# How to entangle two particles?

I have notions of the mathematics involving (entangled state is one which is not a product state of two qubits, etc) the idea of entanglement. However, still can not figure out how to, let us say, pick a particle (which type of particle we can use as qubit? protons? electrons? ununumbiums!?!?) A in one laboratory of Max Planck and a particle B in Boston and then entangle them. How can we entangle 2 arbitrary particles?

Entanglement is what generally happens when things interact with each other in a particular way. So if you want to entangle two particles, you must first bring them together so they can interact.

vanhees71
How? can you bring me an example?

DrChinese
Gold Member
How? can you bring me an example?

There are a number of methods, and most use some/all of the following concepts:

a) The particles must be indistinguishable on the basis they are entangled. They can be distinguishable on other bases.
b) There is some conserved quantity or relationship between the particles.
c) The number of objects entangled can be 2 or more, with no theoretical upper limit.

1. Photons are most often entangled via a process called Parametric Down Conversion, in which a single photon is split into 2 entangled ones.
2. A helium atom has 2 electrons, which are entangled when in the same lowest shell together.

a) The particles must be indistinguishable on the basis they are entangled. They can be distinguishable on other bases.

I don't see how that is necessary. First of all, you don't even need several particles for entanglement. A single particle with entangled spin and position is very common. Secondly, I don't see what practically speaks against entangling very different quantities between two different particles.

b) There is some conserved quantity or relationship between the particles.

I'm not sure what you mean with this. Can you give an example of how you think this applies?

c) The number of objects entangled can be 2 or more, with no theoretical upper limit.

With the restriction of the monogamy of entanglement: http://www.quantiki.org/wiki/Monogamy_of_entanglement

Allow me to add that two indistinguishable particles are 1) always entangled if they are fermions 2) entangled if they're in different states and bosons.

Cheers,

Jazz

DrChinese
Gold Member
1. A single particle with entangled spin and position is very common. Secondly, I don't see what practically speaks against entangling very different quantities between two different particles.

2. I'm not sure what you mean with this. Can you give an example of how you think this applies?

With the restriction of the monogamy of entanglement: http://www.quantiki.org/wiki/Monogamy_of_entanglement

3. Allow me to add that two indistinguishable particles are ... always entangled ... if they're in different states and bosons.

We are both speaking in general terms. However your objections are themselves objectionable. :-)

1. There is no meaningful way to say a single particle is entangled. Entanglement is represented by a system of 2 or more particles.

2. As to conservation: PDC photons are entangled as to frequency with conserved total momentum (from the input photon) according to a common formula. Spin (polarization) is conserved as a constant in type II PDC.

3. Could you give me an example of this? Specifically, what does the "different states" criteria have to do with anything? And just to be clear about MY objection to your concept about entanglement of fermions versus bosons: there is no requirement that entangled particles be the same kind of particle at all.

1. There is no meaningful way to say a single particle is entangled. Entanglement is represented by a system of 2 or more particles.

I agree with @Jazzdude. Entanglement is a general property that states in tensor product spaces may have (with respect to that particular factorization); any collection of independent degrees of freedom can be entangled together. That tensor product can be between any two Hilbert spaces, not just those corresponding to separate particles. Entanglement is just more striking when the factor spaces correspond to separate particles because then you can observe the 'spooky' spatial non-locality for which entanglement is famous. But it's perfectly sensible to talk about, say, the spin of a single particle becoming entangled with its angular momentum in spin-orbit coupling. It's exactly the same phenomenon and that is the language widely used.

2. As to conservation: PDC photons are entangled as to frequency with conserved total momentum (from the input photon) according to a common formula. Spin (polarization) is conserved as a constant in type II PDC.

Given that, again, there's no need for the things being entangled to even be particles, there's no fundamental relationship between entanglement and conservation. Generally, the physical processes that mediate interactions between different degrees of freedom have certain conservation laws associated with them, but that's completely general and nothing to do with entanglement.

Could you give me an example of this? Specifically, what does the "different states" criteria have to do with anything?

