Griffiths Quantum Mechanics 2nd edition Chapter 8 equation is confusing

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Homework Statement:
Griffiths Quantum Mechanics 2nd edition
Relevant Equations:
Griffiths second edition equations 8.24 and 8.25
In Griffiths Quantum Mechanics 2nd edition, in Chapter 8 he calculates the following integral on page 323
Screen Shot 2020-10-10 at 7.52.17 PM.png

and he gets
Screen Shot 2020-10-10 at 7.57.06 PM.png

I disagree with this result, I think the integral should be
Screen Shot 2020-10-10 at 7.55.18 PM.png

since

Screen Shot 2020-10-10 at 8.17.16 PM.png

Maybe somebody can explain why I am wrong? Also, from equation 8.24 to 8.25, he makes the assumption that sinϵ ≅ ϵ, but how does that imply sin^(-1)ϵ ≅ ϵ, which seems to be what he assumes to get from 8.24 to 8.25. Any insight is greatly appreciated!
 

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Answers and Replies

  • #2
TSny
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I disagree with this result, I think the integral should be
View attachment 270765
Note that the integral must go to zero as ##r_2## approaches ##r_1##. But your result doesn't satisfy this condition.

Check the following:
1602385130516.png


Also, from equation 8.24 to 8.25, he makes the assumption that sinϵ ≅ ϵ, but how does that imply sin^(-1)ϵ ≅ ϵ, which seems to be what he assumes to get from 8.24 to 8.25. Any insight is greatly appreciated!
Take the inverse sine of both sides of sinϵ ≅ ϵ. Or, do a Taylor series of ##\sin^{-1} x## about ##x=0## .
 
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Thank you, I get it now!
 
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