Time - Dependent Wave Functions

Particle in an Inpenetrable Box (0<x<1)

(*Normalized energy eigenstates*)
phi[n_,x_]=Sqrt[2]Sin[n Pi x]
Plot[{Abs[phi[0,x]]^2,Abs[phi[1,x]]^2,Abs[phi[2,x]]^2},{x,0,1}]

(*State 1 at t=0 describes a function that is flat in the interal [0.4,0.6] and zero elsewhere *)
psi1[x_]=1/Sqrt[0.2]HeavisideTheta[x-0.4] HeavisideTheta[0.6-x];
Plot[psi1[x],{x,0,1}]

(*State 2 at t=0 describes a function that is flat in the interal [0,0.2] and zero elsewhere *)
psi2[x_]=1/Sqrt[0.2]HeavisideTheta[x] HeavisideTheta[0.2-x];
Plot[psi2[x],{x,0,1}]

(*State 3 at t=0 describes a function that is flat in the interal [0,0.1] and zero elsewhere *)
psi3[x_]=1/Sqrt[0.1]HeavisideTheta[x] HeavisideTheta[0.1-x];
Plot[psi3[x],{x,0,1}]

(*Projection of State 1 on energy eigenstate n*)
c1[n_]=Integrate[phi[n,x]/Sqrt[0.2],{x,0.4,0.6}]

(*Projection of State 2 on energy eigenstate n*)
c2[n_]=Integrate[phi[n,x]/Sqrt[0.2],{x,0,0.2}]

(*Projection of State 2 on energy eigenstate n*)
c3[n_]=Integrate[phi[n,x]/Sqrt[0.1],{x,0,0.1}]

(*State 1 at t=0 as series of n states, but series terminated at m-th term*)
psi1[m_,x_]:=Sum[c1[n]phi[n,x],{n,1,m}]

(*Time-dependent state 1 as series of n states, but series terminated at m-th term*)
psi1[m_,x_,t_]:=Sum[Exp[I t n^2]c1[n]phi[n,x],{n,1,m}]

(*State 2 at t=0 as series of n states, but series terminated at m-th term*)
psi2[m_,x_]:=Sum[c2[n]phi[n,x],{n,1,m}]

(*Time-dependent state 2 as series of n states, but series terminated at m-th term*)
psi2[m_,x_,t_]:=Sum[Exp[I t n^2]c2[n]phi[n,x],{n,1,m}]

(*State 3 at t=0 as series of n states, but series terminated at m-th term*)
psi3[m_,x_]:=Sum[c3[n]phi[n,x],{n,1,m}]

(*Time-dependent state 2 as series of n states, but series terminated at m-th term*)
psi3[m_,x_,t_]:=Sum[Exp[I t n^2]c3[n]phi[n,x],{n,1,m}]

(*Pictures for time-dependence of the modulus squared of state 1*)
tt1=Table[Plot[Abs[psi1[10,x,t]]^2,{x,0,1},PlotRange->{0,8}],{t,0,0.79,0.01}];

(*Pictures for time-dependence of the modulus squared of state 2*)
tt2=Table[Plot[Abs[psi2[10,x,t]]^2,{x,0,1},PlotRange->{0,8}],{t,0,6.3,0.025}];

(*Pictures for time-dependence of the modulus squared of state 2*)
tt3=Table[Plot[Abs[psi3[10,x,t]]^2,{x,0,1},PlotRange->{0,8}],{t,0,6.3,0.025}];

(*Pictures for time-dependence of the modulus squared of state 2*)
tt3=Table[Plot[Abs[psi3[10,x,t]]^2,{x,0,1},PlotRange->{0,8}],{t,0,6.3,0.025}];

ListAnimate[tt1,10]

ListAnimate[tt2,10]

ListAnimate[tt3,10]

Gaussian Wave Packet

We take the result for a coherent state from the lecture notes Chap 3.2 and plot a movie for its time dependence.  (delta=0,omega=1)

PsiZsquaredGaus[x_,t_,v0_,sigma_,deltav_]=
1/(2Pi)^(1/2) Exp[-(x-v0 t)^2/(sigma^2+(deltav t )^2)]/Sqrt[sigma^2+(deltav t )^2]