Fermions are subject to the Pauli exclusion principle; so, anti-symmetrizing the state of a collection of fermions necessarily produces an entangled state since you cannot start the tensor product of identical pure states. The state of a collection of bosons needs to be symmetric under particle exchange, so each particle is allowed to be in the same pure state giving an overall product state. On the other hand, if you start out with at least two particles in different pure states and then symmetrize the whole thing you get an entangled state as with the fermions.

Last edited:
1. There is no meaningful way to say a single particle is entangled. Entanglement is represented by a system of 2 or more particles.

This is not accurate. Entanglement happens between tensor factor spaces, not particles. A single nonrelativistic Schroedinger particle of spin 1/2 lives in a space that factors into a Hilbert space spanned by particle positions and a two dimensional spin space. The particle's state does not necessarily factor into states in those two factor spaces however, and the particle can be entangled. A state like ## \left| A \right\rangle \left| \uparrow \right\rangle + \left| B \right\rangle \left| \downarrow \right\rangle## where ##A## and ##B## are different position states is position-spin entangled.

2. As to conservation: PDC photons are entangled as to frequency with conserved total momentum (from the input photon) according to a common formula. Spin (polarization) is conserved as a constant in type II PDC.

Obviously, conserved quantities must also be conserved for processes that entangle subsystems. I don't see how the existence of a conserved quantity is required for entanglement though. Of course, in the Hamiltonian formalism energy is always conserved under unitary evolution, but that's surely not what you mean.

3. Could you give me an example of this? Specifically, what does the "different states" criteria have to do with anything? And just to be clear about MY objection to your concept about entanglement of fermions versus bosons: there is no requirement that entangled particles be the same kind of particle at all.

No, entangled particles do not have to be of the same kind. However indistinguishability is sufficient for entanglement if the states are different. That follows from the (anti)symmetrisation of multi-particle states. Start with a two particle state ##\mathcal{H}^{\otimes 2}## and the two single particle states ##\left| A \right\rangle## and ##\left| B \right\rangle##.

The fermionic state is then ## \left| A \right\rangle \left| B \right\rangle - \left| B \right\rangle \left| A \right\rangle## where the order of the kets indicate the space. This state only exists if ##\left| A \right\rangle## and ##\left| B \right\rangle## are different. But in this case the state does not factor and the combined state is entangled.

The bosonic state is ## \left| A \right\rangle \left| B \right\rangle + \left| B \right\rangle \left| A \right\rangle##. This state does exist even if ##\left| A \right\rangle## and ##\left| B \right\rangle## are the same single particle state. If they are the same state the state can trivially be written as a product state. If they're not the same the state does not factor and again the combined state is entangled.

Cheers,

Jazz

vanhees71
DrChinese
Gold Member

I agree with this statement. And I also stand by my post #4 as written.

And again you and I disagree on how to address an answer to the OP. What you and Jazzdude are saying is a deeper level than appropriate, in my opinion (and I realize you are a professor). Any answer given can necessarily be argued with on some level. I didn't shred your answer in post #2 as I could have (as I am sure you know perfectly well that entanglement is possible of particles that never even existed at the same time). But that detail wouldn't be of much help to the OP.

My experience here is that some posters benefit from one person's manner of addressing a question, and some from another's. It is common that different approaches are taken to get to that point. It is more effective to assist the OP that to critique someone else's style or approach in the name of "correctness". For the OP's purposes, I believe my answer is better than what you or Jazzdude have said so far. I would encourage you to provide something more for the OP.

To the OP regarding your original question: there are many ways to entangle particles/systems/properties. As you can see from the responses, the fundamental issues can be perceived in a variety of ways. You might enjoy a few of the following references, in which laboratory entanglement creation is discussed.

http://arxiv.org/abs/quant-ph/0205171
http://www.nature.com/nature/journal/v409/n6822/full/409791a0.html
http://arxiv.org/abs/quant-ph/0303018
http://lanl.arxiv.org/abs/1006.4344[/user]

DrChinese
Gold Member
However indistinguishability is sufficient for entanglement ...

And again my point, we are mixing the general and the specific. This is actually quite nearly what I said earlier: "The particles must be indistinguishable on the basis they are entangled." You objected to that, saying "I don't see how that is necessary." Almost any experiment with entanglement will mention indistinguishably (as I did) and/or show a related equation with something conserved.

