Position Eigenstates: Measurement & Imperfect Performance

[Dale]

When a particle's position is measured, does the wavefunction collapse to the eigenstate of the measurement (a delta function), or do you account for the accuracy and precision of the measurement device by making the state be a mixture of eigenstates corresponding to the device's imperfect performance (e.g. a gaussian spike)?

 

[atyy]

Although loosely we say that after a position measurement, the state collapses into a position eigenstate, this is strictly speaking not true since the position eigenstate is unphysical (not normalizable). One way to treat this is to take the resolution of the detector into account.

An example of such a treatment is given in http://arxiv.org/abs/1211.4169 (section 4.1).

 

[Matterwave]

To me, the only sensible option is the latter. A delta function itself can not be physically achievable since its position uncertainty is 0, meaning its momentum uncertainty is infinite. An infinite momentum uncertainty is just as unphysical to me as an infinite position uncertainty. And no matter how much introductory physics books like to maintain that the "uncertainty principle is still maintained because you get a zero times infinity", I do not buy such arguments since they have no backing behind them to take this limit.

However, as I have not read any peer reviewed articles on this particular subject, I can't vouch for my answer.

 

[WannabeNewton]

When a particle's position is measured, does the wavefunction collapse to the eigenstate of the measurement (a delta function), or do you account for the accuracy and precision of the measurement device by making the state be a mixture of eigenstates corresponding to the device's imperfect performance (e.g. a gaussian spike)?

The measuring device is assumed to be ideal just as in classical mechanics.

 

[dextercioby]

There are valid formulations without collapse and never in the theory are measuring devices supposed to be imperfect.

 

[jostpuur]

I don't believe that a perfect answer to the question would exist because the usual philosophical problems (though sometimes of practical kind...) are related to this. However, I believe that a somekind of answer, which probably hints in a right direction, does exist, and it is that in a quantum mechanical measurement the state collapses through an orthogonal projection to a subspace spanned by some collection of the eigenvectors. In other words, the guess about Gaussian weight looks little suspicious, but if it was replaced by a characteristic function (indicator function), then I would say yes.

For example, suppose that some particle detecting camera has pixels, and that the pixels can be modeled as some sets \Omega_1,\Omega_2,\Omega_3,\ldots\subset{R}^3. They could be little cubes or balls, for example. Then the camera works so that it will tell if a particle was found in some pixel. If a particle was found in a pixel \Omega_k, in the measurement process the wave function was projected like this:

<br /> \psi \mapsto \chi_{\Omega_k}\psi<br />

where

<br /> \psi:\mathbb{R}^3\to\mathbb{C}<br />

is some wave function, and

<br /> \chi_{\Omega_k}:\mathbb{R}^3\to\{0,1\},\quad \chi_{\Omega_k}(x) = \left\{\begin{array}{c}<br /> 1,\quad &amp; x\in \Omega_k \\<br /> 0,\quad &amp; x\notin\Omega_k \\<br /> \end{array}\right.<br />

is the characteristic function corresponding to the spatial region of the pixel.

The usual explanation about the quantum mechanical collapse is that the state is projected like this:

<br /> |\psi\rangle \mapsto |a\rangle \langle a |\psi\rangle<br />

where |a\rangle is some normalized eigenstate. The normalization means \langle a&#039;|a\rangle = \delta_{aa&#039;} with a Kronecker delta. I think it is very reasonable to generalize this by stating that perhaps the collapse happens like this:

<br /> |\psi\rangle\mapsto \sum_{a\in\Omega} |a\rangle \langle a|\psi\rangle<br />

where the set \Omega is related to the practical resolution of the measurement device. Also I might ask that if some measurement device cannot separate some eigenvalues, which are too close to each other, what else would the collapse be? Then with observables of continuous spectra this is naturally interpreted as

<br /> |\psi\rangle\mapsto \int\limits_{\Omega}|x\rangle\langle x|\psi\rangle dx<br />

where the normalization is achieved with a Dirac delta: \langle x&#039;|x\rangle = \delta(x-x&#039;).

 

[tom.stoer]

... and no matter how much introductory physics books like to maintain that the "uncertainty principle is still maintained because you get a zero times infinity" ...

the well-know proof for the uncertainty relation does not work for position eigenstates b/c it uses expressions like |ψ|², <ψ|ψ> and similar expressions which are ill-defined for ψ(x) ~ δ(x); I do not know whether this proof can be generalized, e.g. using rigged Hilbert spaces

 

[atyy]

There are valid formulations without collapse and never in the theory are measuring devices supposed to be imperfect.

