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Classical uncertainty principles

  1. Oct 10, 2004 #1
    Please check my work.

    Using the first uncertainty principle:

    [tex]
    \Delta x \Delta k \sim 1
    [/tex]

    derive the second uncertainty principle:

    [tex]
    \Delta \omega \Delta t \sim 1
    [/tex]


    My work:

    [tex]
    \Delta x \Delta k \sim 1
    [/tex]

    [tex]
    \frac {\Delta x} {\Delta t} = V \Rightarrow \Delta x = V \Delta t
    [/tex]

    [tex]
    V \Delta t \Delta k \sim 1
    [/tex]

    [tex]
    V_\textrm {group} = \frac {\Delta \omega} {\Delta k}
    [/tex]

    [tex]
    \frac {\Delta \omega} {\Delta k} \Delta t \Delta k \sim 1
    [/tex]

    [tex]
    \frac {\Delta k} {\Delta k} \Delta \omega \Delta t \sim 1
    [/tex]

    [tex]
    \Delta \omega \Delta t \sim 1
    [/tex]

    What do you think?
     
  2. jcsd
  3. Oct 11, 2004 #2
    Looks good to me. Right on man. What were you uncertain about when you wrote this?
     
  4. Oct 13, 2004 #3

    I wasn't sure if my use of

    [tex]V_\textrm {group} = \frac {\Delta \omega} {\Delta k}
    [/tex]

    was legal. But then I realized that the uncertainty principles are used for wave packets so it was fine.
     
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