Expectation value for electron in groundstate

[Cp.L]

Homework Statement

Show that the expectation value for r for an electron in the groundstate of a one-electron-atom is:
<r>=(3/2)a$_{0}$/Z

Homework Equations

Expectationvalue:
<f(x)>=∫$\psi$*f(x)$\psi$dx, -∞<x>∞

$\psi$$_{100}$=C$_{100}$ exp(-Zr/a$_{0}$), $a_{o}\ =\ 0.5291\ \times\ 10^{-10}m , h\ =\ 6.626\ \times\ 10^{-34}\ J\ s$

The Attempt at a Solution

Im stuck..

 

[Ray Vickson]

Homework Statement

Show that the expectation value for r for an electron in the groundstate of a one-electron-atom is:
<r>=(3/2)a$_{0}$/Z

Homework Equations

Expectationvalue:
<f(x)>=∫$\psi$*f(x)$\psi$dx, -∞<x>∞

$\psi$$_{100}$=C$_{100}$ exp(-Zr/a$_{0}$), $a_{o}\ =\ 0.5291\ \times\ 10^{-10}m , h\ =\ 6.626\ \times\ 10^{-34}\ J\ s$

The Attempt at a Solution

Im stuck..

What is stopping you from doing the integral? Of course, first you must determine the correct value of C₁₀₀; do you know how to do that?

RGV

 

[Cp.L]

Hi, yes i used that C$_{100}$ =(1/$\sqrt{\pi}) (\frac{z}{a_{0}}$)$^{3/2}$

Then i do the integral, but i must be doing it wrong cause i end up with e$^{\frac{-2zr}{a_{0}}}$ in the answer and also a problem of e$^{∞}$

 

[vela]

Show your work.

 

[Cp.L]

Ok, i found a mistake, but still don't get it right.
I write:

<r>=$\frac{1}{\pi}$$\frac{z_{0}^{3}}{a_{0}^{3}}$∫e$^{\frac{-2Zr}{a_{0}}}$r dr

Where the limits are from -∞ to ∞

Integrating i let U= r, du=1, dv= e$^{\frac{-2Zr}{a_{0}}}$, V=$\frac{a_{0}}{-2Z}$e$^{\frac{-2Zr}{a_{0}}}$

this integration leaves me with e$^{∞}$, or is there some trick for e$^{-∞}$ - e$^{∞}$

 

[vela]

The integral should be from 0 to infinity. You also have a mistake somewhere else as your expression has units of 1/length.

 

[Cp.L]

Oh yes that's logical since its the radius, thanks. Yes and the integration by parts should be
u=r
du=dr
dv=e$^{\frac{-2Zr}{a_{0}}}$
v=$_{}\frac{a_{0}}{-2Z}$e$^{\frac{-2Zr}{a_{0}}}$ dr

integrating $\frac{z^{3}}{\pi a_{0}}$∫e$^{\frac{-2Zr}{a_{0}}}$ r dr from 0 to ∞ with these vaues i get
<r>= -$\frac{Za_{0}}{4\pi}$

I wonder if my C$_{100}$ might be wrong..

 

[vela]

How did you find C₁₀₀?

 

[Cp.L]

It was listed in my book, for the groundstate of an hydrogen atom so it should actually be correct.

 

[vela]

You need to go back and review spherical coordinates so you can set up the integrals correctly. In particular, you need to fix your limits and get the volume element right.

 

[Ray Vickson]

It was listed in my book, for the groundstate of an hydrogen atom so it should actually be correct.

You should verify for yourself the value given in the book. If you cannot get the book's value, that is a signal that you may be doing something wrong---or possibly that the book made an error---but you should go with the first hypothesis unless you have overwhelming evidence to the contrary.

Once you are able to get the correct value of C₁₀₀ you will be in a better position to find <r>.

RGV

 

[Cp.L]

Hi, thanks a lot. It really helped to use spherical coordinates :) got the correct answer now :)