Do you mean something like the model in http://arxiv.org/abs/quant-ph/0307057 (section VII.B, Eq 310-338)?

 

[Dale]

Thanks everyone, this is helpful.

atyy, I think that I understood your point about position states not being normalizable, I had forgotten about that when I asked. Does the same thing apply for any non-quantized eigenvalues. E.g. if you are measuring energy for an unbound state (so that it is not quantized), presumably the same issue would arise that the detector would give a number with limited precision. Or does that problem only arise for position since the eigenstate is not a good function?

 

[USeptim]

If the particle has a delta-function position, the momenta would be spread over the infinite, when you integrate the square of this you will get an infinite kinetic energy which is un-physical.

 

[atyy]

atyy, I think that I understood your point about position states not being normalizable, I had forgotten about that when I asked. Does the same thing apply for any non-quantized eigenvalues. E.g. if you are measuring energy for an unbound state (so that it is not quantized), presumably the same issue would arise that the detector would give a number with limited precision. Or does that problem only arise for position since the eigenstate is not a good function?

In my post I was only thinking that the post-measurement state cannot be a position eigenstate because it is not normalizable and hence not in the Hilbert space, and not that position is a continuous variable.

In the first link I gave http://arxiv.org/abs/1211.4169 Distler and Paban say that when an observable has a continuous spectrum, the resolution of the detector must be specified as part of what it means to measure the variable. However, WannabeNewton and dextercioby seem to disagree with this, as I think http://arxiv.org/abs/quant-ph/0307057v1 Ozawa and http://arxiv.org/abs/0706.3526 Busch (citing Ozawa) also do. So I'd like to know whether WannabeNewton and dextercioby would agree with Ozawa and Busch, and disagree with Distler and Paban (or whether they agree with neither, and prefer some other formulation for continuous variables).

 

[Nugatory]

When a particle's position is measured, does the wavefunction collapse to the eigenstate of the measurement (a delta function), or do you account for the accuracy and precision of the measurement device by making the state be a mixture of eigenstates corresponding to the device's imperfect performance (e.g. a gaussian spike)?

For most problems it doesn't matter, as the forward evolution in time of the delta function leads immediately to more reasonable states (such as that gaussian spike). You don't have to ascribe physical significance to "the wave function at the exact moment of collapse" (phrasing it this way makes it easier to understand the question as an interpretational issue) to get sensible results for any time after that... and that delta function does make for a simple initial condition.

 

[vanhees71]

Despite the fact that I don't think that one needs any collapse postulate and that the minimal statistical interpretation of quantum theory is the only solid interpretation we have about this most successful theory invented by man so far, it's also mathematically clear that there are no position eigenvectors in the literal sense but only generalized ones in the sense of distribution theory. Since the spectrum of the position operator is continuous only, there is no true eigenvalue and no true eigenstate. This is made explicit in the position representation, where the generalized eigenvector is represented by the Dirac \delta distribution:
u_{\vec{x}&#039;}(\vec{x})=\delta^{(3)}(\vec{x}-\vec{x}&#039;).
This is not a square integrable function and thus cannot represent a physically realizable state for a particle.

This is also reflected in the already mentioned Heisenberg-Robertson uncertainty relation \Delta x_j \Delta p_k \geq \hbar/2 \delta_{jk}: There is no state, where the particle has sharp position or sharp momentum.

 

[strangerep]

When a particle's position is measured, does the wavefunction collapse to the eigenstate of the measurement (a delta function), [...]

First, one must let go of the standard "collapse" nonsense. Although it makes approximate sense (well, sort of) for filter-type measurements, it does not hold water in general. Think of a photon striking a detector screen: the photon is absorbed and does not even exist after the "measurement".

A more realistic approach is to model the measurement via an interaction between system and apparatus, such that the final state of the apparatus is correlated with the initial state of the system. See Ballentine ch9 for a discussion of this.

Regarding an observable with continuous spectrum, one can a construct suitable model using POVM (positive operator-valued measure). The resolution of unity provided by a complete set of (usually non-orthogonal) coherent states is an example of this sort of thing.

