1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Radial Distribution Function

  1. Mar 6, 2014 #1
    1. The problem statement, all variables and given/known data
    Hey Guys!

    Here's my problem: ψ=2(Z/a)^3/2*e^ρ/2 for the 1s orbital of a hydrogen atom. Write down the radial distribution function expression (P) of a 1s electron and determine the most likely radius.

    Z nuclear charge
    r radius
    a Bohr's radius

    2. Relevant equations

    P=(ψ*)(ψ)∂τ= 4(Z/a)^3*e^ρ*4π*r^2*∂r

    3. The attempt at a solution

    We need to calculate ∂P/∂r

    My professor solves r=a/Z which is all well and fine, but in an intermediate step he goes from P=(stuff)*e^(-2Zr/a)*r^2*∂r to ∂P/∂r=∂/∂r((stuff)*e^(-2Zr/a)*r^2)

    To me it seems that he has neglected a ∂r and it should read ∂P/∂r=∂/∂r((stuff)*e^(-2Zr/a)*r^2*∂r), in which case I am not sure how to calculate something like this.

    What would help me solve this is to know if I am making some stupid mistake, or if there is some rule in the form x=y∂r ---> ∂x=∂(y∂r)=? (my guess would be ∂x=∂y∂r+y(∂^2)r, but this doesn't agree with my professor's tricks, and I haven't taken a PDE class to know my way around)

    Thanks guys!
  2. jcsd
  3. Mar 6, 2014 #2
    Nevermind, realized my professor's mistake was P=4(Z/a)^3*e^ρ*4π*r^2 and not P=(ψ*)(ψ)∂τ (i.e. P=(ψ*)(ψ)∂τ/dr)
  4. Mar 6, 2014 #3


    User Avatar
    Science Advisor
    Homework Helper
    2017 Award

    Note that P is a probability density function! i.e. ##P(r)\ dr## is the probability to find something between ##r## and ##r+dr##.

    In other words, the ∂r should not be in your P expression. (there should also be a minus sign in the exponent).

    And you don't want ∂P/∂r (it is negative definite).

    If you want to calculate the most likely radius, that means integrating ψ*ψ dτ over a shell with radius r and thickness dr to get a radial probability density distribution. Differentiate that wrt r to find the most likely r.

    (This works if your ψ is normalized and also if it is not)

    Note that for the mean radius, you need the expectation value for ##|\vec r|##. that means integrating ψ* r ψ dτ and gives a different answer!
    (This works if your ψ is normalized and if it is not, divide by ∫ψ*ψ dτ )

    [edit] crossed your second post -- took me a while to type this together!
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted