Here, I reproduce results of a recent arXiv preprint: Enrique G. Neyra, Marcelo F. Ciappina, Lorena Rebón, “Free temporal evolution of a superbandwidth wave packet: The quantum tale of the hare and the tortoise”, https://doi.org/10.48550/arXiv.2501.06001. The paper explores the free evolution of superbandwidth wave packets in quantum mechanics, drawing parallels with laser pulse propagation in dispersive media. It reveals that such wave packets can exhibit super- and sub-oscillatory local momenta and achieve spatial localization during free propagation, linked to the momentum distribution. Fore more detailed discussion on superoscillations, the reader can refer to this series of papers: Michael Berry et al 2019 J. Opt. 21 053002
The initial wave function in momentum space:
In[]:=
ϕ0=Exp-
2
k-k0
Δk
-αExp-
2
k-k0
Δk/2

Out[]=
-
2
(k-k0)
2
Δk

-
-
4
2
(k-k0)
2
Δk

α
Find the normalization constant (ϕ/ as a normalized function)
In[]:=
=Sqrt@Integrate[ϕ0^2,{k,-∞,∞},Assumptions->{Δk>0}]//PowerExpand//FullSimplify
Out[]=
1/4
π
2-4
2
5
α+
2
α
Δk
3/4
2
Find the initial wave function in position space:
In[]:=
ψ0=FourierTransform[ϕ0,k,x]//PowerExpand//FullSimplify
Out[]=
-
k0x-
2
x
2
Δk
4

-2+
3
2
x
2
Δk
16

αΔk
2
2
Solve the time-dependent Schrodinger equation with a given initial state:
In[]:=
DSolveValue[{D[u[x,t],t]==-1/2D[u[x,t],{x,2}],u[x,0]==ψ0},u[x,t],{x,t},Assumptions->{k0∈Reals,t>=0,Δk>0}]
Out[]=
2
-2
2
k0
t+4k0x+
2
x
2
Δk
-4+2t
2
Δk

πΔk
2+t
2
Δk
-
2
8
2
k0
t-16k0x-
2
x
2
Δk
16-2t
2
Δk

παΔk
8+t
2
Δk
2π
Define the time-dependent wave function:
In[]:=
ClearAll[ψ]​​ψ[x_,t_]:=
2
-2
2
k0
t+4k0x+
2
x
2
Δk
-4+2t
2
Δk

πΔk
2+t
2
Δk
-
2
8
2
k0
t-16k0x-
2
x
2
Δk
16-2t
2
Δk

παΔk
8+t
2
Δk
2π

Check the wave function satisfies Schrodinger equation:
In[]:=
D[ψ[x,t],t]==-
1
2
D[ψ[x,t],{x,2}]//FullSimplify
Out[]=
True
Calculate gradient of arg of wave function:
In[]:=
D[ψ[x,t],x]/ψ[x,t]//FullSimplify
Out[]=
-
2
-2
2
k0
t+4k0x+
2
x
2
Δk
-4+2t
2
Δk

πΔk(-2k0+x
2
Δk
)
3/2
(2+t
2
Δk
)
-
2
8
2
k0
t-16k0x-
2
x
2
Δk
16-2t
2
Δk

παΔk(-8k0+x
2
Δk
)
8+t
2
Δk
(-8+t
2
Δk
)
2
-2
2
k0
t+4k0x+
2
x
2
Δk
-4+2t
2
Δk

πΔk
2+t
2
Δk
-
2
8
2
k0
t-16k0x-
2
x
2
Δk
16-2t
2
Δk

παΔk
8+t
2
Δk
Define a function as the gradient of arg of wave function:
In[]:=
gradArg[x_,t_]:=
-
2
-2
2
k0
t+4k0x+
2
x
2
Δk
-4+2t
2
Δk

