Wave functions and probabilities

AI Thread Summary
The discussion revolves around calculating the probability associated with a wave function, specifically using the equation P = |Ψ|²dV. The user initially miscalculates the volume element dV and is corrected to use 4/3 instead of 4 for the area in spherical coordinates. After adjusting the probability equation to P = A²e^{-2αr²}4πr²dr, the user seeks clarification on how to handle the dr term when taking the derivative to find the maximum probability. Participants suggest treating dr as a constant during differentiation. The conversation emphasizes the importance of correctly applying mathematical principles in quantum mechanics.
kidsmoker
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Homework Statement



http://img200.imageshack.us/img200/9268/29360438.jpg


Homework Equations



P=\left|\Psi \right| ^{2}dV

The Attempt at a Solution



Okay, so r^{2} = x^{2}+y^{2}+z^{2} and \left|\Psi \right| ^{2} = A^{2}e^{-2\alpha r^{2}} .

The volume of the of the bit we're interested in should be

dV = 4\pi(r+dr)^{3} - 4\pi r^{3} \approx 12\pi r^{2}dr if we ignore the (dr)^{2} and (dr)^{3} terms. Have I done something wrong here, as I was expecting to just end up with 4\pi r^{2}dr?

Assuming it's correct, the probability is then

P = A^{2}e^{-2 \alpha r^{2}}12\pi r^{2}dr .

To find where this has a maximum value, would I set \frac{dP}{dr}=0 and then find the corresponding r values? But how do I take the derivative when there's a dr term in there? :confused:

Thanks for any help!
 
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Hi kidsmoker! :smile:
kidsmoker said:
The volume of the of the bit we're interested in should be

dV = 4\pi(r+dr)^{3} - 4\pi r^{3} \approx 12\pi r^{2}dr if we ignore the (dr)^{2} and (dr)^{3} terms. Have I done something wrong here, as I was expecting to just end up with 4\pi r^{2}dr?

Yup! :biggrin:

4 is for areas

try 4/3 ! :wink:

(or just multiply the area by dr)
 
tiny-tim said:
Hi kidsmoker! :smile:


Yup! :biggrin:

4 is for areas

try 4/3 ! :wink:

(or just multiply the area by dr)


Oh yeah, hahaha, i always get that wrong!

So we have P = A^{2}e^{-2 \alpha r^{2}}4\pi r^{2}dr then what happens to the dr when i take the derivative? :s

Thanks.
 
kidsmoker said:
… what happens to the dr when i take the derivative?

Just ignore it :wink: … it's a constant

think of it as called something other than dr! :rolleyes:
 
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