Size of He atom at 10^(-9)K

1. May 18, 2006

mmh37

Can anyone help me to find an approximation to the size of a He atom at very low temperature (10^(-9)K)?

My attempt:

At this very low temperature all electrons will be in the ground state, thus using the equipartition theorem:

$$1/2kT = h*c/lambda$$

now, for the zero point energy, lambda is 1/2 the diameter of the Helium atom.
However, this approach does not yield the correct answer.

I have had a look at several textbooks, but couldn't find any hints in there.
Thanks a lot for your help!

2. May 19, 2006

Andrew Mason

Temperature, being a statistical concept, is not defined for an atom. I don't see how temperature would affect the size of any atom.

But, assuming this question is asking what the uncertainty of a He atom's position would be if it had a kinetic energy in the range of He atoms in He gas at 10^-9 K, you would have to apply the Heisenberg uncertainty principle:

$$\Delta x \Delta p = \hbar/2$$

You would have to work out $\Delta p[/tex] from the energy range - which is roughly 0 < E < kT. Using [itex]E = p^2/2m$ would give $\Delta p = \sqrt{2mE} = \sqrt{2mkT}$

So

$$\Delta x = \frac{\hbar}{\sqrt{8mkT}}$$

AM

3. May 19, 2006

Gokul43201

Staff Emeritus
I believe the question may be refering to the de Broglie wavelength (this might be the preamble to a discussion of BECs), which is of the same order as the number calculated by AM.

4. May 20, 2006

mmh37

it is indeed, as the question then asks to compare this with BEC.

However, the solution manual says that d = 60 um, but with the uncertainty principle I got 2.45 *10^(-7)m ?

5. May 20, 2006

Gokul43201

Staff Emeritus
I get a number that's pretty close to 60um (I get p ~ 10^{-29} Ns). You must have made a numerical error. If you still can't find the error, post your calculation here, and someone will point it out.