Expectation value r^2 for a radial wave function

[blaksheep423]

Homework Statement

The ground state (lowest energy) radial wave function for an electron bound to a proton to form a hydrogen atom is given by the 1s (n=1, l=0) wave function:

R₁₀ = (2 / a^3/2) exp(-r / a)

where r is the distance of the electron from the proton and a is a constant.

a) Sketch the wave function, and
b) What is the expectation value of <r²> for the electron?

Homework Equations

i think the equation for <r²> is

\int R*r²R dr
from -infinity to infinity

The Attempt at a Solution

for a) i am completely blanking on how to sketch the wave function. for b) i was able to use an integral table to evaluate <r²> and i got 8 / \sqrt{}a

however, i have a feeling that my original equation for <r²> is wrong, and should involve some sort of Y and \phi and \Theta
also, should i be integrating from 0 - a, or -infinity to infinity?

 

[cepheid]

for a) i am completely blanking on how to sketch the wave function.

It decays exponentially. It's an exponential function. You just have to sketch it, not even plot it. You should have some idea of what it looks like.

for b) i was able to use an integral table to evaluate <r²> and i got 8 / \sqrt{}a

however, i have a feeling that my original equation for <r²> is wrong, and should involve some sort of Y and \phi and \Theta
also, should i be integrating from 0 - a, or -infinity to infinity?

It's true that in general you need to evaluate〈ψ|r²|ψ〉using the entire wavefunction, which in this case means doing the full 3D integral. However, in this case, because l = 0, the angular part of the wavefunction, Y₀₀ is constant (i.e. it is independent of θ and ϕ), making things really easy. You can look up what it is.

 

[cepheid]

also, should i be integrating from 0 - a, or -infinity to infinity?

Well, you should integrate over all possible r values. But you can't have a negative radius, can you?

 

[jdwood983]

The Attempt at a Solution

for a) i am completely blanking on how to sketch the wave function. for b) i was able to use an integral table to evaluate <r²> and i got 8 / \sqrt{}a

however, i have a feeling that my original equation for <r²> is wrong, and should involve some sort of Y and \phi and \Theta
also, should i be integrating from 0 - a, or -infinity to infinity?

You're off by a little bit. In the 1s state, there is actually no angular dependence, but you do need the 4\pi r^2 coefficient of dr. Thus, your integral should be

<br /> \langle r^2\rangle=\int_0^\infty R^*r^2R\,4\pi r^2dr<br />

 

[blaksheep423]

OK, so i understand that the second r² term came from r d\Theta r sin\Theta d\phi, but where does the 4\pi come from? \phi goes from 0-2\pi, but what about the extra factor of 2? Is it because the function is symmetric, so you're now integrating from 0- infinity and multiplying the result by 2?

And what happens to sin\Theta d\Theta?

man, i need a laTex tutorial...

 

[blaksheep423]

ok, i think I've got it now. if anyone sees an error in this post, let me know.

the integral above comes from integrating sin(\Theta) from 0 to \pi = 1, and d \phi from 0 - 2\pi = 2\pi. then, integrating from 0 - infinity gives the extra factor of 2, yielding

<r²> = (4 \pi)(4/a³) \int r⁴ exp(-2r/a) dr -------------- from 0 to infinity

for this particular radial wave function. Right?

Well, with my handy table of integrals, i evaluated this to an obscenely complicated function, which because of the exponential ended up evaluating fairly easily from 0 to infinity. i got

<r²> = 12 \pi a²

ignore the fact that the pi's are all superscripts. i don't know why that's happening.

is this the correct answer?

 

[Pengwuino]

The sin(\theta) d\theta contributes a 2 which results in the 4 \pi

 

[jdwood983]

<r²> = 12 \pi a²

ignore the fact that the pi's are all superscripts. i don't know why that's happening.

is this the correct answer?

I get that value as well. As for your work not aligning correctly, this is because you are not writing everything in the tex environment. Rather than writing "tex \int /tex r sup4/sup etc "etc (with braces, I'm dropping them so as to show you what i mean) write "tex \int_{bottom limit}^{upper limit} r^4e^{-2r/a}dx /tex" and it'll look pretty:

<br /> \langle r^2\rangle=\frac{3\pi}{a^3}\int_0^\infty r^4\exp\left[-\frac{2r}{a}\right]dx=12\pi a^2<br />


Also:

<br /> \int_0^\pi\sin[\theta]d\theta\int_0^{2\pi}d\phi=\int_{-1}^1d(\cos\theta)\int_0^{2\pi}d\phi=(1-(-1))\cdot(2\pi-0)=4\pi<br />