Solving Kevin's QM Mystery: Sketchy Eigenfunctions

[homology]

So What's the dealy-do with the eigenfunctions of the position operator x and the momentum operator p? As a blossoming mathematician the thought of using functions that don't even reside in the space as a basis gives me the chills. Moreover, not only using functions that are not normalizable, don't reside in the space, but the eigenfunctions of x aren't real valued functions at all, they're delta functions which are distributions.

I'm hammering my way through Von Neumann's text on the Mathematical Foundations of QM, but I wouldn't object to a little help right now. Can someone justify this for me?

Thanks,
Kevin

 

[Haelfix]

Get used to it, its ugly.

What you really should do if you're careful is to break up the functions into a limiting sequence, and that will converge into something well defined on your hilbert space.

In QFT its even worse, and we don't even talk about what 'space' things live in anymore, as its just too painful. Instead, we live with what we call normal ordering of operators, and *assume* that the object generated is well behaved (and they seem to be).

 

[homology]

I'm not sure that I should get used to it. As I said I'm working through Neumann's text and he claims, essentially, that he can work it all out without what he calls "mathematical fiction". I'm sure there has been a resolution to using eigenfunctions that don't even live in the space.

Now, it may be the case that physicists still use such notions because they represent a shorthand of a justified method, just like the delta function is used in integrals when, technically its a distribution and should only act upon schwarz functions, but nevertheless I'd like to know the resolution.

But you're right, its ugly and somewhat disturbing, all the more disturbing that more physics text don't point out the disturbing nature of it...but oh well.

Kevin

 

[Eye_in_the_Sky]

Now, it may be the case that physicists still use such notions because they represent a shorthand of a justified method, just like the delta function is used in integrals when, technically its a distribution and should only act upon schwarz functions, but nevertheless I'd like to know the resolution.

In that case, I quote for you what Robphy suggested to me in another thread.

In that case, I think the remarks on pages 46 and 47 of
http://www.math.sunysb.edu/~leontak/book.pdf
are relevant here.

If that's too big a leap, try starting at Chapter 2, page 35.

 

[Eye_in_the_Sky]

Taming the BEAST

So What's the dealy-do with the eigenfunctions of the position operator x and the momentum operator p? As a blossoming mathematician the thought of using functions that don't even reside in the space as a basis gives me the chills ... not only using functions that are not normalizable ... Can someone justify this for me?

Let |f> be a "true" resident of the space; i.e. <f|f> < infinity.

Let |x> and |p> be the infamous "chilling" nonresidents.

Write: <x|f> = f(x), a "tame creature".

Write: <p|f> = F(p), another "tame creature".

Now, enter the "danger zone" and calculate:

<p|x> = [2 pi h_bar]^-1/2 e^-ipx/h_bar , a "wild monster".

Now stick the number 1 = Integral { |x><x| dx } into the middle of <p|f> to change from one "wild-monster-basis" of "tame creatures" to another to get:

F(p) = <p|f> = <p| [Integral { |x><x| dx }] |f>

= Integral { <p|x><x|f> dx }

= [2 pi h_bar]^-1/2 Integral { e^-ipx/h_bar f(x) dx } .


... Why, it's just a little, itty-bitty "Furry"-transform.

 

[homology]

The problem is, and I do appreciate your reply Eye_in_the_Sky, is that the delta function cannot be integrated since it is not a function of a real variable. I'm afraid I'll just have wait out the following academic year in which I'll be taking measure theory and functional analysis to get to the bottom of this.

Again thanks,

Kevin

 

[Eye_in_the_Sky]

Yes, Homology I agree: with the currently available tools, we have basically reached the limits of what can be said. However, I still do feel compelled to try to "hammer in" this one last point ... which is really only going to be helpful if Fourier transforms are already in the "tool box".

The application of a Fourier transform followed by the inverse Fourier transform gives back the original function. Thus, this "double" transformation in effect "behaves" like a single Dirac delta-function integration, and hence, it gives a possible rigorous definition of the delta-function. Moreover, matching the Fourier "language" to that of QM, one sees that (modulo the appearance of some constants in trivial places) the "forward" transformation is accomplished by <p|x>, whereas the "inverse" transformation is accomplished by <x|p>.

(In my days and place of study, Fourier tranforms were already being "hung on my belt" at the time of first encountering "|x> and |p>".)

 

[homology]

Don't get me wrong Eye_in_the_Sky, I use delta functions in integrals, I differentiate them, and I've done quite a few Fourier transforms (though QM has brought new light on them, I have previously used them in PDEs) and 'proved' the common table entries (including integrating exp(ikx) over R). Of course I also have use delta functions in the physicist's manner so that I can work problems and build a better understanding of QM.

I was hoping that the 'answer' to my original question would be something I could grasp now. I see that it will take some time to reach that goal. For the present I appreciate your efforts and enjoyed your last few posts. They will not go by without some thought on my part.

Cheers,

Kevin

 

[Hurkyl]

Well, it pays to think a bit more abstractly here. The fact that wavefunctions are functions isn't that important; it's that they're vectors. The whole dealy is that you need to switch to a larger vector space!

Recall that it's never about how a wavefunction behaves at a point; it's about how it behaves with respect to dot products and the various operators.

 

[Haelfix]

Well, you can make things rigorous in one step by replacing the delta function with say a sequence of gaussians. But it doesn't buy you anything, except cluttering up the notation.

The real crux of the problem lies in the measure that you are considering.. In the case of vanilla quantum mechanics, you will find that the Wiener measure is applicable and basically is found to be rigorous. You can formulate all of quantum mechanics with it, and at the end of the day end up with something that looks and feels acceptable to any die hard mathematician.

