# Heisenberg Uncertainty Derivation

1. Apr 2, 2009

### XJellieBX

1. The problem statement, all variables and given/known data
Using the uncertainty relation for momentum and position, show that the quantum-mechanical uncertainty in the position of a particle at temperture T is
$$\Delta x$$~$$\sqrt{\frac{h^{2}}{4mkT}}$$
where T is the temperature and k is the Boltzmann's constant.

2. Relevant equations
$$\Delta p\Delta x\geq h/2$$, h being Planck's constant
K.E.=0.5 mv2=0.5 kT

3. The attempt at a solution
I isolated $$\Delta x$$ and subbed $$\Delta p$$=mv=kT/v.
So, $$\Delta x$$~h/2$$\Delta p$$ ~ h/2mv ~ hv/2kT.
I've tried subing in a whole bunch of stuff for v but I can't seem to get the equation. Any insight?

2. Apr 2, 2009

### rl.bhat

Delta p[/tex] = sqrt(2mE) where E is the kinetic energy which is equal to 1/2*kT

3. Apr 2, 2009

Thanks =)