 

[Dale]

For most problems it doesn't matter, as the forward evolution in time of the delta function leads immediately to more reasonable states (such as that gaussian spike). You don't have to ascribe physical significance to "the wave function at the exact moment of collapse" (phrasing it this way makes it easier to understand the question as an interpretational issue) to get sensible results for any time after that... and that delta function does make for a simple initial condition.

This was particularly helpful.

 

[tom.stoer]

I think it makes no sense to "discuss away" the position eigenstates.

You do have this problem in QM simply b/c in practice you use position- or momentum eigenstates to do calculations. Yes, you may argue that a momentum measurement does not collapose your state to a plane wave; this is certainly correct. But fapp you use plane waves ;-)

 

[atyy]

Regarding an observable with continuous spectrum, one can a construct suitable model using POVM (positive operator-valued measure). The resolution of unity provided by a complete set of (usually non-orthogonal) coherent states is an example of this sort of thing.

Yes. If we want a concrete proposal to discuss, how about http://arxiv.org/abs/quant-ph/0307057? Ozawa proposes a measurement model for a POVM for sharp position in section VII.B, Eq 310-338, with the post-measurement state of the system given in Eq 325. As far as I can tell, the post-measurement state is not a delta function.

Busch http://arxiv.org/abs/0706.3526 agrees that Ozawa's measurement model does sharp position measurement, but Busch does not explicitly comment on the post-measurement state given by Ozawa.

There is an interesting comment in the discussion of Ozawa's paper. "(I) Can every measuring apparatus be described by an indirect measurement model? ... Although some measuring apparatuses, especially in the attempts for quantum nondemolition measurements [26], allow indirect measurement model descriptions, it is still difficult to convince any schools of measurement theory of the affirmative answer to question (I) above."

 

[vanhees71]

I think it makes no sense to "discuss away" the position eigenstates.

You do have this problem in QM simply b/c in practice you use position- or momentum eigenstates to do calculations. Yes, you may argue that a momentum measurement does not collapose your state to a plane wave; this is certainly correct. But fapp you use plane waves ;-)

I heavily disagree. There are no eigenstates for spectral values of a self-adjoint operator in the continuous part of the spectrum. That's a mathematical fact and has nothing to do with the physical interpretation of quantum theory. The statement that pure states are represented by rays defined by true normalizable Hilbert-space vectors must be taken seriously.

Of course, also "momentum eigenstates", i.e., "plane waves" in the position representation
u_{\vec{p}}(\vec{x})=\frac{1}{(\sqrt{2 \pi})^{3}} \exp(\mathrm{i} \vec{p} \cdot \vec{x})
are obviously no wave functions representing a pure state in quantum mechanics, because they are not normalizable.

They are "generalized" states, i.e., they belong to the dual of the nuclear space, which is an appropriate dense subspace, where the self-adjoint operators are defined. The dual of this smaller subspace of Hilbert space is always much larger and contains generalized functions (distributions).

The physicist's sloppy use of these concepts of the "rigged Hilbert space" works so well, because the Hilbert space is extremely nice to us ;-)). My math prof. used to say that the separable Hilbert space is so well-mannered that it is almost like a unitary space of finite dimension. The emphasis lays on "almost". A very nice paper on the fact, that sometimes you can get nonsense with the naive physicist's handling of these issues can be found here

http://arxiv.org/abs/quant-ph/9907069

An example, where any serious student should get worried once in his/her quantum mechanics lecture when it comes to the evaluation of cross sections from the S matrix. There, the physicists happily square the "momentum-conserving" \delta distribution and then discussing this mathematical nonsensical result away with a lot of handwaving (sometimes called "Fermi's 2nd trick"; nobody could tell me so far, what's Fermi's 1st trick then ;-)). Here, it's easily cured by using wave packets, i.e., true states, in the initial state, and this elucidates a lot the physical meaning of what a cross section is and how real-world scattering experiments are to be understood quantum mechanically. You find a very good explanation in the good old textbook by Messiah and (for the relativistic case) in Peskin/Schroeder, Intro to Quantum Field Theory.

Other sloppyness is not cured yet: There is no mathematical proof for the mathematical existence realistic relativistic interacting field theories like QED or the Standard Model although these are among the most successful theories concerning the agreement between theory and observations ;-)).

