Uncertainty principle between Kinetic energy and Potential energy

1. Jun 2, 2010

J.Asher

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. Jun 2, 2010

rhoparkour

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:

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

which goes to:

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

Hope this was helpful, be happy and good luck.

Last edited: Jun 2, 2010