- #1
dweeegs
- 12
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Homework Statement
Calculate the uncertainty product ΔrΔp for the 1s electron of a hydrogen-like atom with atomic number Z. (Hint: Use <p> = 0 by symmetry and deduce <p^2> from the average kinetic energy)
Homework Equations
All I have is the wavefunction. For a 1s, it takes the form:
wavefunction = (Z/a0)^(3/2)*2exp(-Zr/a0)
Where Z is the atomic number, a0 is the Bohr radius
Other equations that I think I need include:
The energy for a hydrogen-like atom: E = (-13.6 eV)(Z^2/n^2)
Δr = √(<r^2> - <r>^2)
Δp = √(<p^2> - <p>^2)
<f> = ∫f(r)*P(r)dr
That integral is from 0 to ∞, and the <> are supposed to denote averages.
The Attempt at a Solution
I can find Δr no problem. Just use the above formula, where f(r) is r^2 for <r^2> and f(r) is r for <r>^2. The problem is finding the uncertainty in momentum. The above formula for momentum uncertainty should reduce down to
Δp = √(<p^2>)
And that's where I'm stuck; the hint isn't helping me much.
For another problem, I found the uncertainty in KE by finding the uncertainty in <U> for a hydrogen atom, and then used <KE> + <U> = <E>, where <E> = -13.6/n^2.
I'm not sure what the potential for a hydrogen-like atom is. Also, how can I get KE into momentum? I'm just confused on this part as a whole.
Any help is appreciated! Thank you!