 

[tom.stoer]

I heavily disagree. There are no eigenstates for spectral values of a self-adjoint operator in the continuous part of the spectrum. That's a mathematical fact and has nothing to do with the physical interpretation of quantum theory.

I agree in principle, but I disagree in practice.

We use plane waves, that's a fact!

You are invited to re-write 99% of the QM literature.

 

[jostpuur]

First, one must let go of the standard "collapse" nonsense. Although it makes approximate sense (well, sort of) for filter-type measurements, it does not hold water in general. Think of a photon striking a detector screen: the photon is absorbed and does not even exist after the "measurement".

The standard collapse explanations are often too distant from reality. I'll put forward a real question of practical relevance here: We assume that a time evolution of a wave function is known under the assumption, that there is nothing on its way (nothing that would cause a wave function collapse). The wave function is

<br /> \psi:[0,\infty[\times\mathbb{R}^3\to\mathbb{C},\quad (t,x)\mapsto \psi(t,x)<br />

We assume that this is a non-relativistic case. We can also assume that at time t=0 we have \langle x\rangle = 0 and \langle p\rangle = (0,0,p_3) with p_3&gt;0 so we know roughly where the packet is going. Then we modify the situation by placing a two dimensional plate \mathbb{R}^2\times \{R\} on the way with R&gt;0, and we assume that this plate functions as a measurement device. It absorbs the hitting particle, and informs us about the hitting point. What is the probability density

<br /> \rho:\mathbb{R}^2\to [0,\infty[<br />

that describes the particle's hitting point? What principles imply what this probability density ends up being?

A more realistic approach is to model the measurement via an interaction between system and apparatus, such that the final state of the apparatus is correlated with the initial state of the system. See Ballentine ch9 for a discussion of this.

Does Ballentine answer my question about the probability density on a two dimensional detector plate?

 

[bhobba]

The standard collapse explanations are often too distant from reality.

The point of what Strangerep and Vanhees said is that the formalism QM does not have collapse. That is an interpretational thing.

Does Ballentine answer my question about the probability density on a two dimensional detector plate?

That's trivial.

Its a two dimensional wavefunction, so you take the absolute value squared it to get the probability density.

Here is a correct QM analysis of the situation you are getting at - in one dimension - but easily extended to two
http://arxiv.org/ftp/quant-ph/papers/0703/0703126.pdf

But before going any further you need to understand what a wave-function is and the exact postulates that go into QM.

For that you need to read the first 3 chapters of Ballentine.

Since you are a math type Gleason's theorem may also help in understanding the underlying principles:
http://kof.physto.se/cond_mat_page/theses/helena-master.pdf

Basically if you require non contextuality Gleason's theorem shows Born's rule is the only measure that can be defined on a Hilbert space.

For a simpler more modern version see post 137:
https://www.physicsforums.com/showthread.php?t=763139&page=8

Thanks
Bill

 

[strangerep]

[...] What principles imply what this probability density ends up being?

The best reference for that (afaik) is Mandel & Wolf.

There was a long thread about detector clicks and single photons in which more precise reference within M+W were given. See in particular my posts #7 and #73 -- but the whole thread is probably worth (re-)reading.

BTW, (Bhobba), I don't think this is trivial. M+W spend a lot of time on this sort of thing. Especially, the extra detailed stuff in ch14, iirc. I think I'll augment my signature line to include a mention of M+W.

 

[jostpuur]

That's trivial.

Its a two dimensional wavefunction, so you take the absolute value squared it to get the probability density.

The wave function has the form \psi:[0,\infty[\times\mathbb{R}^3\to\mathbb{C}, and the probability density \rho:\mathbb{R}^2\to [0,\infty[. It makes no sense to "set" \rho=|\psi|^2.

 

[tom.stoer]

The wave function has the form \psi:[0,\infty[\times\mathbb{R}^3\to\mathbb{C}, and the probability density \rho:\mathbb{R}^2\to [0,\infty[. It makes no sense to "set" \rho=|\psi|^2.

Both the wave function and the probability density depend on time; so it's simply

\rho(x,t) = |\psi(x,t)|^2;\;\forall x \in \mathbb{R}^3\,\wedge\,\forall t \in \mathbb{R}^3

If the wave function is a time-dependent wave packet, then the probability density is time-dependent as well. Associated with this time-dependent probability density you get a time-dependent probability current as well.

