Position basis in Quantum Mechanics

[pabloweigandt]

TL;DR Summary

Can I conceive a countable position basis in Quantum Mechanics? How can I talk about the position basis in the separable Hilbert space?

Can I conceive a countable position basis in Quantum Mechanics? How can I talk about the position basis in the separable Hilbert space?

 

[andresB]

I'm not sure to understand the question, but no. The position eigenkets are not countable, they are not contained in the Hilbert space of state vectors, yet they are a basis for a separable Hilbert space.

From an informal point of view, as long as we can write normalizable vectors as a linear combination of position kets, why care if they are countable or not?

From a formal point of view, this issue is treated rigorously in the theory of rigged Hilbert spaces, a formalism that not many physicist care about.

 

[pabloweigandt]

If they are a basis for a separable Hilbert space, how come they are not countable? I was thinking that maybe you can generate all the uncountable position eigenkets with a countable basis. And also I am trying to understand how it is possible to avoid rigged Hilbert spaces to treat QM.

Separability is supposed to be very important in QM.

About the informality, how can I explain the usage of distributions like the delta functional if we don't include the rigged Hilbert space from the beginning? And even in an informal manner, you can't just replace summations with integrations and delta Kroneckers with delta functional and so on.

 

[andresB]

And also I am trying to understand how it is possible to avoid rigged Hilbert spaces to treat QM.

I would say that some presentations of the mathematics of QM are convincing enough that most people don't need to go beyond them (I like Messiah's for example).

If you prefer complete rigor, then feel free to fully go into functional analysis, distribution theory, and rigged Hilbert spaces. But I have to point out that it is like saying "how can you use calculus to solve problems in electromagnetism if you avoid the gory details of mathematical analysis"

Separability is supposed to be very important in QM.

The space of allowed states is indeed separable. Position kets do not belong in the Hilbert space.

 

[Demystifier]

If they are a basis for a separable Hilbert space, how come they are not countable?

A good analogy to have in mind is the relation between countable and uncountable sets in pure math. A separable countable basis is like the set of rational numbers, while the uncountable position basis is like the set of real numbers. Your question is analogous to the following one: How come that any rational number can be represented by a real number? The answer should be easy.

 

[martinbn]

Think about Frourier series and transform. For $L^2(\mathbb R/\mathbb Z)$, the functions $e^{inx}$ form a basis and any function can be expressed in terms of them $f(x)=\sum c_ne^{inx}$. For $L^2(\mathbb R)$, which is also seperable, you have analogously $e^{i\lambda x}$, which are uncountable and not in the space, but every function is still expressable as $f(x)=\int \hat {f}(\lambda)e^{i\lambda x}d\lambda$.

 

[vanhees71]

If they are a basis for a separable Hilbert space, how come they are not countable? I was thinking that maybe you can generate all the uncountable position eigenkets with a countable basis. And also I am trying to understand how it is possible to avoid rigged Hilbert spaces to treat QM.

Separability is supposed to be very important in QM.

About the informality, how can I explain the usage of distributions like the delta functional if we don't include the rigged Hilbert space from the beginning? And even in an informal manner, you can't just replace summations with integrations and delta Kroneckers with delta functional and so on.

You are right, the position basis is not a basis in the strict sense of Hilbert-space theory. For a separable Hilbert space the bases are all countable. An important example in quantum mechanics is the energy eigenbasis of the harmonic oscillator.

The "position eigenbasis", as physicists call it, consists indeed not of square integrable functions but are distributions. Indeed in position representation the "generalized position eigenfunctions" are Dirac-$\delta$ distributions,
$$u_{x'}(x)=\delta(x-x').$$
For simplicity here I assume motion in one spatial dimension (the generalization to motion in arbitrary dimensions is obvious).

You should note that unbound self-adjoint operators on a Hilbert space are only defined on a dense subspace of Hilbert space. The generalized eigenfunctions for values in the continuous spectrum are distributions defining the dual space of this dense subspace, which is "larger" than the Hilbert space itself. The dual of the Hilbert space is isomorphic to the Hilbert space itself.

Of course you can live completely without the idea of the rigged Hilbert space. Then a rigorous treatment of (non-relativistic first-quantization) QT is pretty clumsy. A famous treatment is in the famous book by von Neumann. The rigged-Hilbert space formalism on the other hand is very convenient, because it makes the "robust mathematics of physicists", introduced by Dirac, rigorous. Here a good intro is the PhD thesis

http://galaxy.cs.lamar.edu/~rafaelm/webdis.pdf

 

[jbergman]

Hall's book, Quantum Theory for the Mathematician, goes over this in detail, especially in chapter 6 and 7, without the rigged hilbert space formalism.

In short, he defines projection operators for measurable subsets and then develops a "functional calculus" for multiplication operators that essentially reproduces the results if we treat the position delta functions as an eigenbasis.

It's a bit technical and messy, but rigorous.

 

[mattt]

Any good book about functional analysis will explain the gory details (Brian Hall, Valter Moretti, Walter Rudin, Eberhard Zeidler, Kadison-Ringrose, Shlomo Sternberg, Reed-Simon, Gerald Teschl...).