(*Pictures for time-dependence of the modulus squared of state 1*)
ttgaus=Table[Plot[PsiZsquaredGaus[x,t,0.5,0.1,0.02],{x,0,7},PlotRange->{0,5},
AspectRatio->1/6,ImageSize->700],{t,-1,14,.25}];

ListAnimate[ttgaus,10,ImageSize->700]

Coherent State for Harmonic Oszillator

We take the result for a coherent state from the lecture notes Chap 3.2 and plot a movie for its time dependence.  (delta=0,omega=1)

PsiZsquared[x_,t_,x0_,xtilde_]=1/((Pi)^(1/2)xtilde) Exp[-(x-x0 Cos[t])^2/xtilde^2]

(*Pictures for time-dependence of the modulus squared of state 1*)
tt3=Table[Plot[PsiZsquared[x,t,1,0.4],{x,-2,2},PlotRange->{0,Automatic}],{t,0,2Pi-0.1,.1}];

ListAnimate[tt3,10]

Scattering of Wave-Packet on a Potential Step

We have V (x) = V0 HeavisideTheta[x].  (Potential zero for negative x and V0 for positive x). [ We set hbar = 1, so p = k (k, wave number) and  2 m = 1, V0 = 1. ]

Energy eigenstates of FREE incoming waves (from negative x) with positive momenta p=k, i.e. V(x)=0:  (k must be always smaller than 1 !!)

Phifree[x_,k_]=(Exp[I k x]);

Energy eigenstates of incoming waves (from negative x) with positive momenta p=k, having E < V0:  (k must be always smaller than 1 !!)

Phibelow[x_,k_]=(Exp[I k x] + (k-I Sqrt[1-k^2])/(k+I Sqrt[1-k^2])Exp[-I k x] )HeavisideTheta[-x]+
(2k)/(k+I Sqrt[1-k^2])Exp[-Sqrt[1-k^2]x]HeavisideTheta[x];

Phibelow[x_,k_]=(Exp[I k x]);

Energy eigenstates of incoming waves (from negative x) with positive momenta p=k, having E > V0:  (k must be always larger than 1 !!)

Phiabove[x_,k_]=(Exp[I k x] + (k-Sqrt[k^2-1])/(k+Sqrt[k^2-1])Exp[-I k x] )HeavisideTheta[-x]+
(2k)/(k+ Sqrt[k^2-1])Exp[I Sqrt[k^2-1]x]HeavisideTheta[x];

Phiabove[x_,k_]=(Exp[I k x]);

Wave packet in mom space:  (Gaussian for simplicity - we only integrate it in an interval [-2 deltak, 2 deltak])

g[k_,deltak_]=1/((Pi)^(1/2)deltak) Exp[-k^2/deltak^2];

Time-dependent configuration space wave paket:  ( It starts at x = - 120 at t = 0 )

Psibelow[x_,t_,k0_,deltak_]:=NIntegrate[ g[k-k0,deltak]Exp[I k 120-I k^2 t]Phibelow[x,k],{k,k0-2deltak,k0+2deltak},PrecisionGoal->3]

Time-dependent configuration space wave paket:  ( It starts at x = - 120 at t = 0 )

Psiabove[x_,t_,k0_,deltak_]:=NIntegrate[ g[k-k0,deltak]Exp[I k 100-I k^2 t]Phiabove[x,k],{k,k0-2deltak,k0+2deltak},PrecisionGoal->3]

(*Pictures for time-dependence of the modulus squared of state 1*)
tt4=Table[Plot[  Abs[Psibelow[x,t,0.5,0.1]]^2,{x,-150,50},PlotRange->{0,1.2},Epilog->{Line[{{-Abs[-120+t],0},{-Abs[-120+t],1.2}}]}],{t,0,240,3}];

ListAnimate[tt4,4]

(*Pictures for time-dependence of the modulus squared of state 1*)
tt5=Table[Plot[  Abs[Psiabove[x,t,1.3,0.1]]^2,{x,-150,150},PlotRange->{0,1.2},
Epilog->{Line[{{-100+2 1.3 t,0},{-100+2 1.3 t,1.1}}],Line[{{-Abs[-100+2 1.3 t],0},{-Abs[-100+2 1.3 t],1.1}}]}],{t,0,100,1}];

ListAnimate[tt5,5]