What Book Best Explains Fourier Transforms for Quantum Mechanics?

[Avatrin]

Hi

To properly understand introductory quantum mechanics, I want to understand what the Fourier transform actually gives me mathematically. What book do you recommend? I found one book, but it doesn't get to Fourier transformations until after seven long chapters. Is that what I have to expect?

In case you are wondering what I already know. Here it is:
Single- and multivariable calculus
Linear algebra
Complex analysis (with some topology of complex spaces)
Real analysis (metric spaces, measure theory, Lebesgue integration, some functional analysis)
Abstract algebra (group, ring and field theory)

I just need to understand what the solution of the Schrodingers equation for the free particle actually tells me (for which I have to use the Fourier integral). The "intuitive" explanations are not satisfactory for me.

 

[DrClaude]

You can have a look at a book such as Mary Boas' Mathematical Methods in the Physical Sciences. For a more indepth view, you need to look at harmonic analysis. A good resource is the classic book by Lanczos, Applied Analysis. It's chapter IV in that book, but you have enough backgroud to start reading from there (and the first three chapters are not especially relevant).