Time dependent QM: Two dimensional tunneling
Classically, if the kinetic energy Ek of a particle is less than the height V0 of the barrier, the particle bounces back. In quantum mechanics the particle can enter the forbidden region.

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

Rectangular potential barrier width: 3 Bohr
Rectangular potential barrier height: 0.46 Hartree

Description of phenomena

When the 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 wave packet splits into two. The reflected wave packet is returning to the left. The second one, the transmitted wave packet of probability propagates towards the right, demonstrating that there is a finite chance for the particle to tunnel through the classically forbidden region.

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 256x200 size (for fast network links) and at reduced 64x50 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, 320x200MPEG (310k), 160x120XING MPEG (231k).
...Abs2[ Psi[ x, t ] ] animation, 320x200 MPEG (265k), 160x120 XING MPEG (233k).

Click here to download the PC animation (FLI format, compressed with ZIP, 1.5M) !
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