Don't understand continuous basis

[Alesak]

Hi,
I'm beginning to learn QM, and I've never seen any treatment of vector spaces with infinite bases. Countable case is quite digestible, but uncountable just flies over my head.

Can anyone recommend me place where to learn this more advanced part of linear algebra, with focus on stuff needed for QM? I already know a lot of finite-dimensional linear algebra algebra, but I can't find anything that would skip this and go into this more intermediate-level stuff directly.

 

[dextercioby]

The standard advice is to take a university level class on introductory functional analysis. If this is out of your reach, then get your hands on a more mathematical textbook on QM such as the text by Galindo & Pascual.

 

[Fredrik]

I like Axler for linear algebra, and I think Kreyszig seems to be the best choice by far for functional analysis.

I haven't read Kreyszig myself, but I know how ridiculously hard some other books are. For example, Conway has a nice selection of topics, and it handles the stuff about orthonormal bases of infinite-dimensional Hilbert spaces better than any other book I've seen, but it's impossible to read that book unless you're already very good at topology. (Even if you are, you will probably find it very hard to follow his proofs). Kreyszig doesn't assume that you know topology already.

I should probably warn you that this stuff is very, very hard. If you can learn it (up to and including the spectral theorem for normal, not necessarily bounded operators) in less than a year, you're a lot smarter than me.

 

[Dickfore]

I would suggest you remember the following rules:
$$(e_{m} \vert e_n) = \delta_{m,n} \rightarrow (e_{\nu} \vert e_{\rho}) = \delta(\nu - \rho)$$
for the orthogonality, and:
$$\sum_{m}{\vert e_{m} )( e_{m} \vert} = 1 \rightarrow \int{d\nu \, \vert e_{\nu} )( e_{\nu} \vert} = 1$$
for the completeness relation.

The analogy between the two is similar to the analogy between Fourier series and Fourier transforms.

 

[Alesak]

The standard advice is to take a university level class on introductory functional analysis. If this is out of your reach, then get your hands on a more mathematical textbook on QM such as the text by Galindo & Pascual.

Unfortunately I study something quite far from physics, so no functional analysis course for me! Thanks for the reference.

I should probably warn you that this stuff is very, very hard. If you can learn it (up to and including the spectral theorem for normal, not necessarily bounded operators) in less than a year, you're a lot smarter than me.

Challange would be Accepted! if I had the time. Anyway, I'm reading Elementary Functional Analysis by MacCluer and after reading appendix(on measurable spaces) and first chapter(on Hilbert spaces), I have to say this little brochure is very good. It even has C* chapter! After I finish this, then maybe I'll attempt some horrible tome like Conway. They should bind such books in black leather.



A few other unrelated questions:
They say a quantum states exists in some vector space, regardles of coordinates, from which all relevant information can be extracted. To get predictions for some observable(hermitian operator), we put it into basis which consists of eigenfunctions of that operator. But, some operators seem to have countable basis(such as energy) and some other uncountable(such as position and momentum). How is that even possible?

I've hear expressions like "position space" and "momentum space". I suppose they are really the same space, only with different coordinates. Correct?

How much of other physics should I know for QM? I know pretty much only high school stuff right now. Eventualy I'd like to get into QFT and statistical mechanics.

 

[Fredrik]

But, some operators seem to have countable basis(such as energy) and some other uncountable(such as position and momentum). How is that even possible?

The position and momentum operators are unbounded operators, whose domains are actually dense proper subsets of the Hilbert space, rather than the entire space. These operators simply do not have any eigenvectors. If you want to understand why we can pretend that they do, the key term is "Rigged Hilbert space". This concept is discussed in other threads here. Some of them contain references. So you can start with a search.

The starting point of the Rigged Hilbert space idea goes like this: If you take an arbitrary square-integrable function ψ and have a position or momentum operator act on it, the result may not be square integrable. (This is why these operators can't have the entire space as their domains). However, there's a proper subspace K that consists of all square-integrable ψ such that no matter how many position and momentum operators you apply to it, the result is still square integrable. Now you can define two kinds of generalized state vectors: "bras" are linear functionals on K. "kets" are antilinear functionals on K. Somehow the set of kets will be large enough to contain kets that correspond to the original state vectors, as well as kets that we can think of as eigenvectors of position and momentum. (The same goes for the set of bras. The bras defined this way are only needed to make sense of bra-ket notation, as it appears in e.g. Sakurai).

Edit: I should also mention that a perfectly valid option is to not study this stuff. You will be able to understand QM quite well without knowing the rigorous mathematics.

I've hear expressions like "position space" and "momentum space". I suppose they are really the same space, only with different coordinates. Correct?

Wavefunctions are members of the semi-inner product space of square-integrable complex-valued functions on ℝ³. The wavefunction and its Fourier transform are sometimes referred to as the "position-space wavefunction" and the "momentum-space wavefunction" respectively.

