What Hilbert space for a spinless particle?

[David Olivier]

What, if the particle you want to describe is located on the moon?

No problem. Coefficient $b$ can be arbitrarily small; it must just not be zero.

It might seem that the condition $\exists a>0, b>0, \forall x \in \mathbb R, |\psi(x)|^~ < a e^{- b x^2}$ gives a special role to the origin $x = 0$. But this is not so. It is equivalent to $\exists a>0, b>0, \forall x \in \mathbb R, |\psi(x)|^~ < a e^{- b (x-A)^2}$ for any real $A$. It is really only a condition about how the function behaves at infinity (+ or - infinity), which is what we need to ensure that it is quadratically integrable (and remains so after all iterations of $X$ and $P$).

 

[Physics Footnotes]

To get back to basics: I still don't know what the one-dimensional state space of a spinless particle can look like. Let's say we don't try to squeeze any $\delta$'s into it, nor any $e^{ikx}$'s. What do we have? The space of continuous $\mathbb R \to \mathbb C$ functions? That's certainly too large. We want the dot product, so we need the functions to be quadratically integrable. We also want $P = -i \frac \partial {\partial x}$ to be an endomorphism, hence they must be derivable. Actually, for this they must be $C^\infty$ (because the resulting function must satisfy the same conditions). For $X$ to be an endomorphism, we need not only our $\psi(x)$ to be quadratically integrable, but also $x \psi(x)$, $x^2 \psi(x)$ and so on. How do we impose this on our functions?

Short Answer: The space of admissible (pure) states of a spinless particle in one dimension (in the position representation) is taken to be the Hilbert Space $L^2[\mathbb R]$ of (Lebesgue) square-integrable functions from $\mathbb R$ into $\mathbb C$. (Strictly speaking there are some nuances about equivalence classes, but that would be a distraction from your real question.)

Longer Answer: The square-integrability requirement secures, as you might guess, the statistical interpretation of the wave function. The use of a fancy Lebesgue integral, as opposed to a more pedestrian Riemann one, ensures you get a well-behaved Hilbert Space (which is partly justified physically and partly mathematically).

The sorts of constraints you are playing around with are not at all necessary for a wave function to represent a plausible physical state, however they do become the sorts of things you have to think about when specifying the domains of individual operators.

The momentum operator $P=-i\frac{d}{dx}$ you mention, for example, is indeed not defined on the entire Hilbert Space (which not only contains non-smooth functions but also wildly discontinuous ones, by the way!). However, even the plain old position operator $Q=x$ is not defined everywhere on $L^2[\mathbb R]$, since multiplication by the independent variable will send some functions out of the Hilbert Space.

The main point I would make here though, is that the restriction of the domain of an operator corresponding to an observable is not due to certain states being unphysical (which seems to be the way you're thinking), but rather it is a necessary technical requirement to make the operator self-adjoint, which ensures it has the right measure-theoretic properties to yield consistent expectation values. And that, by the way, is a very subtle and difficult task; PhDs have been granted for working out the domain of self-adjointness of a single operator!

 

[David Olivier]

Thanks. I was not really trying to impose restrictions to ensure physicality; rather just trying to get things to work mathematically; and in particular ensure that the operators map the set to itself.

I'm not sure my solution works, though; the "Hilbert space" I described may not be complete.

 

[Physics Footnotes]

Thanks. I was not really trying to impose restrictions to ensure physicality; rather just trying to get things to work mathematically; and in particular ensure that the operators map the set to itself. I'm not sure my solution works, though; the "Hilbert space" I described may not be complete.

That's right. Sticking to well-behaved functions will not, in general, give you a Hilbert Space.

By the way, following on from what I explained above, the 'endomorphism' requirement you mention is correct, but only insofar as an operator must map functions in the Hilbert Space (linearly) to other functions in the Hilbert Space. When an operator is only defined on a proper subset $\mathcal D \subset \mathcal H$, it is perfectly ok if the operator projects wave functions out of $\mathcal D$, so long as they stay in $\mathcal H$. This happens especially when you start needing to impose boundary conditions on wave functions to make an operator self-adjoint.

Also, keep in mind that domains always matter when an operator is unbounded, since no such operator can be defined on the entire Hilbert Space, while remaining self-adjoint.

Again, if you like sorting out these kinds of subtleties, I think you'd enjoy von Neumann's book.

 

[DrDu]

No problem. Coefficient $b$ can be arbitrarily small; it must just not be zero.

When you consider a free particle in an initial wavefunction $\sim \exp(-bx^2)$, this wavefunction will remain Gaussian, but b will decrease with time t like 1/t . So for any fixed lower bound on b, an initially arbitrary localized wavefunction will violate it, for sufficiently long times. This is one of the fundamental observations to proove the validity of scattering theory.

 

[David Olivier]

When you consider a free particle in an initial wavefunction $\sim \exp(-bx^2)$

I was just giving an upper bound on the absolute value of the function; not saying it had to be a Gaussian!

 

[DrDu]

I think a Gaussian is a valid counterexample. And one counterexample is sufficient to disprove a theory.

