Time dependent QM: Two dimensional

    hard core potential
A hard core potential is such that the quantum particle can't penetrate into it, i.e. its probabiltity density is zero under the region of the potential. Mathematically this can be achieved by an infinite high potential barrier.

Initital conditions

The initial wave function is a two dimensional Gauusian wave packet.
initial position { x0, y0 } : { -30, -30 } Bohr
initial impulse { px0, py0 } : { 0.6349, 0.6349 } Bohr-1
DeltaX = 3.53553391 Bohr

Calculated properties of the initial wave packet:
p = Sqrt[ px02 + py02 ] = 0.8979 Bohr-1
vGroup = p = 0.8979 Bohr-1
DeltaP = 0.141421356 Bohr-1
EKin = 0.846245263 Hartree
T2 = 21.6506351 au[time]

Where:
DeltaX = Sqrt[ < x2 > - < x >2 ]
DeltaP = Sqrt[ < p2 > - < p >2 ]
EKin = < p2 > / ( 2 m )
T2 = the time while DeltaX doubles

The potential

Radial hard core potential radius: 8 Bohr
Radial hard core potential height: Infinity

Description of phenomena

When the incoming wave packet is moving toward the potential barrier in the constant (zero) potential region its shape does not change, except it is spreading out. When the leading edge of the wave packet reaches the potential barrier oscillations occur in the wave packet caused by the interference between the incident and reflected waves. After a transitory period the outgoing wave packet emerges. The outgoing state is a superposition of the incident packet with a radial symmetric wave front.

Details of the calculation

Size of the calculation box: 256 x 256 Bohr (512x512 points)
Size of the presentation box: 128 x 128 Bohr (256x256 points)
Calculation time step: 0.3 au[time]
Presentation time step: 1.2 au[time]
Total calculation time: 150 au[time]

Images

The Re[ Psi[ x, t ] ] and Abs2[ Psi[ x, t ] ] phase images are available at full 256x256 size (for fast network links) and at reduced 64x64 size (for slow network links).
Click here to see the full size images, or
click here to see the reduced size images!

Animation

...Re[ Psi[ x, t ] ] animation, 256x256 MPEG (202k).
...Abs2[ Psi[ x, t ] ] animation, 256x256 MPEG (158k).

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