This is standard textbook QM (which is the translation of bhobba's "trivial)

 

[jostpuur]

Recall this:

That is what the observations look like. If you want a theory to describe that, you need a theory that gives some probability density \rho:\mathbb{R}^2\to [0,\infty[. Your probability density \rho:\mathbb{R}^4\to [0,\infty[ is something else, and doesn't look very useful.

 

[bhobba]

It makes no sense to "set" \rho=|\psi|^2.

Gleason say's otherwise.

Thanks
Bill

 

[bhobba]

If you want a theory to describe that, you need a theory that gives some probability density \rho:\mathbb{R}^2\to [0,\infty[.

Again - see Gleason.

You may not have seen the Born Rule (that's basically the squaring rule for probabilities from the wave function) in the form of given an observation, O, the expected value of the observation is E(O) = Trace (PO) where P is a positive operator of unit trace, by definition called the state of the system.

A few points here. First, note the state of the system is in general an operator not an element of a vector space. A state of the form |u><u| is called pure. A convex sum of pure states is called mixed. It can be shown all states are mixed or pure. The |u> in the pure state can be mapped to a vector space - but not uniquely ie |u><u| = |cu><cu| for any complex number c.

When people speak of states from a vector space they mean pure states. For pure states its simple to see E(O) = <u|O|u>. Now the observable of position that gives one for position x and zero otherwise is O = |x><x|. Then of course E(O) = probability of getting x. And from that its again easy to see E(O) = <u|x><x|u> = |<u|x>|^2 which is how the squaring rule comes into it.

Thanks
Bill

 

[bhobba]

(which is the translation of bhobba's "trivial)

Indeed it is.

But on reflection I am actually going to side with Strangerep on this.

On the surface it looks 'trivial' from the viewpoint of the formalism of QM ie its a simple expression of the Born rule - but a closer look showes it is a bit more subtle.

Thanks
Bill

 

[jostpuur]

The wave function has the form \psi:[0,\infty[\times\mathbb{R}^3\to\mathbb{C}, and the probability density \rho:\mathbb{R}^2\to [0,\infty[. It makes no sense to "set" \rho=|\psi|^2.

Gleason say's otherwise.

You must be stating this accidentally, while not having read my posts properly.

If the psi has the form \psi:\mathbb{R}^4\to\mathbb{C}, and if the rho was defined by \rho=|\psi|^2, then also the rho would have the form \rho:\mathbb{R}^4\to [0,\infty[. However, the rho, about which I have been asking about, has the form \rho:\mathbb{R}^2\to [0,\infty[. Therefore it makes no sense to attempt to define it by \rho=|\psi|^2.

 

[bhobba]

If the psi has the form \psi:\mathbb{R}^4\to\mathbb{C}

Where do you get R^4 from - you talked about R^3 previoiusly?

So let's be careful about what it is. ψ(x,t) is the expansion of the state in terms of position x at a particular time t. As I explained |ψ(x,t)|^2 gives the probability of observing the particle at position x and time t.

But no, I read it correctly. Gleason explains that.

I suggest you study it.

The essence is non-contextuality.

Specifically the wavefunction gives the probability of observing a particle at a point. You place a screen there and the points in space you are observing are those on the screen. That is the observation you are making so its the observable that applies. The wave function at other points would be the probability if the screen was there - it isn't - so its simply a by-product of the calculation.

For the precise analysis based on that see the link I gave previously:
http://arxiv.org/ftp/quant-ph/papers/0703/0703126.pdf

If you want to discuss this situation can you please reference the link above as it gives the correct explanation.

Thanks
Bill

 

[dextercioby]

You must be stating this accidentally, while not having read my posts properly.

If the psi has the form \psi:\mathbb{R}^4\to\mathbb{C}, and if the rho was defined by \rho=|\psi|^2, then also the rho would have the form \rho:\mathbb{R}^4\to [0,\infty[. However, the rho, about which I have been asking about, has the form \rho:\mathbb{R}^2\to [0,\infty[. Therefore it makes no sense to attempt to define it by \rho=|\psi|^2.

What is \rho ?

 

[Jano L.]

