How Does a Gaussian Wave Packet Spread Over Time?

[Shackleford]

We finally started modern quantum mechanics. I'm definitely feeling a bit lost or fuzzy, especially in the more rigorous book Quantum Physics by Gasiorowicz. At any rate, we have two more problem sets before the last exam.

As for the problem, I'm not sure what to. Is it asking to calculate the fractional change in size of a wave packet when it spreads to 10^-10 m from 10^-16 m? I guess I could also call the given dimensions L, such as in delta-x = L/2 or something. Sorry for the poor scan.

 

[fantispug]

Well to start with what is the wavefunction (as a function of space and time) of a Gaussian representing a free particle.

What could we mean by the 'size' of the wavepacket?

(I think the question is asking you to calculate the fractional change in size of the wavepacket over a time period of one second for the case when its initial size is 10^-10, and then separately calculate it for the case when its initial size is 10^-16)

 

[Shackleford]

Well to start with what is the wavefunction (as a function of space and time) of a Gaussian representing a free particle.

What could we mean by the 'size' of the wavepacket?

(I think the question is asking you to calculate the fractional change in size of the wavepacket over a time period of one second for the case when its initial size is 10^-10, and then separately calculate it for the case when its initial size is 10^-16)

The Gaussian wave function is (sorry I still haven't learned latex yet)

psi-(x,t) = integral from negative infinity to positive infinity of dk A(k)e^(kx-wt)

From remembering what's in my Modern Physics textbook, the size of the wave packet could be half of the wavelength. And there's a relation I think that applies to all wave functions generally: delta-k * delta-x = 1/2