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Trace of operator with continuous spectrum

  1. Nov 6, 2013 #1
    Greetings,

    I must be missing something obvious but how is Tr{} defined exactly in case of contunuous spectrum operators? Everywhere I look I see it defined as a sum of [possibly infinite sequence of] eigenvalues. Is the following correct:
    Given [itex]Q = \int f(q) \left| q\right\rangle \left\langle q\right| dq[/itex], where [itex]\left\langle q' | q'' \right\rangle = \delta (q'-q'')[/itex], then [itex]Tr \{\rho Q \} = \int f(q) \left\langle q | \rho | q \right\rangle dq [/itex] ?

    Thanks, DK
     
  2. jcsd
  3. Nov 6, 2013 #2

    DrDu

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  4. Nov 6, 2013 #3
    The definition here (and elsewhere) is only for discrete sequences of basis vectors, not for continuum. Is it just a matter of replacing sums with integrals and that's it? What about convergence criteria and suchlike?
     
  5. Nov 6, 2013 #4

    kith

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    I would say yes. If the integral doesn't converge, the operator is simply not in the trace class.

    The density operator is in the trace class by definition. Since tr{AB}=tr{BA} => tr{[A,B]}=0 but [x,p]=i*hbar, at least xp and px can't be in the trace class.
     
  6. Nov 6, 2013 #5

    DrDu

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    A necessary condition for a hermitian operator to be trace class is that it has only a discrete spectrum.
     
  7. Nov 6, 2013 #6
    What about [itex]\rho = \int_0^1 |x\rangle \langle x | dx[/itex], where [itex]\langle x' |x'' \rangle = \delta(x'-x'')[/itex]? Is this a valid state operator? What about its spectrum?
     
  8. Nov 6, 2013 #7

    dextercioby

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    The trace of an operator which is not necassarily self-adjoint is defined only wrt a countable orthonormal set in a Hilbert space. There's always a self-adjoint operator for which this set is its (true) eigenvectors set.
     
  9. Nov 6, 2013 #8

    rubi

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    The definition also works in case of an uncountable basis. In this case, one can show that at most countably many terms of the sum can be non-zero (otherwise, the sum diverges).
     
  10. Nov 6, 2013 #9
    Thanks for your replies but I am even more confused now.
    Can someone please show me (or point me to) an example of state operator ρ for a spin-0 particle in free space which corresponds to a gaussian wavepacket?
     
  11. Nov 6, 2013 #10

    bhobba

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    Dextercioby gave the correct answer IMHO.

    To the original poster this is a VERY mathematically non trivial issue (when you hear those words the translation is its HARD).

    The correct way to handle continuous spectrums, while it can be done the way Von Neumann did in his classic Mathematical Foundations using Stieltjes integrals and so called resolutions of the identity, is not the way its usually done in physics, and you need what are called Rigged Hilbert Spaces:
    http://arxiv.org/pdf/quant-ph/0502053v1.pdf

    But my advice is, if you are just starting out in QM, is not to get caught up in this stuff. I did, and it lead me on a sojourn into some very esoteric areas of pure math. I emerged with a good understanding of this stuff, but its not really germane to the physics which is what you should be concentrating on at the start.

    Thanks
    Bill
     
    Last edited: Nov 7, 2013
  12. Nov 6, 2013 #11

    kith

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    A wavepacket corresponds to a pure state so the state operator is simply ρ=|ψ><ψ|.
     
  13. Nov 6, 2013 #12
    Bill,

    It's actually my third attempt. First time was long long time ago in a galaxy far far away and I skipped most of the lectures as I had more important stuff to occupy myself with :) Second time I skimmed through some introductory texts and then Griffith but I took a great deal of it for granted and it didn't sink in.
    This time, following suggestions on this forum and yours in particular, I'm going thought the Ballentine text pretending I forgot everything I read on the subject before. I do not mind gory details, in fact I'm looking for them.

    Ballentine starts with postulating the existence of a state operator ρ with Tr{ρ} = 1. Nothing at all is said about the vector space on which the operator is hmm... operating, apart from it is being some sort of [rigged] Hilbert space. OK. Then at some point positional operator Q is introduced as Q|ψ> = x|ψ>. This makes ψ a continuous function of x: ψ=ψ(x). I take it as a hint that we are now operating in a space of functions of x with inner product defined as [itex]\langle \phi | \psi \rangle = \int \phi^* (x) \psi(x) dx[/itex]. Hence the question how do I calculate trace for an operator on this space.