How much of other physics should I know for QM? I know pretty much only high school stuff right now. Eventualy I'd like to get into QFT and statistical mechanics.

You can get away with knowing only a little. It helps to have a solid understanding of the basics of classical mechanics, just so you'll know what QM isn't.

 

[Alesak]

I got a small related question: In the book I'm reading(Shankar), he writes things like:

$<f|g>=\int^{\infty}_{-\infty}<f|x><x|g>dx=\int^{\infty}_{-\infty}f(x)^*g(x)dx$,

which is just the usual definition of dot product. Why does he bother using the completeness relation in the middle? Is that just for educational reasons, to indicate that |x> is basis?

Also, what is the completeness relation used for? It doesn't have a wikipedia page, so I'm lost... :)

Thanks

 

[Alesak]

Edit: I should also mention that a perfectly valid option is to not study this stuff. You will be able to understand QM quite well without knowing the rigorous mathematics.

It seems to me that the best way is to use the sloppy physicists' mathematics at first, then learn some QM and then learn the more advanced mathematics rigorously, with QM in mind.

 

[dextercioby]

I got a small related question: In the book I'm reading(Shankar), he writes things like:

$<f|g>=\int^{\infty}_{-\infty}<f|x><x|g>dx=\int^{\infty}_{-\infty}f(x)^*g(x)dx$,

which is just the usual definition of dot product. Why does he bother using the completeness relation in the middle? Is that just for educational reasons, to indicate that |x> is basis?

Also, what is the completeness relation used for? It doesn't have a wikipedia page, so I'm lost... :)

Thanks

Using the completeness relation for the operator x, he basically projected from an abstract space of state vectors (in which the kets and bras f,g live) onto the L^2 space of functions of variable x, aka wave functions in the position representation.

 

[Fredrik]

I got a small related question: In the book I'm reading(Shankar), he writes things like:

$<f|g>=\int^{\infty}_{-\infty}<f|x><x|g>dx=\int^{\infty}_{-\infty}f(x)^*g(x)dx$,

which is just the usual definition of dot product. Why does he bother using the completeness relation in the middle? Is that just for educational reasons, to indicate that |x> is basis?

Also, what is the completeness relation used for? It doesn't have a wikipedia page, so I'm lost... :)

Thanks

What you have on the left is the result of the bra <f| acting on the ket |g>, or equivalently, the inner product of the two kets |f> and |g>. What you have on the right is the inner product of the functions f and g defined by f(x)=<x|f> and g(x)=<x|g>. So you're dealing with two different inner products.

 

[mattt]

Hi,
I'm beginning to learn QM, and I've never seen any treatment of vector spaces with infinite bases. Countable case is quite digestible, but uncountable just flies over my head.

Can anyone recommend me place where to learn this more advanced part of linear algebra, with focus on stuff needed for QM? I already know a lot of finite-dimensional linear algebra algebra, but I can't find anything that would skip this and go into this more intermediate-level stuff directly.

I would recommend "Mathematical Foundations of Quantum Mechanics" ( Von Neumann, 1932 ).

The only problem may be his notation and nomenclature that is a bit dated.

If you want more modern books about the mathematics behind non-relativistic quantum mechanics I would recommend: "Functional Analysis" (Walter Rudin), "Fundamentals of the Theory of Operator Algebras" (Kadison and Ringrose, two volumes), "Methods of Modern Mathematical Physics" (Reed and Simon, four volumes).

In short: you choose a Separable Complex Hilbert Space. The Pure States of your system will be represented by (equivalence classes of) unit vectors. The Observables will be represented by Self-Adjoint Operators (Bounded or not Bounded) on this Complex Separable Hilbert Space. (The Domain of the not-Bounded ones will be a dense strict subspace).

To every Self-Adjoint Operator on a Separable Complex Hilbert Space corresponds a unique Spectral Resolution of the Identity. This unique Spectral Resolution of the Identity (for each Self-Adjoint Operator) gives you, for each unit vector state, a Probability Measure on the Borel Sets of the real line.

So, given ANY Borel Set of the real line, the Probability Measure of that Set (as I said, the Probability Measure given for each unit vector state, by the unique Spectral Resolution of the Identity that corresponds to each Self-Adjoint Operator) represents the relative frequency for an infinite number of identical systems identically prepared in the same state, for the experimental value when measuring the Observable (that corresponds to that Self-Adjoint Operator) to be in that given Borel Set.

So, mathematically speaking, you don't even need to talk about "bases" if you don't want to.

But if you are not already a mathematician, I would recommend you better to get used to the mnemotechnic bra-ket Dirac notation and rules, because it is much faster to more or less understand and start doing things with it.

After that, if you want to actually understand what is really happening behind many "apparent paradoxes" in Quantum Mechanics, then you need to understand it in a more rigorous mathematical way.