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Calculating uncertainty

  1. Sep 27, 2009 #1
    1. The problem statement, all variables and given/known data
    For a object of mass m, Heisenberg’s uncertainty principle relates the uncertainty in the object’s position Δx to the uncertainty in the object’s speed Δv:
    (Δx)(Δv) ≥ (h divided by (4)(pi)(m))
    where h is Planck’s constant.
    Calculate the minimum uncertainty in the speed of a tennis ball of mass 0.058 kg, assuming that the uncertainty in its position is approximately equal to its own diameter of 6.5 cm. If you assume the tennis ball has a speed equal to the uncertainty value you calculated, how long would it take for the ball to travel a distance equal to its own size? Based on this, do you feel we can ever say where a tennis ball is with a reasonable uncertainty?
    Repeat all of the above analysis for a hydrogen atom of mass 1.67 × 10−27 kg with diameter 1.06 angstrom.

    2. Relevant equations
    h = 6.62606896× 10e-34 J·s


    3. The attempt at a solution
    First I plug in the numbers to figure out velocity v:
    (6.5cm)(Δv) ≥ (6.62606896× 10e-34 J·s divided by (4)(pi)(0.058kg))

    (6.5cm)(Δv) ≥ (about) 9

    Using basic algebra:

    (Δv) ≥ 9/6.5

    (Δv) ≥ 1.4

    So then I plug 1.4 into (Δv)? What do I solve for? I'm not sure where to go from here.
     
  2. jcsd
  3. Sep 28, 2009 #2
    You already have the answer. You forgot about the [itex] 10^{-34} [/itex] in your calculation however. Make sure you give the right units.
     
  4. Sep 28, 2009 #3
    Hmmm I don't understand.
    What do you mean by I forgot about the 10[tex]^{-34}[/tex] in your calculation? Do you mean I need to use that unit (seconds)?

    If so, then I think the complete answer is that "it would take 1.4 seconds for the ball to travel a distance equal to its own size."
     
  5. Sep 28, 2009 #4
    you used h = 6.62606896 Js, instead of h=6.62606896× 1e-34 Js

    The first part of the question was: Calculate the minimum uncertainty in the speed of a tennis ball That is what you did (except for a factor of 1e-34)

    the answer should be a speed, so the units should be in m/s. Convert the size of the ball to meters also.
     
  6. Sep 28, 2009 #5
    OK I converted to meters(6.5cm=.065m). Now I have:

    (.065m)(Δv) ≥ (6.62606896× 10e-34 J•s divided by (4)(pi)(0.058kg))

    (.065m)(Δv) ≥ (about) 9Js

    Using basic algebra:

    (Δv) ≥ 9J•s /.065m

    (Δv) ≥ 138.46m/s

    The minimum uncertainty for the speed of a tennis ball is 138.46m/s

    Now I must figure out how long it will take for the ball to travel a distance of its own size at a velocity of 138.46m/s.

    Can I just divide .065m by 138.46m/s?

    If so, then the ball will travel 4.6 × 10^-4m
     
  7. Sep 29, 2009 #6
    you still use h = 6.62606896 Js, instead of h=6.62606896× 1e-34 Js

    why do you think

    (.065m)(Δv) ≥ (6.62606896× 10e-34 J•s divided by (4)(pi)(0.058kg))

    implies

    (.065m)(Δv) ≥ (about) 9Js

    do you really think the minimum velocity uncertainty of tennis balls is 138 m/s ?
     
  8. Sep 29, 2009 #7
    Ahh, ok, I was dividing with the correct number, but I didn't see the "e-4" at the end for a reason that is more suitable for a TI-84 technical forum. :)
    Thanks for catching that!
    0.0009

    (.065m)(Δv) ≥ .0009Js
    (Δv) ≥ 0.014m/s

    divide .065m by 0.014m/s:
    4.64s
     
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