Normalization and orthogonality of wavefunctions

[jessedupont]

I have two wavefunctions that I need to normalize but I cannot figure out how to get them into an acceptable integrable form...
the first is psi=(2-(r/asub0))*e^(-r/asub0)
the second is psi=rsin(theta)*cos(phi)*e^(-r/2asub0)
I know these need to be in the form (where psi will be name y for simplicity sake)
1=int(Y*Ydtao). and this will eventually come to the form (let ~ be the symbol for integral where 1~2 is the integral from 1 to 2 for example.)
1=0~2pi 0~pi 0~inf r^2dr*sin(theta)*dtheta*dphi

then i need to confirm that these two functions are mutually orthogonal but as long as I can integrate them I should be able to figure out all that.

Thanks!

 

[jfy4]

$$Now I got to be honest here, I'm a little shaky on this, but since r, \theta, \phi are not functions of each other you can just integrate both those functions over all space just like you said. That is, you can do the r part times the \theta part times the phi part and if you get 0 then they are orthogonal.$$

 

[strangerep]

(jessedupont: it's usual on this forum to use latex to make one's equations more
readable. Since this is your first post, I've converted some of your stuff to latex to
help you get started. To see the latex code for the equations, just click on them.
That will also give you a link to a quickstart latex guide...)

I have two wavefunctions that I need to normalize but I cannot figure out how to get them into an acceptable integrable form...
the first is

$$\psi ~=~ \left(2 - \frac{r}{a_0}\right) e^{-r/a_0}$$

the second is

$$\psi ~=~ r \sin(\theta) \cos(\phi) e^{-r/2a_0}$$

I know these need to be in the form (where psi will be name y for simplicity sake)
$$1 ~=~ \int \bar{Y} Y d\tau$$

and this will eventually come to the form (let ~ be the symbol for integral where 1~2 is the integral from 1 to 2 for example.)
1=0~2pi 0~pi 0~inf r^2dr*sin(theta)*dtheta*dphi

by which I assume you mean:

$$1 ~=~ \int_0^{2\pi} d\phi \int_0^\pi sin(\theta) d\theta<br /> \int_0^\infty \bar{\psi} \psi \; r^2 dr ~~~~~~~~ (?)$$

then i need to confirm that these two functions are mutually orthogonal but as long as I can integrate them I should be able to figure out all that.

So what exactly is your problem? Just perform the integrals...

 

[jessedupont]

thanks for the help strangerep. I'm just not too sure how to integrate it or what it should look like before integrating. I only had through cal II so I think I missed out on some of the helpful stuff for this situation. I tried to integrate what I had on wolfram but it gave me another variable in my answer (refering to the first wavefunction).

 

[jtbell]

I'm just not too sure how to integrate it

Make friends with a good table of integrals.

(Unless your professor really expects you to work out all your integrals completely from scratch, of course.)

 

[jessedupont]

Any tips on some good integral tables. (for really complex ones) all of the ones I've found don't have anything that complex...

 

[dextercioby]

Grashteyn and Ryzhik's book ((any edition, probably the latest has more infomation) is a great source for integrals (you need definite integrals) of all sorts of function, Laguerre polynomials included.

 

[strangerep]

thanks for the help strangerep. I'm just not too sure how to integrate it or what it should look like before integrating. I only had through cal II so I think I missed out on some of the helpful stuff for this situation. I tried to integrate what I had on wolfram but it gave me another variable in my answer (refering to the first wavefunction).

OK. Concentrate on your first wavefunction and try to normalize that.
Substitute your $$\psi$$ into the latex integral expression I gave.
Since it has no angular dependence those parts of the integral are trivial.
(Do you agree? If not, then you really do need to take a remedial calculus
course. :-)

The radial part of the integral is done using integration by parts.
E.g., use:

$$e^{-r} \, dr ~=~ - d(e^{-r})$$

and then integrate by parts. Then repeat this technique until you
get rid of the r's.

If that's not enough explanation, then maybe you should move this
question over to the calculus forum since it's only math at this point.

 

[jtbell]

Any tips on some good integral tables. (for really complex ones) all of the ones I've found don't have anything that complex...

Ever since I was an undergraduate (almost 40 years now), I've used the https://www.amazon.com/dp/1584882913/?tag=pfamazon01-20. Your example reduces (after multiplying out the polynomials) to a sum of integrals of the form

$$\int {r^n e^{-ar}}$$

which I'm pretty sure are in those tables, although I don't have my copy at home to check. I know I've done integrals like those in the past.