πΔk(-2k0+x
2
Δk
)
3/2
(2+t
2
Δk
)
-
2
8
2
k0
t-16k0x-
2
x
2
Δk
16-2t
2
Δk

παΔk(-8k0+x
2
Δk
)
8+t
2
Δk
(-8+t
2
Δk
)
2
-2
2
k0
t+4k0x+
2
x
2
Δk
-4+2t
2
Δk

πΔk
2+t
2
Δk
-
2
8
2
k0
t-16k0x-
2
x
2
Δk
16-2t
2
Δk

παΔk
8+t
2
Δk

α=1vsα=0

Plot the momentum distribution:
In[]:=
With[{var={k0->2π,α->1,Δk->.5}},​​Plot[(ϕ0//.k->k0kk/.var)^2,{kk,0.5,1.5},PlotRange->All,AxesLabel->{"k/
k
0
","|
ϕ
0
|^2"},GridLines->{{1+3Δk/k0/.var,1-3Δk/k0/.var},None},GridLinesStyleDirective[Red,Thin],PlotLabel->"Momentum distribution for α=1,
k
0
=2π and Δk=0.5"]]
Out[]=
Plot the position distribution and the normalized local momentum:
In[]:=
With{var={k0->2π,α->1,Δk->.5},var0={k0->2π,α->0,Δk->.5}},​​ManipulatePlotAbs[ψ[x,t]/.var],Abs[ψ[x,t]/.var0],Im
1
k0
gradArg[x,t]/.var,1+3Δkk0/.var,1-3Δkk0/.var,Im
1
k0
gradArg[x,t]/.var0,{x,-20,50},PlotRange->All,Filling->{1->0,3->1},PlotLegends->{"|ψ
2
|
","|
ψ
α=0
2
|
","local phase momemtum/
k
0
","bandwidth of momentum distribution",None,"local phase momemtum/
k
0
when α=0"},PlotStyle->{Automatic,Dashed,Automatic,Directive[Red,Thin],Directive[Red,Thin],Directive[Black,Dashed]},PlotLabel->"Position distriubution and local momentum",{t,Range[4],SetterBar},SaveDefinitions->True
Out[]=
​
t
1
2
3
4
|ψ
2
|
|
ψ
α=0
2
|
local phase momemtum/
k
0
bandwidth of momentum distribution
local phase momemtum/
k
0
when α=0
In[]:=
With{var={k0->2π,α->1,Δk->.5},t1=2.8,t2=3,t3=3.2},​​PlotAbs[ψ[x,t1]/.var],Abs[ψ[x,t2]/.var],Abs[ψ[x,t3]/.var],Im
1
k0
gradArg[x,t1]/.var,Im
1
k0
gradArg[x,t2]/.var,Im
1
k0
gradArg[x,t3]/.var,{x,0,40},PlotRange->All,PlotStyle->{Blue,Black,Red,Directive[Blue,Thickness[.003]],Directive[Black,Thickness[.003]],Directive[Red,Thickness[.003]]},PlotLegends->{"|
ψ
t=2.8
2
|
","|
ψ
t=3.0
2
|
","|
ψ
t=3.2
2
|
","local phase momemtum/
k
0
at t=2.8","local phase momemtum/
k
0
at t=3.0","local phase momemtum/
k
0
at t=3.2"},PlotLabel->"Position distriubution and local momentum"
Out[]=
|
ψ
t=2.8
2
|
|
ψ
t=3.0
2
|
|
ψ
t=3.2
2
|
local phase momemtum/
k
0
at t=2.8
local phase momemtum/
k
0
at t=3.0
local phase momemtum/
k
0
at t=3.2
In[]:=
With[{var={k0->2π,α->1,Δk->.5},xPL=15.0,xPR=22.9},​​Plot[{Abs[ψ[xPL,t]]^2Im[gradArg[xPL,t]]/.var,Abs[ψ[xPR,t]]^2Im[gradArg[xPR,t]]/.var},{t,2,4},PlotRange->{-.2,All},PlotLabel->"local momemtun current density",AxesLabel->{"t","J(x,t)"},PlotLegends->{"J(15.0,t)","J(22.9,t)"}]]
Out[]=
J(15.0,t)
J(22.9,t)

Madelung trajectories

In[]:=
pnds=ParametricNDSolveValue[{x'[t]==Im[gradArg[x[t],t]],x[0]==x0},x,{t,0,f},{x0,f,k0,α,Δk}]
Visualize the position density:
Find Madelung trajectories given some initial conditions and parameters:
Visualize them:
Plot the momentum distribution:
Plot the position distribution and the normalized local momentum:

Madelung trajectories

Visualize the position density:
Find Madelung trajectories given some initial conditions and parameters:
Visualize them:

CITE THIS NOTEBOOK

Quantum tale of the hare and tortoise​
by Mohammad Bahrami​
Wolfram Community, STAFF PICKS, January 17, 2025
​https://community.wolfram.com/groups/-/m/t/3358663