For most problems it doesn't matter, as the forward evolution in time of the delta function leads immediately to more reasonable states (such as that gaussian spike). You don't have to ascribe physical significance to "the wave function at the exact moment of collapse" (phrasing it this way makes it easier to understand the question as an interpretational issue) to get sensible results for any time after that... and that delta function does make for a simple initial condition.

This was particularly helpful.

Actually, time evolution of delta distribution leads to oscillating function of position and time $G(x,t)$ whose modulus squared is the same everywhere and decreases in time as $1/t$, which does not conform to the Born interpretation of the regular wave functions $\psi$. $G$ is just a propagator of the Schroedinger equation (a kernel function in an integral giving the evolution operator).

 

[atyy]

I've been trying to understand Ozawa's model for a sharp position measurement and state reduction. Although at least the measurement part is agreed on by Busch, it is very formal that I don't know whether there is consensus about whether sharp position measurements are possible.

I have found another reference supporting the claim of Distler and Paban that sharp position measurements are not possible.

Caves & Milburn, Quantum-mechanical model for continuous position measurements, Phys Rev A, 1987
"In the standard formalism of quantum mechanics the change in the state of a system produced by an instantaneous, precise measurement may be calculated by using projection operators. In the case of position measurements such an approach is inappropriate. Formally this is because there are no normalized position eigenstates. Physically it is because an arbitrarily precise measurement of position requires arbitrarily strong coupling to the system and an arbitrarily large amount of energy. Thus in the case of position (or momentum) measurements one must generalize the standard formalism to include imprecision in the measurements. ..."

 

[Dale]

Thus in the case of position (or momentum) measurements one must generalize the standard formalism to include imprecision in the measurements. ..."

I assume that this imprecision is not from the uncertainty principle, but is an additional imprecision above and beyond that.

 

[atyy]

I assume that this imprecision is not from the uncertainty principle, but is an additional imprecision above and beyond that.

Yes, in Caves and Milburn's and Distler and Paban's papers, the additional imprecision is due to the measuring device.

Having glanced at Caves and Milburn's paper, what they do actually seems very similar to Ozawa. They have a measurement apparatus which interacts briefly with the system, and the give an explicit Hamiltonian for the interaction. After the interaction, they do a projective measurement of the position of the apparatus.

What I wasn't sure about was whether the projective measurement of the apparatus is justified. Caves and Milburn explain that although it is impossible, taking the projective measurement of position on the apparatus instead of the system itself is enough to introduce noise into the measurement, so they allow themselves a projective measurement on the apparatus as an approximation.

Interestingly, Wiseman makes a similar comment in http://arxiv.org/abs/quant-ph/0302080 "If one attempts to apply a projection postulate directly to the atom, one will obtain nonsensical predictions. However, it can be shown that assuming a projective measurement of the field will yield results negligibly different from those assuming a projective measurement at any later stage, because of the rapid decoherence of macroscopic objects [48]. For this reason, it is sufficient to consider the field as being measured. Because the field has interacted with the system, their quantum states are entangled. If the initial state of the field is assumed known (which is not unreasonable in practice, because it is often close to the vacuum state), then the projective measurement of the field results in a measurement of the atom. Such a measurement however is not projective. Instead, we need a more general formalism to describe such measurements."

So I think within a formalism that allows projective measurements of position on the apparatus but not the system, Ozawa does correctly present a Hamiltonian that will execute sharp position measurements, and I think his state reduction to a wave function that isn't a delta function is probably also correct.

At any rate, it is definitely correct that a delta function is not a physical state. But it seems much more involved to answer the question about what the post measurement state is (when there is one), and one way of doing that seems to be to explicitly propose a measurement model and see what happens.

 

[tom.stoer]

I assume that this imprecision is not from the uncertainty principle, ...

As I said, the proof of the uncertainty principle fails for generalized states

the well-know proof for the uncertainty relation does not work for position eigenstates b/c it uses expressions like |ψ|², <ψ|ψ> and similar expressions which are ill-defined for ψ(x) ~ δ(x); I do not know whether this proof can be generalized, e.g. using rigged Hilbert spaces

 

[vanhees71]

I cannot repeat this often enough: The uncertainty principles holds for all states a quantum system can have. Generalized eigenstates for spectral values of self-adjoint operators in the continuous part of the spectrum do not represent physical states of the system!

 

[Jano L.]

