# Standard derivation and probability density

## Homework Statement

the question asks you to calculate the standard derivation for the mean distance of an electron from the nucleus.
you are given the mean distance (<r>), and the probability density

## Homework Equations

delta r = sqrt (<r^2> - <r>^2)
<r> = 3.a/2

## The Attempt at a Solution

We had to calculate <r^2> by integrating a given value.
after integrating by parts 4 times i ended up with -3a^2
putting this into the standard derivation lead to the square root of minus 21.a^2/4
this obviously gives a complex value.
i was wondering if this can be correct, or if the negative sign has no meaning in the calculation.
i have read from some physics books but i cant find an answer

Matterwave
Gold Member
<r^2> can never be negative (otherwise, it would imply that measuring r would tend to yield an imaginary number!). Check your calculations again.

i've checked my calculations and i still get a value of -3a^2

i suppose there could be a typo somewhere, but apart from that my answer seems correct for the integral i did.
you end up with;
4/a^3 X (24/(-2/a)^5) = -3a^2

thanks anyway :)

Matterwave
Gold Member
What can I say? I mean, thousands of people, students and professors alike, have done this calculation without it being negative, and even using logic itself, <r^2> cannot be negative.

I mean, all I can say is to check your calculations again, there has to be an error somewhere. Either you used the wrong wave function, or you didn't multiply by the complex conjugate, or you didn't integrate correctly, or something...

I'm sure if you put your full calculation here someone will point out to you where you went wrong. That's all I can advise.

thanks for your help, i can understand it better now

i have just found the error in the calulation (this bit wasnt actucally done by me), i guess i just glanced over it and forgot to correct it

thanks again :)