Mathmetics in Introductory Quantum Mechanics book

In summary: In this case, the book Introductory quantum mechanics, written by liboff, recommends studying the integrals [Intergral(-inf to +inf) e^(-x^2)dx, Intergral(-inf to +inf) (x^2) e^(-x^2)dx, and other many integrals] and the Gaussian integrals [http://en.wikipedia.org/wiki/Gaussian_integral], which are used extensively in quantum mechanics.
  • #1
rar0308
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I'm reading Introductory quantum mechanics written by liboff.
When I solve problems, I stuck with calculation such as Intergral(-inf to +inf) e^(-x^2)dx, Intergral(-inf to +inf) (x^2) e^(-x^2)dx, and other many integrals.
I studied thomas' calculus but I think I haven't seen these in the book. So can't do it. Are there math books about these integrals? What math subject is related to these (high level?)integrals?
 
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  • #2
These are examples of Gaussian integrals:

http://en.wikipedia.org/wiki/Gaussian_integral

I don't recall seeing them in any of my undergraduate math courses. I think most physics students (in the USA at least) learn about them in their physics courses.
 
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  • #3
<Intergral(-inf to +inf) e^(x^2)dx> and the other one. I think you meant them with a minus e^(-x^2).

It depends on the curricula. An advanced course of calculus is normally taken before quantum mechanics or statistical mechanics.
 
  • #4
The first one, Integral(-inf to +inf) e^(x^2)dx, is not in Thomas because it does not converge. But [itex]\int_{-infty}^\infty e^{-x^2}dx[/itex], as dextercioby suggests, certainly is in Thomas, and the second is a variation. You may be trying to find an anti-derivative formula and not finding that- neither integrand has an "elementary" anti-derivative. Both are used extensively in probability ([itex]e^{-x^2}[/itex] is the "bell shaped curve") and so in quantum mechanics.

Here is a simple way to get the first integral:
Let [itex]I= \int_{-\infty}^\infty e^{-x^2}dx[/itex]. Since the integrand is symmetric about x=0, we also have [itex]I/2= \int_0^\infty e^{-x^2}dx[/itex].

And, we can write [itex]I/2= \int_0^\infty e^{-y^2}dy[/itex]. Multiplying those together, [itex]I^2/4= \left(\int_0^\infty e^{-x^2}dx\right)\left(\int_0^\infty e^{-y^2}dy\right)[/itex]. By Fubini's theorem, we can write that product of integrals as a double integral:
[tex]I^2/4= \int_{x= 0}^{\infty}\int_{y=0}^\infty e^{-(x^2+ y^2}dydx[/tex]

Now, change to polar coordinates: [itex]x^2+ y^2= r^2cos^2(\theta)+ r^2sin^2(\theta)= r^2 and [itex]dydx= r drd\theta[/itex]. The area of integration, with both x going from 0 to infinity is the first quadrant. In polar coordinates, r goes from 0 to infinity while [itex]\theta[/itex] goes from 0 to [itex]\pi/2[/itex]. The integral becomes
[tex]I^2/4= \int_{\theta= 0}^{\pi/2}\int_{r=0}^\infty e^{-r^2} rdrd\theta= \frac{\pi}{2}\int_0^\infty e^{-r^2} rdr[/tex]

That extra 'r' in the integrand now allows us to make the change of variable [itex]u= r^2[/itex] so [itex]du= 2r dr[/itex] and the integral becomes
[tex]I^2/4= \frac{\pi}{4}\int_0^\infty e^{-u}du[/tex]
which is easy.

It was necessary, to make that change to polar coordinates, that the integral be over the entire first quadrant. Again, that function, [itex]e^{-x^2}[/itex], has no elementary anti-derivative. In fact, its anti-derivative is typically written "erf(x)", the "error function", and it values are got by a numerical integration.
 
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  • #5
Thank you very much for your helps and I'm glad to finally locate it.
Trying to find that integral, I have flipped over pages of thomas so many times.
Even I borrowed from library advanced calculus written by bucks. I found gamma function and Integral(-inf to +inf) e^(-x^2)dx in this book. Now I'm about to read both of them. Thanks again.
P.S. I'm curious about how to input mathematical notations at the post.
 
  • #6
rar0308 said:
I'm curious about how to input mathematical notations at the post.

https://www.physicsforums.com/showthread.php?t=386951 [Broken]
 
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  • #7
For the records, just because it's <Introductory> QM, it doesn't mean it uses simple mathematics. That's why serious professors always provide their students with the so-called <prerequisites> before attempting any of their courses. As anybody should know, complex analysis and multi-variable differential and integral calculus are pre-requisites for a quantum mechanics course.

Of course, books don't have prerequisites, but merely opening one should get you informed on the necessary mathematics you need to comprehend its content.
 

1. What is the purpose of studying mathematics in an introductory quantum mechanics book?

The purpose of studying mathematics in an introductory quantum mechanics book is to provide a foundation for understanding the mathematical principles and concepts that underlie quantum mechanics. These mathematical tools are essential for solving problems and making predictions in quantum mechanics.

2. What types of mathematics are typically covered in an introductory quantum mechanics book?

Introductory quantum mechanics books typically cover topics such as linear algebra, complex numbers, differential equations, and calculus. These mathematical concepts are essential for understanding the principles of quantum mechanics and for solving problems related to quantum mechanics.

3. How important is a strong mathematical background for understanding quantum mechanics?

A strong mathematical background is crucial for understanding quantum mechanics. The principles and concepts in quantum mechanics are described using mathematical equations, and a good understanding of mathematics is necessary for interpreting and using these equations effectively.

4. Are there any specific mathematical prerequisites for studying introductory quantum mechanics?

While there are no specific mathematical prerequisites for studying introductory quantum mechanics, a solid foundation in basic mathematics, including algebra, geometry, and trigonometry, is highly recommended. Additionally, a good understanding of calculus and linear algebra can also be beneficial.

5. How can I improve my mathematical skills for studying introductory quantum mechanics?

To improve your mathematical skills for studying introductory quantum mechanics, it is recommended to practice solving problems and familiarize yourself with the mathematical concepts covered in the book. Additionally, seeking help from a tutor or joining a study group can also be beneficial in improving your understanding of mathematics in the context of quantum mechanics.

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