Uncertainty Product for a hydrogen-like atom,

[dweeegs]

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!

 

[andrien]

you may probably calculate <p²> by using
<p²>=∫ψ*p²ψd³r,then apply first quantized form as p=-ih^-∂/∂r and then you have to do some by part or a simple straightforward way without by part and done.On the other hand you might want to use
k.E.=p²/2m

 

[wotanub]

Am I too late?

It's no problem to calculate the expectation values of the position operators with a position-space wavefunction right? So it should be just as easy to calculate the expecation values of momentum operators with a momentum-space wavefunction!

If you find the momentum-space wavefunction, you are in the clear. My method completely ignores the hint though.