I cannot repeat this often enough: The uncertainty principles holds for all states a quantum system can have. Generalized eigenstates for spectral values of self-adjoint operators in the continuous part of the spectrum do not represent physical states of the system!

I think everybody here agrees with that. The sad thing is Dirac made the impression that $|x\rangle$ (or $|p\rangle$) is just another ket - that is to be interpreted as a "state of the quantum system". I am afraid most teachers and textbooks still leave this impression on the students.

 

[tom.stoer]

You cannot ignore that fact that these states are used in calculations! My statement is not about "states a system can have" but about calculations we make!

 

[atyy]

What about the square-root of a delta function? It's discussed at the start of Gagen et al, Physical Review A, 48, 132-142 (1993) http://www.millitangent.org/pubs/q_meas/08_double_well_zeno.pdf.

Is the square-root of a delta function a position eigenstate, or is only the delta function a position eigenstate (but not in the Hilbert space)?

Is the square-root of a delta function normalizable?

I found another thread on this https://www.physicsforums.com/showthread.php?t=344902.

 

[atyy]

There are valid formulations without collapse and never in the theory are measuring devices supposed to be imperfect.

If I understand dextercioby correctly, maybe this is the most elegant solution - to deny that the question makes sense. One way of getting rid of collapse, even with Copenhagen is to couple the system to measuring ancillae, and push all measurements (something that yields a "classical" definite outcome) to the end of the experiment ("Principle of Deferred Measurement" http://en.wikipedia.org/wiki/Deferred_Measurement_Principle). Since there is only one measurement, there is no need for a post-measurement state.

There are two disadvantages, but I think they are not relevant for this question. The first is that measurement cannot be used as state preparation, but this can be argued to be not a general principle. The second is that in relativity, if a set of measurements is simultaneous in one frame, they will not be simultaneous in another frame (unless we also add that spatially separated measurements are impossible, ie. we do the measurements at the end of the experiment in the same place at the same time, so no Bell tests are allowed). However, there is no relativistic position operator, so that solves the problem, ie. the question of an exact position measurement only makes sense in non-relativistic quantum theory.

Within this framework of deferred measurements, although there is no post-measurement state, there is a post-interaction state. I think Ozawa's Hamiltonian for a sharp position measurement allows that calculation with nothing that is equivalent to a collapse to a delta function. (I use the term "equivalent" because Ozawa's original presentation doesn't use state reduction as a fundamental postulate, but I think the postulate can be derived from his equations).

 

[jostpuur]

What is \rho ?

I explained it in post #20 first. In post #25 I linked a related picture, which cannot be missed when glancing through the thread.

What is the probability density

<br /> \rho:\mathbb{R}^2\to [0,\infty[<br />

that describes the particle's hitting point? What principles imply what this probability density ends up being?

Here I put forward an interesting and relevant question, and it still remains unanswered. Is this because the physicists of modern times do not know the answer to this extremely basic question of the most basic quantum mechanics?

The wave function has the form \psi:[0,\infty[\times\mathbb{R}^3\to\mathbb{C}, and the probability density \rho:\mathbb{R}^2\to [0,\infty[. It makes no sense to "set" \rho=|\psi|^2.

Both the wave function and the probability density depend on time; so it's simply

\rho(x,t) = |\psi(x,t)|^2;\;\forall x \in \mathbb{R}^3\,\wedge\,\forall t \in \mathbb{R}

If the wave function is a time-dependent wave packet, then the probability density is time-dependent as well.

If the psi has the form \psi:\mathbb{R}^4\to\mathbb{C}

Where do you get R^4 from - you talked about R^3 previously?

You are deliberately not understanding what I write. This is probably because you don't want to admit to yourself that you don't know the answer to my question.

 

[vanhees71]

You cannot ignore that fact that these states are used in calculations! My statement is not about "states a system can have" but about calculations we make!

Of course not. I love the \delta distribution, particularly when I have to do an integral over it ;-)).

Nevertheless you must keep in mind what's representing something in the real world you describe in physics, and a \delta distribution or a plane-wave solution of the momentum-eigenvalue problem do not represent a pure state in quantum mechanics. If you assume this, you run into many contradictions within the theory itself, and if you claim that such generalized eigenvectors represent a state you should also be able to describe how you prepare it (at least in principle) for particles in the real world. I don't see, how you would do that.

