Learning Gaussian Integrals for Quantum Mechanics

[Jimmy84]

Good evening

Im starting to learn quantum mechanics from Griffith's book however I am having problems when dealing with Gaussian integrals in the first chapter.
What book should I read in order to understand this subject? are there resources about gaussian integrals out there?

Thanks a lot.

 

[MarneMath]

Well, the Gaussian Integral is a fairly common thing. Have you tried looking it up in a mathematical physicist book? What exactly about it do you not understand?

 

[Jimmy84]

Im being asked to find this integral

 

[Jimmy84]

I already know how to find the integral of e^-x^2 but I would like to improve my knowledge of this kind of integrals.

 

[Jimmy84]

I haven't seen a book about mathematical physicists that covers that subject.

 

[Stephen Tashi]

Is what you want a collection of examples where the techique of "integration by substitution" is applied to covert Gaussian-like integrals to integrations of the standard Gaussian distribution?

I don't know of an entire book devoted to that topic.

 

[Jimmy84]

Any book touching the subject of gaussian integrals would be ok.

 

[Stephen Tashi]

I don't have Griffith's book so it is unclear to me if the topic of "Gaussian integrals" entails anything beyond exercises in integration by substitution. Are there examples of problems you want to solve that don't amount to integration by substitution?

 

[DeIdeal]

I don't have Griffith's book so it is unclear to me if the topic of "Gaussian integrals" entails anything beyond exercises in integration by substitution. Are there examples of problems you want to solve that don't amount to integration by substitution?

I don't have Griffiths' QM book either, but I could see it including things like the Schrödinger equation for the Gaussian wave packet, which would mean that while the integrals would still technically amount to substitutions, the subs would be complex, so OP could need material about contour integration as well, depending on what (s)he knows and wants to know.

If certain hand-waviness is not a problem, most Gaussian integrals in basic QM I've seen can be evaluated with naive use of substitutions and \intop_{-\infty}^{\infty}e^{-x^2}\mathrm{d}x=\sqrt{\pi}

It's hard to tell, but I'd use something like Arfken & Weber (or pretty much any books on mathematical methods for physicists, really) as a reference. Depends a lot on what OP actually needs.

 

[Jimmy84]

I don't have Griffiths' QM book either, but I could see it including things like the Schrödinger equation for the Gaussian wave packet, which would mean that while the integrals would still technically amount to substitutions, the subs would be complex, so OP could need material about contour integration as well, depending on what (s)he knows and wants to know.

If certain hand-waviness is not a problem, most Gaussian integrals in basic QM I've seen can be evaluated with naive use of substitutions and \intop_{-\infty}^{\infty}e^{-x^2}\mathrm{d}x=\sqrt{\pi}

It's hard to tell, but I'd use something like Arfken & Weber (or pretty much any books on mathematical methods for physicists, really) as a reference. Depends a lot on what OP actually needs.

Thanks, do you mean using as a reference the last chapter of Arfken & Weber ?
I just finished a course on complex variable and we had contour integration.

 

[DeIdeal]

I don't know about the chapter number, the edition I have (5th) doesn't deal with integrals at all in its final chapter, but the section on contour integration (chapter name is "Functions of complex variable") should have everything you might need when it comes to them, the method of steepest descent presented in the same chapter might also be useful. Fourier transformations are also often used in QM and they are in the "Integral transformations" chapter.

Now, Gaussian integrals themselves are mainly governed in the chapter "Gamma-factorial function", so if you just want to learn about them, that chapter should do fine. As the name suggests, it deals with things not directly related to Gaussian integrals, but you should be able to skip unnecessary parts. It seems to lack tables you might want to use once you've got the gist of the idea, but you can find those on Wikipedia, for example.

 

[Jimmy84]

I don't know about the chapter number, the edition I have (5th) doesn't deal with integrals at all in its final chapter, but the section on contour integration (chapter name is "Functions of complex variable") should have everything you might need when it comes to them, the method of steepest descent presented in the same chapter might also be useful. Fourier transformations are also often used in QM and they are in the "Integral transformations" chapter.

Now, Gaussian integrals themselves are mainly governed in the chapter "Gamma-factorial function", so if you just want to learn about them, that chapter should do fine. As the name suggests, it deals with things not directly related to Gaussian integrals, but you should be able to skip unnecessary parts. It seems to lack tables you might want to use once you've got the gist of the idea, but you can find those on Wikipedia, for example.

Thanks a lot, I ll try to review some complex integrals and residues in order to start with the "Gamma factorial function" chapter.