Axiomatic derivation of [X,P]?

Gerenuk

How do you derive what quantum mechanical momentum is, from some axioms about reality?
Therefore how do you justify one of the following more or less equivalent statements:
[X,P]=ih
<x|p>=exp(ikx)
psi(x,k)=exp(ikx)

I've seen argumentations why quantum mechanics is set up with the mathematical framework it has (Hilbert space etc.), but no an explanation for momentum.

strangerep

It comes from translations that we can perform to move from x="here" to x="there".

Less flippantly, one can start with the Galilean group of transformations (for the nonrelativistic case), and note that the probabilities we measure in experiments are invariant under these transformations.

Ballentine's textbook does a reasonably good job of developing (nonrelativistic) QM from this perspective.

(For the relativistic case, start with the Poincare group instead.)

Gerenuk

Sounds good! I'll surely get this book.
Could you give me a hint, why translation operators and momentum in reality as we know it are related?

Translations probably yield the above form, but where does reality turn into translations? And can I derive classical mechanics from translations as well?

Now I even wonder... what is the essence of momentum anyway? Is it more than just postulating that it is conserved?

meopemuk

The operator of momentum P is *defined* as a generator of space translations. Then the translation by a finite distance [itex]a [/itex] is represented by the exponential function of the generator [itex] e^{\frac{i}{\hbar}Pa} [/itex]. The action of this transformation on the position operator X should result in a shift

[tex] e^{-\frac{i}{\hbar}Pa} X e^{\frac{i}{\hbar}Pa} = X-a [/tex]

From this postulate you can get the desired commutator

[tex] [X,P] = i \hbar [/tex]

Eugene.

espen180

This may be interesting:
http://www.hep.upenn.edu/~rreece/doc...esentation.pdf
The equivalence you mentioned is arrived at midway.

Gerenuk

OK, from these ideas I understand why translation operations give the quantum mechanics form of momentum.
So how are translation generators now consistent with the momentum we learned at school?

espen180

I think that requires some knowledge about Lie Groups. However, I found this short explanation. I haven't looked through it properly, but I looks like it answers your question.
http://www.dfcd.net/articles/momentumop.pdf

dextercioby

[x,p]=ihbar 1 is the cornerstone of quantum mechanics. It's a postulate, unless one starts off with the Galilei group. So axiomatic derivation would mean to go along Ballentine's arguments and refine them using functional and harmonic analysis.

strangerep

Ballentine also covers this. In sect 3.3 he covers the generators of the Galilei group, one of which is the P mentioned above, but at that point its meaning is merely geometrical, and not yet identified with the "momentum" concept from classical dynamics.

Then in sect 3.4 he proceeds to identify these generators the dynamical variables from classical mechanics, using the postulate that the dynamics of a free particle are invariant under the full Galilei group.

(Sorry, but the entire argument is too long to reproduce here. Maybe Amazon or Google Books will let you read some of those sections of Ballentine.)

Gerenuk

It's OK. I'll get the book, so no need to reproduce what is well written somewhere else ;)

So what is actually the definition of the classical momentum? I don't know what it means (yet), but you say momentum is a complete and direct consequence of the Galilei group, which itself is more or less simple translations?

For the relativistic case I do the same argument with the Poincare group?

Does Ballentine give all the neccessary maths I need to understand full how that emerges?

A more philosophical question:
Does all this treatment mean, that everything has to be a particle and not some sort of field?

Isn't it that when you have invariants, it can mean that your coordinate system is overdetermined? As an example take x, y, z coordinates for a sphere which are overdetermined and contraints, in contrast to euler angles. Does such an idea of other coordinate exists where the Galilei invariance is a natural consequence of the mathematical representation?

strangerep

For the free nonrelativistic case, we just use p = mv = m dq/dt. To get a
velocity operator from Galilean generators, we just use
[tex]
V = i[H,Q]
[/tex]
(since d/dt of an operator corresponds to commutation with the Hamiltonian H).
But this is a bit of an oversimplification.

For less trivial interacting systems, the distinction between "position" and
"momentum" gets a bit blurred -- if you've done any Hamiltonian dynamics
perhaps you've heard of canonical transformations which mix position and
momentum, but preserve Hamilton's dynamical equations? But I was only talking
about the free case here.

In the relativistic case, things are trickier since there's no position
operator in the basic algebra. One can be built up, but not for all
combinations of mass and spin. E.g., a position operator for the photon
remains problematic to this day.

But the basic idea is the same: determine the unitary irreducible
representations of the Poincare group. That's the "Wignerian" approach.

