# Orders of magnutide in Heisenberg's uncertainty principle

1. Dec 14, 2008

### Heirot

Is there any a priori connection beetween the orders of magnitude of e.g. momentum, and its uncertainty? Why do we always assume that the momentum is the same order of magnitude as its uncertainy? I'm refering to all those "back of the envelope" calculations.

Thanks

2. Dec 14, 2008

### Hootenanny

Staff Emeritus
I'm not quite sure what you mean, could you perhaps expand? Under which circumstances are you referring to?

3. Dec 14, 2008

### Heirot

For example, let's put an electron in a box of length L. Since the electron is in the box, we know that delta x = L (or L/2, I'm not quite sure). Using uncertainty relations, we have delta p >= hbar / 2L. As L gets smaller, delta p gets bigger. So, the textbooks conclude, at one point delta p gets so big that there's enough energy to create a positon - electron pair! But energy increases with p, not delta p! For all we now, p could be zero all the time. Why do we assume that p increases as delta p increases?

4. Dec 14, 2008

### Hootenanny

Staff Emeritus
The uncertainty in momentum can be related to the expectation value (average) of the momentum and momentum squared, which in turn can be related to the expectation value of the kinetic energy. It turns out that for a particle in a box, the expectation value of the kinetic energy of a particle is directly proportional to the expectation value of the square of the momentum and hence also directly proportional to the uncertainty in momentum.