Introduction to Quantum Mechanics - Third Edition, by Richard Liboff

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pmb
"Introduction to Quantum Mechanics - Third Edition," by Richard Liboff

Does anyone here have "Introduction to Quantum Mechanics - Third Edition," by Richard Liboff?

I'd like some help with the way he explains Hermite functions. In the problems he says to prove a the relation

exp(2zt - t^2) = sum(k = 0 - infinty) H_n(z)t^n/n!

Which, of course, is done with a Taylor series expansion. It appears that Liboff wants the reader to prove this with the inormation that he gives in that section regarding H_n(z). But I don't see that its at all obvious.

Anyone?

Pete
 

Answers and Replies

  • #2
Liboff question

I will look at in on Monday. I don't have the text with me now.
 
  • #3
pmb


Originally posted by sdeliver645
I will look at in on Monday. I don't have the text with me now.
Thank you. Much appreciated.

Side question: Do you mind if I ask you something? Do you know the subject matter very well? I.e. have you gone through such a coursesequence or text? I'd enjoy discussing the contents of this book with someone. Are you up for it? Liboff some some weird stuff and there are errors in this verion - alot of errors I think. He also gets into some stuff which is pretty specialized especially in the problems.

Thanks

Pete
 
  • #4
Liboff

Yes, I did go through parts of Liboff.

I used it for my second quantum course and part of my atomic physics course.

I am not an expert, I only have an undergrad., although I did do quite well in the courses. We weren't assigned many of the problems, but I wouldn't mind thinking along side you for some of them.
 
  • #5
pmb


Originally posted by sdeliver645
Yes, I did go through parts of Liboff.

I used it for my second quantum course and part of my atomic physics course.

I am not an expert, I only have an undergrad., although I did do quite well in the courses. We weren't assigned many of the problems, but I wouldn't mind thinking along side you for some of them.
That'll be great - I'm sure that if we have the same text then one might see what the other might not be picking up on. And I hope this is the case on the Hermite polynomials

Thanks

Pete
 
  • #6
pmb


In the mean time maybe someone can help me with this -

There is a relation which goes like this

F(z+a) = exp[-(z+a)^2]

Then they claim that the Taylor series expansion of this is

F(z+a) = sum(n=0 - Inf) a^n/n! F^n(z)

F(z+a) = sum(n=0 - Inf) a^n/n! (-1)^n H_n(z) exp(-z^2)

Now how in the world do they get that?

Thanks

Pete
 
  • #7
Hurkyl
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exp(-(z+a)^2) = exp(-z^2 - 2az - a^2)
= exp(-z^2) exp(2z(-a) - (-a)^2)
 
  • #8
pmb
Originally posted by Hurkyl
exp(-(z+a)^2) = exp(-z^2 - 2az - a^2)
= exp(-z^2) exp(2z(-a) - (-a)^2)
I'm sorry. If that was supposed to be helpful then I don't follow. What is your point?

Pete
 
  • #9
Hurkyl
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Then you can substitute the equation you were originally considering...

F(z+a) = exp(-z^2) exp(2z(-a) - (-a)^2)
= exp(-z^2) * sum(k = 0 - infinty) H_n(z) (-a)^n/n!
= sum(n=0 - Inf) a^n/n! (-1)^n H_n(z) exp(-z^2)


My mistake if you meant that this equation was supposed to be a hint for deriving the equation you were interested in.


Incidentally, what is fair game to use about H_n(z)? I looked up the hermite polys on mathworld, but they essentially start with the egf you're trying to derive. Was hoping to try and help with the original question, but I don't know what you're allowed to use!
 
  • #10
pmb
Originally posted by Hurkyl
Then you can substitute the equation you were originally considering...

F(z+a) = exp(-z^2) exp(2z(-a) - (-a)^2)
= exp(-z^2) * sum(k = 0 - infinty) H_n(z) (-a)^n/n!
= sum(n=0 - Inf) a^n/n! (-1)^n H_n(z) exp(-z^2)


My mistake if you meant that this equation was supposed to be a hint for deriving the equation you were interested in.


Incidentally, what is fair game to use about H_n(z)? I looked up the hermite polys on mathworld, but they essentially start with the egf you're trying to derive. Was hoping to try and help with the original question, but I don't know what you're allowed to use!

I'm sorry but I still don't follow. My question here is how did they get this equation

F(z+a) = sum(n=0 - Inf) a^n/n! (-1)^n H_n(z) exp(-z^2)

from the definition of F(z+a). One can do the differention and set a a to zero etc but it doesn't tell me where the H_n(z) comes from! And that result is messy!

Thank you

Pete
 
  • #11
Hurkyl
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I'm assuming

exp(2zt - t^2) = sum(k = 0 - infinty) H_n(z)t^n/n!

to derive the result for F(z + a)... I'm presuming that since the author has presented this formula in a problem he intends the result to be used to solve other problems, like that of F(z + a)
 
  • #12
jeff
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Originally posted by pmb
Does anyone here have "Introduction to Quantum Mechanics - Third Edition," by Richard Liboff?

I'd like some help with the way he explains Hermite functions.
 
  • #13
pmb


Originally posted by jeff
I think I see now. You're searching my posts to see where I asked a question - then you quote me thus pointing out that I don't know everything.

never claimed I did.

So why are you doing this jeff?

Pmb
 

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