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naima
Gold Member
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another thing.
When we use inner product states must belong to the HS.
HE> does but is it true for TE> ?
When we use inner product states must belong to the HS.
HE> does but is it true for TE> ?
Yes. The range of an unbounded operator is a subset of the Hilbert space.naima said:When we use inner product states must belong to the HS.
HE> does but is it true for TE> ?
tom.stoer said:Fredrik,
if you restrict to finite-norm states in L² you can solve the square well and the harmonic oscillator, but nothing else - not even the free particle case!
tom.stoer said:@Fredrik:
I hope I can explain the problem with x and p:
[tex]1=\langle p|p\rangle=-i\langle p|iI|p\rangle=-i\langle p|[x,p]|p\rangle=-ip\langle p|(x-x)|p\rangle=0[/tex]
Suppose that [x,p]=iI, and that p has an eigenvector.
...
There's nothing wrong with this calculation, so there must be something wrong with the assumptions that went into it. ... The two assumptions can't both be true, so either p doesn't have an eigenvector, or there's no x such that [x,p]=iI.
The problem is obviously the first equation
[tex]1=\langle p|p\rangle[/tex]
You want to show that something is wrong with the second assumption [x,p] but you plug this into an expression that is already ill-defined from the very beginning.
The result of your reasoning is not that [x,p]=iI is wrong, but that p does not have a normalizable eigenvector. For [T,H]=iI this means that you have not shown that T does not exist, but that something may be wrong with |E> being normalizable. That's the reason I am insisting in not sandwiching the operators if you can't be sure that both H and T have discrete spectra.
[you can check this using x, p=-id/dx and plane wave states; the problem is not the commutator but the normalization of the plane wave states]
Can you show me an example where <p|p>=1 holds but where <p|[x,p]p> is ill-defined?bigubau said:It's not <p,p>=1 that is wrong. Under certain conditions, it's right. ... The eigenvectors - when members of the H space - are not in the domain of the commutator.
I would say that the answer is clearly no. If we need it, then the theory based on RHS must be experimentally distinguishable from the one based on Hilbert spaces. (I would take that as the definition of "need"). The RHS just gives us a different way to express the same ideas.bigubau said:It's not on whether we need RHS for QM (the answer is yes),
As I said in at least one of my previous posts, it can't be wrong to "sandwich" the identity operator between two state vectors, because the identity operator is defined on the entire Hilbert space. If the commutator isn't, then it isn't a constant times the identity operator.bigubau said:It's not <p,p>=1 that is wrong. Under certain conditions, it's right. What's wrong in the flow of equalities invoked by Fredrik is the fact that he sandwiched the commutator between the eigenvectors of p. The eigenvectors- when memebers of the H space (RHS left aside)- are not in the domain of the commutator.
tom.stoer said:Can you show me an example where <p|p>=1 holds but where <p|[x,p]p> is ill-defined?
I am referring to not normalizable states. The identity is defined on the entire Hilbert space, but the scalar product <.|.> isn't. In that case <.|1|.> isn't defined, either.Fredrik said:As I said in at least one of my previous posts, it can't be wrong to "sandwich" the identity operator between two state vectors, because the identity operator is defined on the entire Hilbert space.
Fredrik said:I would say that the answer is clearly no. If we need it, then the theory based on RHS must be experimentally distinguishable from the one based on Hilbert spaces. (I would take that as the definition of "need").
Fredrik said:The RHS just gives us a different way to express the same ideas.
Fredrik said:As I said in at least one of my previous posts, it can't be wrong to "sandwich" the identity operator between two state vectors, because the identity operator is defined on the entire Hilbert space. If the commutator isn't, then it isn't a constant times the identity operator.
tom.stoer said:I am referring to not normalizable states. The identity is defined on the entire Hilbert space, but the scalar product <.|.> isn't. In that case <.|1|.> isn't defined, either.
I think we really have a problem with the wording. Let's make a specific example: plane wave states on the real line. Clearly <p|p> does NOT exist; it does not makes sense, even in a Gelfand triple. But of course plane waves are physically relevant.bigubau said:I don't agree with the wording but I got your point. The scalar product is defined on all the Hilbert space. For any vector p in H, <p,1p> is defined and can be made equal to the number 1. You must have meant that p is not normalizable, therefore lies outside the Hilbert space. So your last statement is true only if the 'sandwich' is made with vectors outside the Hilbert space, which would be nonsensical, of course.
So? We don't need to use Dirac's formalism. It's convenient, not necessary.bigubau said:I disagree. We need it for correctness purposes. Without RHS, Dirac's formalism would be full of hand-waving arguments.
I know, but the assumptions that went into the "proof" of 1=0 didn't specify that, so in that calculation it was the actual identity operator.bigubau said:The 1 in the i1 is indeed not the 1 on all the H space, but only a restriction of it to the domain of the commutator.
The inner product on a Hilbert space H is a function from H×H into ℂ that satisfies certain properties.tom.stoer said:The identity is defined on the entire Hilbert space, but the scalar product <.|.> isn't.