 

[David Olivier]

I didn't specify either that $a$ and $b$ must be fixed for all eternity! Just that the objects in my toy (pre-?) Hilbert space $\mathcal H$ were functions of the $x$ spatial coordinate such that there exist, separately for each function, appropriate values $a$ and $b$.

If we start with a Gaussian, it will indeed spread out and we will have to choose smaller and smaller values for $b$. That is not a problem. It remains, at each instance, an element of $H$.

 

[bhobba]

Thanks. I was not really trying to impose restrictions to ensure physicality; rather just trying to get things to work mathematically; and in particular ensure that the operators map the set to itself. I'm not sure my solution works, though; the "Hilbert space" I described may not be complete.

I have to digress into RHS theory a bit which I didn't really want to do but here goes.

Take the physical states to be the space of all vectors of finite length. In practice, especially if you think of the output of measurements as some kind of digital readout, that's what will occcur in practice. Such is called a test space. Now for mathematical convenience define its dual - since this is an advanced level thread you can look up what the dual of a vector space is. It doesn't exist physically - but is useful mathematically. This contains all sorts of weird stuff eg the Hilbert space (the space of all vectors such that ∑|ai|^2 are finite) - in fact it contains any vector finite or infinite. Of course the Hilbert space is isomorphic to Lebesque square intergrable functions. Now let's take some subset of the Hilbert space which has very nice properties eg are good functions - again you can look them up but basically they are continuously differentiable and go to zero fast enough. They are members of the dual I talked about above - but again for mathematical convenience let's assume it as our actual physical space. We then do the same trick - take its dual and use it as our space. Its doesn't exist physically either - for example it contains e^ipx which does not go to zero at infinity like a reasonable function should - but just for mathematical convenience we will do that. That's all that's going on. You take some subset of a Hilbert space with nice properties and that is your test space. You then introduce its dual. This is codified in what's called a Gelfland triple D ⊃ H ⊃ T where T is our test space, H a Hilbert Space and D the dual of T. Again physically the space I talked about at the start is all that can physically exist, but it has proven mathematically useful to extend it to on non-physical states eg you always want to be able to take the derivative because that's part of the momentum operator. The best choice of T is part of the problem to be solved, but its usually good functions and its dual is the Schwartz space. The reason is the Fourier transform of a good function is also a good function. This means its trivial to define the Fourier transform of the dual of good functions - its <F(f)|g> = <f|F(g)>. You can then operate on functions like e^ipx , taking its Fourier transform and showing its the Dirac delta function which is the representation of a state with some exact value like position or momentum. Again - it doesn't exist physically - you can never get a 100% accurate position (again thinking in terms of digital readouts makes such clear) but is very useful in the actual math of QM - and other areas of applied math. This is why an understanding of distribution theory is really a must for any applied mathematician.

Please, please get a hold of the following book and study it - it doesn't give all the detail of RHS's but its a good start:
https://www.amazon.com/dp/0521558905/?tag=pfamazon01-20

All these ideas were worked out by three of the greatest mathematicians of the 20th century - Gelfland, Grothendieck and Schwartz to meet the criticisms of another great 20th century mathematician - Von-Neumann. Dirac's methods in his QM book were beautiful and were readily picked up by physicists, but Von-Neumann could not abide its lack of rigor and came up with his own equally valid, but mathematically rigorous method. It was not as beautiful as Dirac but the fact Dirac worked meant something was going on. These other great mathematicians sorted it out.

But it nowadays has escaped the confines of QM and is used in other areas like White Noise Theory:
http://www.asiapacific-mathnews.com/04/0404/0010_0013.pdf

Thanks
Bill

 

[bhobba]

Again, if you like sorting out these kinds of subtleties, I think you'd enjoy von Neumann's book.

Von Neumann's book is nice, and he in that book gives a big serve to Dirac. By all means give it a study - I have an its well worth it.

But the real answer is in RHS's which is deeper and even more powerful - plus is what physicists usually use even though they don't often realize it.

Ballentine has a nice little section explaining all this but it's far from an actual rigorous development.

The following is also good at your level and explains RHS's with greater detail, as well as Von-Neumann's approach:
https://www.amazon.com/dp/146147115X/?tag=pfamazon01-20

It discusses Schwartz spaces etc and it's relation to the key spectral theorem of Von-Neumann. It really intertwines the two approaches so you get a greater appreciation of what's going on. It extends considerably the comments of Ballentine, but I do suggest you study Ballentine first (for this purpose just the first 3 chapters). I have a copy of both and its well worth it. Then you can undertake Dr Madrid's PhD Thesis which is at a higher level of rigor again.

Thanks
Bill

 

[David Olivier]

Also, keep in mind that domains always matter when an operator is unbounded, since no such operator can be defined on the entire Hilbert Space, while remaining self-adjoint.

