Increase of Uncertainty over Time

[Hornbein]

Looking at the literature it appears that the increase in the variance of the wave function over time is proportional to (1+t^2)1/2. Is that right?

 

[blue_leaf77]

Are you talking about certain form of Gaussian wavepacket?

 

[Hornbein]

Are you talking about certain form of Gaussian wavepacket?

I'm assuming it is a plain vanilla Gaussian, yes.

 

[jtbell]

it appears that the increase in the variance of the wave function over time is proportional to (1+t^2)1/2. Is that right?

More or less. You need another constant in there.

http://quantummechanics.ucsd.edu/ph130a/130_notes/node83.html

(the first result in a Google search for "gaussian wave packet spreading")

 

[blue_leaf77]

I'm assuming it is a plain vanilla Gaussian, yes.

The exact form involves some more constants as is pointed out by jtbell above.
Not all wavepacket evolves in time that way, the temporal behavior depends on the form of the wavepacket itself and on whether the packet is in free space or not. If the packet propagates in a free-potential space, only the position basis wavepacket evolves - the momentum basis wavefunction remains unchanged. If, on the other hand, there is a varying potential, the wavepacket experiences a "force" and consequently the momentum basis wavefunction will also undergo a change in time in addition to the position wavepacket.

 

[Hornbein]

The exact form involves some more constants as is pointed out by jtbell above.
Not all wavepacket evolves in time that way, the temporal behavior depends on the form of the wavepacket itself and on whether the packet is in free space or not. If the packet propagates in a free-potential space, only the position basis wavepacket evolves - the momentum basis wavefunction remains unchanged. If, on the other hand, there is a varying potential, the wavepacket experiences a "force" and consequently the momentum basis wavefunction will also undergo a change in time in addition to the position wavepacket.

Thanks, that's very helpful for a qualitative understanding.