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Homework Help: Uncertainty principle between Kinetic energy and Potential energy

  1. Jun 2, 2010 #1
    1. The problem statement, all variables and given/known data
    Let's say T is kinetic energy and V is Potential.
    Then, find a principle between T and V by using

    dA^2dB^2 (larger or equal) {(1/2i)(<[A,B]>)}^2


    3. The attempt at a solution

    First I try to find commutator of T and V, [T,V]
    then it gives little bit dirty expression..

    [T,V] = -(h^2/2m) [ (d/dx)^2(V) + 2(d/dx)V(d/dx) ]
    (Here h represents h over 2pi)
    Then when I plug it into the general uncertainty principle,
    i on the principle does not cancel out.
    so the inequality cannot hold.

    I thought that mathematically the second deravative of V(x) must be zero to fit the principle
    but there is no clue. Maybe it is wrong also.

    I can't go on further..
    What did I wrong?
     
    Last edited: Jun 2, 2010
  2. jcsd
  3. Jun 2, 2010 #2
    Your principle of uncertainty formula is wrong, just by a hair.

    The expectation value of the commutator is not just squared, it is the modulus squared;
    i.e. it is itself times its conjugate.

    I have not checked the rest, this is all on a quick glance and that's why your i factor does not cancel out.

    I'll try to latex it, the right hand side of the Heisenberg inequatlity should read:

    [itex]
    \left| \frac{1}{2i} < \left[ A,B \right] > \right| ^{2}
    [/itex]

    which goes to:

    [itex]
    \frac{1}{4} \left| < \left[ A,B \right] > \right| ^{2}
    [/itex]

    Hope this was helpful, be happy and good luck.
     
    Last edited: Jun 2, 2010
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