# Use the uncertainty principle for momentum vs. position to e

1. Jan 28, 2016

### FlorenceC

1. The problem statement, all variables and given/known data

Please see the attachment for a better picture

The energy of an electron in a hydrogen atom is: E = p^2/2m - αe2/r; where p is the momentum,
r the orbital radius, me the electron mass, e the electron charge, and α the Coulomb constant.
Use the uncertainty principle for momentum vs. position to estimate the minimum radius and the momentum corresponding to this radius.

2. Relevant equations
E = p^2/2m - αe2/r

I'm not exactly sure what this means conceptually

3. The attempt at a solution

Is this a plug and chug? I feel like it's not that simple.
uncertainty for momentum says

delta x * delta p <= h/2π

so do I just plug in p in delta p?

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2. Jan 28, 2016

### Simon Bridge

The total energy is the sum of kinetic and potential energy.

Does that work? Did you try? How do you work the $\Delta x$?
I would suggest understanding the relation before trying random calculations.

If the electron has energy E, then how does the radius depend on the momentum?
Can you relate $\Delta x$ to radius?

3. Jan 29, 2016

### Twigg

That's just the total (semiclassical) energy of the electron-nucleus system. The first bit on the left is just the kinetic energy of the electron's orbit, written in terms of momentum instead of velocity. The second bit is the electrostatic potential energy due to the attraction between the electron and the positively charged nucleus. The hydrogen nucleus is just a lone proton, so its charge is +e. Might help to rewrite the whole thing like this:
E = p^2 / 2m + a*(-e)*(+e)/r

The problem is part visualization and part plugging and chugging. You can't put p's where delta p's go, and same for x. Interpreting the deltas requires some thinking. Remember that delta x and delta p are the deviations observed in measured values of x and p. So how much can x (the electron's horizontal position) vary as the electron orbits the nucleus in a circle of radius r? Likewise, how much can p (the x-component of the electron's momentum) vary in that orbit?