# Estimate for root-mean-square uncertainty

1. Oct 13, 2014

### terp.asessed

1. The problem statement, all variables and given/known data
In classical mechanism, the lowest possible energy accessible to any system is the minimum potential energy, in this case 0. However, quantum mechanically, one finds that there is a zero-point energy (where ground state energy > classical minimum). Fundamentally, zero-point energy comes from the uncertainty principle, so it is possible to estimate for root-mean square uncertainty in position, by looking at the range of x allowed classically for a given energy. Remember that in the classical mechanics, the total energy is given by:

E = p2/(2mu) + V(x)

so that V(x) > E. Therefore, sketch a a graph of potential energy as a function of x. Estimate root-mean square uncertainty as a function of Energy (E), mu and w.

2. Relevant equations

E = p2/(2mu) + V(x)

3. The attempt at a solution
I drew a graph, V(x) vs. x, and drew a line (Energy) horizontally through the curve, for V(x) = mu*w2x2/2. There are TWO intersecting points where V(x) meets Energy lines--which I set as boundaries. However, I am stuck as how to move from here....any suggestion would be welcome....I've been thinking about using root-mean square2 = <x2> - <x>2, but the question wants ESTIMATE of root-mean-square by LOOKING at the range.....

2. Oct 15, 2014

### vela

Staff Emeritus
You found the turning points. Classically, the particle is confined to the region between those two points. How long is that region? That's what you use as an estimate for $\Delta x$.

3. Oct 15, 2014

Gotcha!