Introduction to Quantum Mechanics - Third Edition, by Richard Liboff

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Discussion Overview

The discussion revolves around the "Introduction to Quantum Mechanics - Third Edition" by Richard Liboff, specifically focusing on the explanation of Hermite functions and related mathematical relations. Participants seek clarification on specific equations and their derivations, particularly involving Taylor series expansions and Hermite polynomials.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant requests help with proving the relation involving Hermite functions and a Taylor series expansion, expressing uncertainty about the clarity of Liboff's explanation.
  • Another participant mentions having gone through parts of Liboff's text and expresses willingness to discuss the book's content, noting potential errors in the text.
  • A participant questions how a specific Taylor series expansion relates to Hermite polynomials, indicating confusion about the derivation process.
  • There is a discussion about substituting equations and how to derive the relation involving Hermite polynomials from the definition of a function.
  • Some participants express uncertainty about what information regarding Hermite polynomials can be used in their discussions.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the clarity of Liboff's explanations or the derivation of the equations involving Hermite functions. Multiple viewpoints and uncertainties remain regarding the mathematical relationships and the use of Hermite polynomials.

Contextual Notes

Participants note potential errors in the text and express uncertainty about the assumptions and definitions related to Hermite polynomials, which may affect their understanding of the problems presented.

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
 
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Liboff question

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


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 - a lot of errors I think. He also gets into some stuff which is pretty specialized especially in the problems.

Thanks

Pete
 
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.
 


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
 


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
 
exp(-(z+a)^2) = exp(-z^2 - 2az - a^2)
= exp(-z^2) exp(2z(-a) - (-a)^2)
 
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
 
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
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
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


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


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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