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Heisenberg Uncertainty Derivation

  1. Apr 2, 2009 #1
    1. The problem statement, all variables and given/known data
    Using the uncertainty relation for momentum and position, show that the quantum-mechanical uncertainty in the position of a particle at temperture T is
    [tex]\Delta x[/tex]~[tex]\sqrt{\frac{h^{2}}{4mkT}}[/tex]
    where T is the temperature and k is the Boltzmann's constant.

    2. Relevant equations
    [tex]\Delta p\Delta x\geq h/2[/tex], h being Planck's constant
    K.E.=0.5 mv2=0.5 kT

    3. The attempt at a solution
    I isolated [tex]\Delta x[/tex] and subbed [tex]\Delta p[/tex]=mv=kT/v.
    So, [tex]\Delta x[/tex]~h/2[tex]\Delta p[/tex] ~ h/2mv ~ hv/2kT.
    I've tried subing in a whole bunch of stuff for v but I can't seem to get the equation. Any insight?
  2. jcsd
  3. Apr 2, 2009 #2


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    Homework Helper

    Delta p[/tex] = sqrt(2mE) where E is the kinetic energy which is equal to 1/2*kT
  4. Apr 2, 2009 #3
    Thanks =)
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