It's not a maths textbook, but anyone with reasonable proficiency in calculus
should be able to cope. He covers the basics of Lie group ideas when
introducing the Galilei group, but some prior exposure to group theory is
never wasted. Since I don't know what your math background is, I can't be more
specific.

Anyway, you can always ask here on PF if you get stuck. :-)

No. We're really finding representations of Lie algebras as operators on a
Hilbert space. In some case these representations turn out to be
finite-dimensional, others infinite-dimensional. Some of the latter are field
theories. Fields can have momentum too... :-)

In the group theoretic development of quantum theory we proceed from the other
direction. One finds a maximal set of commuting operators within the algebra
of dynamical variables, which includes the invariants (called Casimirs) and
one other. Analysis of the spectrum of these operators determines the
necessary dimension of the Hilbert space.

Gerenuk

Oh, it gets more and more interesting and I suppose it's time for me to study books! I've ordered Ballentine, yet I believe it won't outline all ideas you have presented here?!

I have a very good understanding of undergrad physics and math and a few more topics I looked up for interest. I don't know Lie groups and I had only one simple course on group theory.
Could you recommend books I should read to fully understand the answers you have given? It should be a "physicists" book, by which I mean I care about most steps of derivations, but not about mathematical details (convergence, abstract definitions, etc.)
If you give several suggestions, I will pick the most clear for me from the library.

Maybe one more question:
Are dimensions of space in any way predicted by these methods?

strangerep

Start with Ballentine's "QM - A Modern Development", and see whether his treatment of Lie groups via the example of the Galilei group is sufficient for you to get through the rest of the book.

If not, or maybe later, you might take a look at Greiner & Muller, "QM - Symmetries".

If you mean the 3+1 dimensionality of space-time, then no (afaik).

naffin

The first thing Ballentine says about a particle with no internal degrees of freedom (pag.80) is that the operators Q,P form an irreducible set.
I think it is a crucial point, but he doesn't say why we are assuming this.
What exactly means that Q and P are irreducible?
We know only that Q is a position operator and P is the generator of spatial translations. We don't know anything about compatible observables.

Gerenuk

A revivial of the thread... Meanwhile I've had a look at Ballentine and I'm quite pleased with that type of derivation :) However, I'd find it better to see it more general and rigorous derivation (but still understandable for a physicist :) ). Perhaps even one for the Poincare group. Maybe I can grab hold of the other book.

Even better if the derivation was in terms of the density matrix. Could I avoid treating the wavefunction and it's phase invariance this way??

strangerep

I mentioned Greiner & Muller only because it contains some introductory material on Lie groups in a physics-relevant context. I don't think you'll find the same kind of derivation of ##[X,P]## there, nor much about Poincare.

No. The central element in the quantum Galilei algebra (which later gets identified with mass) is a consequence of the requirement that probabilities be invariant. For some groups it can be disposed of by redefining some generators, but not in the Galilei case.

BTW, if you're now interested in delving a bit deeper into this question, you might consider exactly what's going on a bit later in Ballentine when he deals with the external field case. It involves a modification of the Hamiltonian, and some other adjustments.

This leads to a more general perspective that what we're doing is simply finding a quantization of a particular dynamics (meaning that we take classical dynamical variables with their Poisson algebra, and attempt to represent them as operators on a Hilbert space). This yields additional insight into the free case: instead of starting with static Euclidean 3-space, plus time, and then invoking free Newtonian dynamics to identify the geometric quantities with dynamical ones, we could just as well start with the dynamical equations of a free particle and try to find the largest group of transformations that preserve these equations. One finds the Galilei group as a subgroup. Of course, this is to be expected since our intuitive picture of 3D space is constructed by extrapolation of the motion of free particles... :-)

Separately, (and I don't know how much you follow other threads here), but I was recently made aware of how the central-extended Galilei algebra (with ##[X,P] \propto 1##) can be regarded as a consequence of relativistic Poincare invariance -- when one takes the nonrelativistic limit of low velocity. See Kaiser: http://arxiv.org/abs/0910.0352 , sect 4.2, esp p95 et seq. I'm starting to prefer Kaiser's explanation over the standard one.

strangerep

Did you read Ballentine's appendices A & B ?

Complaints about mis-sequencing on this point have been made before: proper understanding of appendix B relies on having studied at least some of ch 4 -- which comes later than p80.

At that point, we know at least that Q and P are linearly independent generators. To go from this to irreducibility, one must understand that irreducibility means that no subspace of the space of states is left invariant by these operators. But detailed understanding of this point needs material on wave functions in the subsequent ch4. Herein lies, perhaps, a weakness in this sequence of presentation -- but a more advanced approach based on general quantization of Hamiltonian dynamics would be too difficult at that stage for the intended readership, imho.