# Uncertainty principle, relating the uncertainty in position to the uncertainty

1. Mar 22, 2007

### ttiger654

Prove the uncertainty principle, relating the uncertainty in position (A=x) to the uncertainty in energy ($$B=p^2/(2m + V)$$):

$$\sigma x\sigma H \geq \hbar/2m |<P>|$$

For stationary states this doesn't tell you much -- why not??

2. Mar 22, 2007

### StatMechGuy

What have you done so far? Do you know the generalized uncertainty relation? Read the guidelines to this forum, please, and we'll be able to help more.

3. Oct 5, 2009

### luck.rajesh

solution-
[x,p2/2m+V]=1/2m[x, p2]+[x,V];

[x, p2]= xp2 − p2x = xp2 − pxp + pxp − p2x = [x, p]p + p[x, p].

using the equation [x,p]=ih{this is known as canonocal commutation relation}

[x, p2]= ihp + pih = 2ihp. and And [x, V ] = 0,
so [x,p2/2m+ V]=1/2m(2ihp) = ihp/m

The generalized uncertainty principle says, in this case,

σ2xσ2H≥{(1/2i)(ih/m)<p>}^2={h/2m<p>}^2⇒ σxσH ≥h/2m|<p>|. QED

For stationary states σH = 0 and p = 0, so it just says 0 ≥ 0.

4. Oct 5, 2009

### luck.rajesh

for reference u can use {griffiths_d.j._introduction_to_quantum_mechanics__2ed.}