Estimate for root-mean-square uncertainty

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Homework Statement


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.

Homework Equations



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

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.....
 

Answers and Replies

  • #2
vela
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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##.
 
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Gotcha!
 

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