 

[vanhees71]

What about the square-root of a delta function? It's discussed at the start of Gagen et al, Physical Review A, 48, 132-142 (1993) http://www.millitangent.org/pubs/q_meas/08_double_well_zeno.pdf.

Is the square-root of a delta function a position eigenstate, or is only the delta function a position eigenstate (but not in the Hilbert space)?

Is the square-root of a delta function normalizable?

I found another thread on this https://www.physicsforums.com/showthread.php?t=344902.

Where precisely is the square root of a \delta distribution (NOT FUNCTION!) described in that paper? This has no sense in standard distribution theory.

Even if you can define something like this in a reasonable way by extending distribution theory somehow, the \delta distribution would not be normalizable, because the integral over a \delta distribution without multiplying it with an appropriate test function doesn't make sense either.

The easiest way to see this is to keep in mind that the position operator as an essentially self-adjoint operator is defined on a dense subset in Hilbert space (the "test functions"). The generalized eigenvalues are in the dual of this subset, which is much larger than Hilbert space. It's containing the \delta distribution, plane waves, etc. But only true Hilbert-space vectors represent pure states!

 

[atyy]

Where precisely is the square root of a \delta distribution (NOT FUNCTION!) described in that paper? This has no sense in standard distribution theory.

Even if you can define something like this in a reasonable way by extending distribution theory somehow, the \delta distribution would not be normalizable, because the integral over a \delta distribution without multiplying it with an appropriate test function doesn't make sense either.

The easiest way to see this is to keep in mind that the position operator as an essentially self-adjoint operator is defined on a dense subset in Hilbert space (the "test functions"). The generalized eigenvalues are in the dual of this subset, which is much larger than Hilbert space. It's containing the \delta distribution, plane waves, etc. But only true Hilbert-space vectors represent pure states!

I don't know either and had many of the same questions as you - the brief mention in the Gagen et al paper was the first time I saw it. The only thing I don't know is whether one really cannot extend the theory so that the the square root of a delta function is in some sense a "normalizable pure state".

 

[tom.stoer]

I explained it in post #20 first. In post #25 I linked a related picture, which cannot be missed when glancing through the thread.

...

Here I put forward an interesting and relevant question, and it still remains unanswered. Is this because the physicists of modern times do not know the answer to this extremely basic question of the most basic quantum mechanics?

...

You are deliberately not understanding what I write. This is probably because you don't want to admit to yourself that you don't know the answer to my question.

The picture you are presenting are spots on a 2-dim. surface; but ρ = |ψ|² is defined over 3-space. I still don't understand your problem.

 

[atyy]

Of course not. I love the \delta distribution, particularly when I have to do an integral over it ;-)).

I think this is the reason why formal collapse of the ancilla (but not the system) to a position eigenstate makes sense (as eg. in Caves and Milburn, and effectively in Ozawa), if we assume that successive measurements are made on the system but not the ancilla. In that case, the ancilla can give a classical result, and collapse formally to a delta function. When the partial trace is taken after collapse of the ancilla and system, the delta function falls under an integral, and the state of the ancilla falls within the Hilbert space.

 

[jostpuur]

The picture you are presenting are spots on a 2-dim. surface; but ρ = |ψ|² is defined over 3-space. I still don't understand your problem.

The problem is that I don't know what formulas or principles imply what the probability density on the two dimensional surface is. Surely you can see that the "spots on a 2-dim. surface" appear to obey some probability density. Something like f(x_1,x_2)\approx C(1+\cos(\frac{2\pi x_1}{\lambda})) in the small region depicted by the pictures. Where does this probability density on the two dimensional surface come from?

I have explained my problem very clearly now, and people here should be able to understand it.

 

[Nugatory]

Where does this probability density on the two dimensional surface come from?

I have a probability density in three dimensions. The set of points that make up the surface of the screen is a subset of that three-dimensional space, so I can evaluate the probability density at those points...

But this seems so obvious that I must not be understanding your question...

 

[jostpuur]

The final answer, which is a probability density on the two dimensional surface, does not depend on time. So an attempt to define it with

<br /> f(x_1,x_2)=|\psi(t,x_1,x_2,R)|^2<br />

doesn't work. We would also like to have the condition

<br /> \int\limits_{\mathbb{R}^2}f(x_1,x_2)dx = 1<br />

in the end.