This is not the case for relativistic quantum mechanics, and one of the main reasons why that theory is so ill defined and problematic.

 

[styler]

Oh homology,
i forgot to mention another cheaper, newer book.
Walter Thirring (a name you will become familar with if you study more mathematical physics) has a cool quantum book. Its pretty rigorous.

http://engineering-books-online.com/search_Walter_Thirring/searchBy_Author.html

You will also benefit from reading Geroch's tiny monograph "mathematical physics"
its little but very enthralling.

Geroch: Mathmatical physics

I read this cool little book ont he train on my way to and from school as an undergrad and it cleared up lots of stuff the physics books were talking about.

 

[homology]

Oh homology,
i forgot to mention another cheaper, newer book.
Walter Thirring (a name you will become familar with if you study more mathematical physics) has a cool quantum book. Its pretty rigorous.

http://engineering-books-online.com/search_Walter_Thirring/searchBy_Author.html
[\QUOTE]

Thanks, I'll check that out.

You will also benefit from reading Geroch's tiny monograph "mathematical physics"
its little but very enthralling.

Geroch: Mathmatical physics

I read this cool little book ont he train on my way to and from school as an undergrad and it cleared up lots of stuff the physics books were talking about.

Really? I've started the book and just haven't had the time to get very far into it. Furthermore, I've always found it kind of funny that a book that is basically about category theory is labelled "mathematical physics" but I'll give a second chance based on your recommendation.

Thanks,

Kevin

 

[styler]

geroch

I don't really think its about category theory it just uses functorial notions to explain some ideas between important objects that come up in physics.
But read it and make your own decisions.

 

[homology]

I don't really think its about category theory it just uses functorial notions to explain some ideas between important objects that come up in physics.
But read it and make your own decisions.

well, I hope we're discussing the same book: the one put out by university of chicago? Its red? that one is certainly about category theory, since they define a category on page 3.

Kevin

 

[styler]

i respectfully disagree...read the book

Yes, that 's the book.
Its not ABOUT category theory.
You're at berkelye right? (i'm jealous) They have the biggest group of foundations/metamatheamtics there of anywhere in the world. If you ask around you'll see that category is a lot more than the def of a category, a morphism and a functor.
The book USES category theory to make some basic associations.
Think of abook on quantum mechanics defining a hilbert space or a frechet space and spending some time developing various facts of functional analysis before it attempts to mke presentation about QM.
The totality of category theory in that book amounts to about 5 definitons and a dozen examples.

 

[homology]

Yes, that 's the book.
Its not ABOUT category theory.

Given the rest of your email, I'll concede your point, if you'll concede the perhaps the title is unusual given its strong use of category theory compared to other books on mathematical physics (which focus more on topology, differential forms and PDEs than the way they're all connected).

You're at berkelye right?

I wish! No, I'm at the university of maine, a place of little theorectical physics and even less related math. I should use the name Crusoe to indicate such isolation as I bear.

Think of abook on quantum mechanics defining a hilbert space or a frechet space and spending some time developing various facts of functional analysis before it attempts to mke presentation about QM.

Are you saying this should be the criteria by which i judge the suitability of a book? I'm just not sure what you're saying, exactly.

Well, I've done as much of Griffith's text as I care to and so I'm going to move on to Shankar, which is better written and much deeper. While he doesn't resolve the issues I have, I can at least ever refine my quantum until I have some measure theory and functional analysis under my belt (this fall!)
I'll definitely read through Geroch, given your high praise.

Thanks again styler

 

[styler]

All i meant by the quote about Qm and hilbert spaces was that it wasn't fair to say geroch was about categories cause it had some categorica stuff in itit just as you wouldn't say griffiths was a functional analysis book just cause it has a chapter on it.

I misunderstood you i thought you were the guy from Berkeley. Sorry i obviously can't read!

As for geroch, what i lied about it was that to many physics students

(and i was one of those first before i got frustruated and decided i needed to learn math ...which of course is s slippery spiral cause either you go into math or you realize that nobody knows enough math to teach you half the things you want to know about physics (you wouldn't believe the fundamental things done in physics so foten taken so for granted that NOBODY has ever justified; you always assume that it will be in some book somewhere til you start really asking...))

physics seems like it uses all sorts of unrelated math. Its nice to see some cohesion (of course if you ever do get into that foundations stuff you'll learn how many holes there are even in pure math, but that's another issue)
and by teachinga bout categories geroch is able to go through all sorts of stuff you'll need if you want to understand GR, QM, QFT and some things in modern theory.
Now that i know you're alone out there i feel for you. All i can say is if you are looking at math/phys i would make sure you learn a lot about
linear algebra and get a good working knowledge of the basics of differetial topology and functional analysis. If you get a clue about these geometry and global analysis will come easier.
Don't for a second underestimate the amount of linear algebra you need.
Many schools only offer two courses in it and rarely does the more advanced one get to the important stuff (triangulation, the spectral thm, theory of determinants, multilinear algebra, inavriant subspaces, numerical linear algebra...etc)
If you learn this crap so much of basic graduate level physics will become easier. I recall seeing so many smart physics students work the butts off to try to remember stuff they could have derived or understood if they knew more math...they spend so much time learning stupid differential equations...and not even in generality. So you will have to learn a lot of it on your own (or depending upon how ambitious and snakey you are...independent study its a great feature of the vestiges of libereal society that in some places you can still get credit for reading a book )

(see the thread on linear algebra starter books and there's lots of e book son the web too)

If you ever want more book reccomendations or course plans just ask.