    Can you be more specific? For a simple wavepacket [itex]|\psi \rangle = e^{-x^2/2a}[/itex], and [itex]\rho | \phi \rangle = (|\psi \rangle \langle \psi|) | \phi \rangle = e^{-x^2/2a} \int \phi(x') e^{-x'^2/2a} dx' [/itex], right? How do I apply textbook definition of Tr{ρ} to it, where do I get the required countable orthonormal basis [itex]|e_k\rangle[/itex] for the space I'm in?
     
  14. Nov 6, 2013 #13

    kith

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    As I wrote in my first post, I would simply use the integral for the trace. tr{ρ} = tr{|ψ><ψ|} = ∫dx<x|ψ><ψ|x> = ∫dx|ψ(x)|² = 1 if ψ(x) is normalized.

    Obviously, the integral doesn't converge if ψ(x) is not a square-integrable function. So the correct Hilbert space is L². The states |x> above are not basis states of this space, they do not even belong to it. So there's no contradiction to what DrDu and rubi said.
     
    Last edited: Nov 6, 2013
  15. Nov 6, 2013 #14

    rubi

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    You just pick some random orthonormal basis for your Hilbert space. In the case of ##\mathcal H = L^2(\mathbb R)##, you can choose the harmonic oscillator eigenstates ##\left|n\right>## for example. Then you just compute ##\mathrm{Tr}\rho = \sum_n \left<n\right|\rho\left|n\right> = \sum_n \left<n|\psi\right>\left<\psi|n\right> = \sum_n \left|\left<n|\psi\right>\right|^2##. The trace is independent of the basis you choose.
     
  16. Nov 7, 2013 #15

    bhobba

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    Well since the test space of the Gelfland triple has a basis of elements for the Hilbert space one can define it in the usual way ie Ʃ <bi|A|bi> which is well defined since A can be applied to the |bi>.

    There is no problem there.

    What you may be worried about is expressions like <x|X|x'> where the state x is represented by the Dirac delta function and you end up with the dreaded square of the Delta function ie ∫ δ (x'-x'') X δ (x - x'') dx''. Making things like this rigorous is far from trivial and you will have to consult advanced treatments on it. The following Phd thesis is the standard treatment:
    http://physics.lamar.edu/rafa/webdis.pdf [Broken]

    The reason you haven't got a straight forward answer, other than what I said above, which is likely to be unsatisfactory to you, is because such doesn't exist - it's a very difficult issue.

    Thanks
    Bill
     
    Last edited by a moderator: May 6, 2017
  17. Nov 7, 2013 #16
    OK thanks, I think I got it (sort of). I'm trying to make sense of this quote:
    So, in the expression <Q> = Tr{ρQ}, ρ=ƩAk|Ek><Ek| is over L2(R) with discrete basis |Ek> but Q=∫x|x><x|dx is over Ω× with continuous basis |x>, do I get this right?

    To compute the trace I can write Tr{ρQ}=ƩAk<Ek|Q|Ek>, then express each |Ek> as an integral in terms of |x> and then massage it until I get Tr{ρQ}=∫x<x|ρ|x>dx. Is this what happens when Ballentine computes the trace in the unnumbered expression just before (2.29)?
     
  18. Nov 7, 2013 #17

    DrDu

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    I think rigorous texts are avoiding to use the trace. However, this is quite a world on it's own.
    The standard method to obtain representations of operators for mixed states is the GNS construction:
    http://en.wikipedia.org/wiki/GNS_construction
    For thermal states, which are maybe of most interest, the Kubo Martin Schwinger construction is standard:
    http://en.wikipedia.org/wiki/KMS_state

    There are quite some text available explaining this C* algebraic approach to QM.
     
  19. Nov 7, 2013 #18

    bhobba

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    Indeed.

    I suspect the trace form of the Born rule still holds even for continuous observables, but its trickier. It cant involve the form I posted previously because if x=x' you get the square of the delta function which is a massive can of worms.

    Conceptually I think it's best to think of such cases as a discrete dust and take limits. Stuff inside the limit cause all sorts of issues like the square of the delta function but outside is OK.

    Thanks
    Bill
     
  20. Nov 7, 2013 #19

    DrDu

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    Nevertheless this KMS condition forms the basis e.g. for the Matsubara formalism in QFT which is quite standard.
     
  21. Nov 7, 2013 #20

    kith

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    I don't know the details of this rigged Hilbert space business but isn't the main result that all pathological functions like exp(ikx) or δ(x) behave nicely if we simply replace H by the extended space Ω× in the formalism? We just have to keep in mind, that state functions need to lie in the Hilbert space which is a subspace of Ω×. If the |x> are eigenstates of the self-adjoint operator X, they should form a basis of Ω×. I don't see a problem with using these states in the trace directly, so I guess this is what Ballentine did in the equation you are asking about. But maybe there are some mathematical subtleties I'm missing.
     
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