I don't know rigged Hilbert spaces well enough to know if this is correct. (I'm guessing that it is). This is why I wanted to keep the discussion about Hilbert spaces. In a Hilbert space, there's no such thing as |p>. In a RHS, |p> is (I think) an antilinear unbounded functional on the subspace of states that are in the domain of x and p, and is closed under a finite number of applications of x and/or p. (For example, if |u> is in that subspace, then pnxmpk|u> is too, for all integers n,m,k).tom.stoer said:Let's make a specific example: plane wave states on the real line. Clearly <p|p> does NOT exist; it does not makes sense, even in a Gelfand triple.
bigubau said:[...] an example is to be found in the article http://arxiv.org/PS_cache/quant-ph/pdf/9908/9908033v4.pdf Section 5, Page 460 from the Journal.
This is something that came up to my mind as well. In all cases you restrict H to something one could ask if H is too small or too big in order to solve a certain problem.strangerep said:This makes me think that Galapon's
example might be mathematically correct, but physically irrelevant.
strangerep said:Now let's consider what all this implies: the [tex]\psi(t)[/tex] in [tex]D_c[/tex] cannot
be sensibly expanded as a linear combination of H eigenvectors because none of
the latter are in that domain. IMHO this is quite a serious problem because such
superpositions are kinda fundamental to QM. This makes me think that Galapon's
example might be mathematically correct, but physically irrelevant.
bigubau said:I don't find it disturbing that the vectors in D_c are not physically relevant. The relevant ones, at least in the common framework quite widely accepted, are the eigenvectors of H.
tom.stoer said:I tried to answer a related question here, but I would like to come back to it.
Looking at x and p it's trivial to define T an H. We use the energy representation with
[tex]H|E\rangle = H|E\rangle[/tex]
and
[tex]\psi(E) = \lange E|\psi\rangle[/tex]
Then we can define
[tex]T = i\frac{\partial}{\partial E}[/tex]
All this can be constructed using well-known relations for x and p. The problem is that we want to relate H to some function on phase space, that means H = H[x,p].
So the question is: for which H[x,p] can one define T using the E-representation? Is this question reasonable? (what about the fact that this seems to fail for discrete E?)
tom.stoer said:[...] We use the energy representation with
[tex]H|E\rangle = H|E\rangle[/tex]
There was a typo in your latex. I guess you meant [tex] \psi(E) = \langle E|\psi\rangle [/tex]?and
[tex]\psi(E) = \lange E|\psi\rangle[/tex]
tom.stoer said:My intention was not to to use the t-representation b/c this is the new ingredient which should be understood based on something that is already known. So my idea was to write down some general formulae in the E- representation.
Then the idea is that a specific H is never given in terms of the t-rep. but in terms of x an p. So the idea should be to check whether for a given H(x,p) a derivation of an operator T in the E-rep. is possible (or if it's not possible, why the derivation fails). That seems to be a natural step b/c it starts with something that is well-known - both theoretically and experimentally.
I'll add the corrected formulas:
[tex]H|E\rangle = E|E\rangle[/tex]
[tex]\psi(E) = \langle E|\psi\rangle[/tex]
[tex]T = i\frac{\partial}{\partial E}[/tex]
strangerep said:[...] was it your intent that the energy spectrum is bounded below?
Sorry; I wasn't clear about that: my intention was to check whether for a given H(x,p) a derivation of an operator T in the E-rep. is possible (or if it's not possible, why the derivation fails); of course for a reasonable given H(x,p) that means that this H indeed is bounded from below.strangerep said:You didn't answer the last question in my post ...
tom.stoer said:Sh...
Yeah, typos; I am sorry. Unfortunately I can't fix it, the post is already locked against editing.
My intention was not to to use the t-representation b/c this is the new ingredient which should be understood based on something that is already known. So my idea was to write down some general formulae in the E- representation.
Then the idea is that a specific H is never given in terms of the t-rep. but in terms of x an p. So the idea should be to check whether for a given H(x,p) a derivation of an operator T in the E-rep. is possible (or if it's not possible, why the derivation fails). That seems to be a natural step b/c it starts with something that is well-known - both theoretically and experimentally.
I'll add the corrected formulas:
[tex]H|E\rangle = E|E\rangle[/tex]
[tex]\psi(E) = \langle E|\psi\rangle[/tex]
[tex]T = i\frac{\partial}{\partial E}[/tex]
I agree,additionalwork is required.A. Neumaier said:This works only if the spectrum of H is purely continuous ...
Of course there's more work tobe done for additonal quantum numbersA. Neumaier said:... and there are no additional quantum numbers, so that the eigenstateis uniquely determined by E.
Having an observable T on a non-dense set would be some progress. But there are physicalexamples w/o where an "observable" is not self-adjoint but only symmetric, i.e. for a particle on R+ with a boundary condition for u(x) at x=0: u(0)=0. The momentum operator -id/dx is only symmetric (as it has index 1), nevertheless I would say it's an observable.A. Neumaier said:Moreover, if H is bounded below (as for all physical Hamiltonians) then T is not self-adjoint, hence not a good observable.
tom.stoer said:there are physical examples w/o where an "observable" is not self-adjoint but only symmetric, i.e. for a particle on R+ with a boundary condition for u(x) at x=0: u(0)=0. The momentum operator -id/dx is only symmetric (as it has index 1), nevertheless I would say it's an observable.