I don't even know about the issue of operators being bounded or not. All I learned of quantum mechanics (30 years ago...) was with bras and kets, and the adjoint of a linear mapping was defined through its matrix on an orthonormal basis ‑ the assumption being that whatever you could do in the finite dimensional case you could do just about equally with infinite matrices, and even with a continuous spectrum. There was no problem going from kets to bras and back again (that's how we would have the “ket $|x>$”). An operator was defined as any self-adjoint linear mapping. They never mentioned restricting domains.

I'm precisely trying to go beyond that approach, into more mathematically rigorous territory.

Again, if you like sorting out these kinds of subtleties, I think you'd enjoy von Neumann's book.

I may well get around to buying it (not cheap!). For the moment, I'm going to continue plowing through my R. Geroch, Mathematical Physics.
I'm only half-way through, and before I get to Hilbert spaces I have measure theory and Lebesgue integrals (which I never studied rigourously).

I've bitten off here more than I can chew for the moment, but I have benefited from this discussion ‑ I didn't know about the von Neumann approach at all, nor about the RHS. If I understand correctly:

• There was the quantum mechanics prehistory, with de Broglie and others calculating wave functions and concluding you needed a Hilbert space.
• Dirac came up with bras and kets, making everything look easy, papering over some nasty details.
• Von Neumann made it mathematically rigorous.
• The RHS came along, allowing it you to keep bras and kets and still be mathematically rigorous.

I'm not so sure I understand what RHS brought into the picture, but for I've already a lot on my agenda.

 

[David Olivier]

Thanks to you too, bhobba. I do understand duals very well. I understand distributions too, and I like them a lot, though lacking a good mastery of measures and integrals (Lebesgue style) I'm a bit more shaky on the subject.

If I understand you well, when you speak of $D \supseteq H \supseteq T$, the $T$ is of very well-behaved functions, for instance $C^\infty$ and bounded in some way ‑ my Geroch suggests having a compact support, for instance; but they don't make up a full Hilbert space. $H$ is the full Hilbert space, that might be $L^2[\mathbb R]$, with functions that are not derivable (as functions) and not square-integrable. Their derivatives (and results of other operations) land in $D$, a space of distributions.

 

[Physics Footnotes]

Dirac successfully unified both Schrödinger's and Heisenberg's approaches to Quantum Theory using a framework which, at the time, was referred to as Transformation Theory.

A mathematician, however, would think of Dirac's theory as mostly symbolic formalism rather than proper mathematics; that is, there are rules for manipulating symbols and then at the end of the day associating selected results with empirical statements. If a mathematician were to stand over Dirac's shoulder, however, and point to various symbols (e.g. an integral involving functions of bras and kets) and say "How is that integral defined exactly?", Dirac would have to respond with some serious hand-waving (or, perhaps, impatiently insist that you wait till the calculation is done, because that's all that matters.)

Von Neumann (judged by most peers to be the best mathematician in the world at the time) tried, but was unable, to make coherent sense of Dirac's formalism (in particular, his reliance on a notoriously self-contradictory Delta Function). He decided that the theory needed to be built in the language of 'operators', with eigenvectors playing a subsidiary role, and succeeded in formulating a mathematically rigorous formulation of QM which agreed perfectly with Dirac's in the finite dimensional case, but differed dramatically in the infinite dimensional case. In this latter sense, von Neumann did not (either in deed or intent) make Dirac's formalism rigorous. Indeed, in his own words:

"It should be emphasized that the correct structure need not consist in a mathematical refinement and explanation of the Dirac method, but rather that it requires a procedure differing from the very beginning, namely, the reliance on the Hilbert theory of operators."​

Rigged Hilbert Space (RHS) is a much more recent invention which aimed to put Dirac's formalism on solid footing in the infinite dimensional case. In doing so, it allows you to talk sensibly about Dirac delta functions by venturing outside the Hilbert space, into the world of distributions, and also sidesteps the subject of domains by doing operator algebra inside a (dense) subspace of the Hilbert space (the three spaces taken together being called a Gelfand Triple or RHS).

Somewhat paradoxically, the price one pays for this is an enormous increase in sophistication of the mathematical machinery needed to understand the theory; well beyond what you need to understand von Neumann's (admittedly often tedious) approach. Still, it shows that Dirac's approach can be made self-consistent.

 

[David Olivier]

Thanks. That makes it clear.

 

[bhobba]

Von Neumann (judged by most peers to be the best mathematician in the world at the time) tried, but was unable, to make coherent sense of Dirac's formalism (in particular, his reliance on a notoriously self-contradictory Delta Function)

Indeed, it took the combined work of other great mathematicians like Grothendieck (who vies with Von-Neumann as one of the greatest- certainly of the 20th centurt) to do it, and it did not come quickly shows just how hard the problem was. But it was worth it - not just for QM but for the other areas of applied math it had now influenced.

To the OP I think the book on QM for mathematicians would be of real value - as I said it compares both methods, something that no other book I know of does.

Its great you understand distribution theory - RHS's are often said to be Hilbert spaces with distribution theory stitched on. That's where the word Rigged comes from - that stitching on of distribution theory is like the rigging or scaffolding of a ship.

Thanks
Bill