Wave Problem? Solve w/Typical Values: T=1500K, m=7amu, d=3mm, L=2m

[GreenLRan]

wave problem??

Homework Statement

An apparatus is used to prepare an atomic beam by heating a collection of atoms to a temperature T and allowing the beam to emerge through a hole of diameter d in one side of the oven. The beam then travels through a straight path of length L. Show that the diamter of the beam at the end of the path is larger than d by an amount of order L*hbar/(d*sqrt(3mkT)) where m is the mass of an atom. Make a numerical estimate for typical values of T=1500K m=7amu(lithium atoms), d=3mm, L=2m.

Homework Equations

no clue

The Attempt at a Solution

I calculated the ratio using the given data and got 2.604*10^-9 meters.
I have no clue where to even begin on this problem. Really all I'm looking for is a hint. Not someone to do it for me.

 

[nrqed]

Homework Statement

An apparatus is used to prepare an atomic beam by heating a collection of atoms to a temperature T and allowing the beam to emerge through a hole of diameter d in one side of the oven. The beam then travels through a straight path of length L. Show that the diamter of the beam at the end of the path is larger than d by an amount of order L*hbar/(d*sqrt(3mkT)) where m is the mass of an atom. Make a numerical estimate for typical values of T=1500K m=7amu(lithium atoms), d=3mm, L=2m.

Homework Equations

no clue

The Attempt at a Solution

I calculated the ratio using the given data and got 2.604*10^-9 meters.
I have no clue where to even begin on this problem. Really all I'm looking for is a hint. Not someone to do it for me.

First step: find at what horizontal speed the beam emerges (that speed will depend on kT and on the mass)

Second step: use that v_x to find how long it takes to travel a distance L

Third step, use the uncertainty principle along the y direction to find the velocity along y (the value of $\Delta y$ is basically the diameter of the hole)

Fourth step: multiply the time of the second step by the y velocity of the third step.

I did it very quickly and got that answer. But the most tricky part is to get correct the factors of 2!

Patrick