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

  1. May 18, 2006 #1
    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:

    [tex] 1/2kT = h*c/lambda [/tex]

    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! :smile:
     
  2. jcsd
  3. May 19, 2006 #2

    Andrew Mason

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    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:

    [tex]\Delta x \Delta p = \hbar/2[/tex]

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

    So

    [tex]\Delta x = \frac{\hbar}{\sqrt{8mkT}}[/tex]


    AM
     
  4. May 19, 2006 #3

    Gokul43201

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    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.
     
  5. May 20, 2006 #4
    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 ?
     
  6. May 20, 2006 #5

    Gokul43201

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    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.
     
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