n,w): E_an = np.zeros(len(x)) for i in range(len(x)): E_an[i] = np.array((n+1/2)*h_bar*w) return E_an '''Numerisk løsning''' d = [v + h_bar**2/(m*dx**2) for v in V_harmosc] e = [-h_bar**2/(2*m*dx**2) for n in range(len(V_harmosc)-1)] E,Psi = linalg.eigh_tridiagonal(d,e) '''LISTENE SOM SKAL PLOTTES''' E_an_harmosc = [[] for _ in range(len(n))] Psi_an_harmosc = [[] for _ in range(len(n))] E_num_harmosc = [[] for _ in range(len(n))] Psi_num_harmosc = [[] for _ in range(len(n))] for i in range(len(n)): E_an_harmosc[i].append(E_harmosc_an(x_harmosc, n[i] - 1, w)) Psi_an_harmosc[i].append(psi_harmosc_an(x_harmosc, n[i] - 1)) E_num_harmosc[i] = np.full((1, len(x_harmosc)), E[n[i] - 1]) Psi_num_harmosc[i].append(Psi[:, n[i] - 1]) xlim_harmosc = (-0.1, 0.1) '''PLOTTER RESULTATENE''' print(plot_V_E(V_harmosc, np.array(E_an_harmosc), np.array(E_num_harmosc), x_harmosc, start_h, stop_h)) print(plot_psi(np.array(Psi_an_harmosc), np.array(Psi_num_harmosc), np.abs(np.array(Psi_an_harmosc))**2, np.abs(np.array(Psi_num_harmosc))**2, x_harmosc, start_h, stop_h, xlim_harmosc))

V0=4.0*1.6E-19 dx=0.1E-10 N = 100 '''Diskretiserer potensialet for endelig brønn''' V = np.asarray([V0]*10*N + [0]*N + [V0]*10*N) x_finwell = np.linspace(start_h, stop_h, len(V))*1.0e9 '''Numerisk løsning''' d = [v + h_bar**2/(m*dx**2) for v in V] e = [-h_bar**2/(2*m*dx**2) for n in range(len(V)-1)] E,Psi = linalg.eigh_tridiagonal(d,e) '''LISTENE SOM SKAL PLOTTES''' E_num_finwell = [[] for _ in range(len(n))] Psi_num_finwell = [[] for _ in range(len(n))] for i in range(len(n)): E_num_finwell[i] = np.full((1, len(x_finwell)), E[n[i]]) Psi_num_finwell[i].append(Psi[:, n[i]]) '''PLOTTING''' plt.figure(1, (15,5)) for i in range(len(n)): plt.plot(x_finwell, E_num_finwell[i][0]) plt.title('Energiverdier',fontsize=20) plt.xlabel('$x$ (nm)',fontsize=20) plt.ylabel('$E(x)$',fontsize=20) plt.grid() plt.show() plt.figure(2,figsize=(15,5)) plt.plot(x_finwell, Psi[:,0],x_finwell,Psi[:,1],x_finwell,Psi[:,2],x_finwell,Psi[:,3]) plt.title('Bundne tilstander',fontsize=20) plt.xlabel('$x$ (nm)',fontsize=20) plt.ylabel('$\psi(x)$',fontsize=20) plt.grid() plt.show() plt.figure(3,figsize=(15,5)) plt.plot(x_finwell, Psi[:,15],x_finwell,Psi[:,16],x_finwell,Psi[:,17],x_finwell,Psi[:,18]) plt.title('Ubundne tilstander',fontsize=20) plt.xlabel('$x$ (nm)',fontsize=20) plt.ylabel('$\psi(x)$',fontsize=20) plt.grid() plt.show()