I suspect we will both be happier addressing the OP's question, which I believe you have yet to weigh in on.

PS Not that I am questioning that it exists, but I don't recall any actual experiments involving single particle entanglement. Can you cite a good example for me to add to my collection?

It is more effective to assist the OP that to critique someone else's style or approach in the name of "correctness". For the OP's purposes, I believe my answer is better than what you or Jazzdude have said so far. I would encourage you to provide something more for the OP.

My reply was intended to address certain, unfortunately quite common, misconceptions about entanglement. They were not specifically directed at the OP but are supposed to stand as a footnote to your contribution. I do agree that answers should be tailored to the level of the original question, but they should also be correct. The fact that you objected to my response clearly shows that the misconceptions are not just simplifications. Also, I've contributed to the OPs question by providing an easy way to "produce" entanglement for identical particles. Frankly, and no offence intended, I think your argument for more suitable answers will appear to others as no more than a distraction from your having been wrong. While that's understandable, a good scientist should stand above this.

Cheers,

Jazz

And again my point, we are mixing the general and the specific. This is actually quite nearly what I said earlier: "The particles must be indistinguishable on the basis they are entangled." You objected to that, saying "I don't see how that is necessary."

There is a difference between sufficient and necessary.

Cheers,

Jazz

Not that I am questioning that it exists, but I don't recall any actual experiments involving single particle entanglement. Can you cite a good example for me to add to my collection?

Stern-Gerlach will do nicely.

Cheers,

Jazz

DrChinese
Gold Member
My reply was intended to address certain, unfortunately quite common, misconceptions about entanglement. They were not specifically directed at the OP but are supposed to stand as a footnote to your contribution. I do agree that answers should be tailored to the level of the original question, but they should also be correct. The fact that you objected to my response clearly shows that the misconceptions are not just simplifications. Also, I've contributed to the OPs question by providing an easy way to "produce" entanglement for identical particles. Frankly, and no offence intended, I think your argument for more suitable answers will appear to others as no more than a distraction from your having been wrong. While that's understandable, a good scientist should stand above this.

Cheers,

Jazz

As always, I stand ready to acknowledge the limits of my knowledge. Your argument for "correctness", on the other hand, will appear to some others as a distraction* for the lack of utility to the question at hand. Sorry, a good scientist should also be helpful.

-DrC

* I would call it splitting hairs as I see no meaningful disagreement.

DrChinese
Gold Member
There is a difference between sufficient and necessary.

Cheers,

Jazz

You know, I almost wrote that you would answer with that exact phrase. But I thought at the time that you wouldn't waste time with that. Thank you for pointing out precisely nothing useful in an attempt to trump someone.

DrChinese
Gold Member
Stern-Gerlach will do nicely.

Cheers,

Jazz

I am sure the OP will be delighted this helpful nugget found its way into the conversation.

All this from: "How can we entangle 2 arbitrary particles?"

I am loath to add this definition, but we've already gone so far down this road that the devil in me can't resist. :) I will say that I plan to step out of this conversation until the OP returns, at which time I will comment further if I have anything USEFUL to add.

Quantum entanglement is a physical phenomenon that occurs when pairs or groups of particles are generated or interact in ways such that the quantum state of each particle cannot be described independently—instead, a quantum state may be given for the system as a whole.

Last edited:
As always, I stand ready to acknowledge the limits of my knowledge.

Yet you don't but rather reiterate something that is wrong:

You know, I almost wrote that you would answer with that exact phrase. But I thought at the time that you wouldn't waste time with that. Thank you for pointing out precisely nothing useful in an attempt to trump someone.

If you get something obvious wrong, you will have to live with the obvious being pointed out to you. This has nothing to do with trumping you. I had already pointed out how you're wrong. You however tried to restate it in a more obfuscated way hoping I would let it stand and make it appear that you have been right from the beginning. That is trumping. But even worse, if I answer and undermine your plan you suggest that it's me and not you who is not providing anything useful to the discussion.

Together with this:

Your argument for "correctness", on the other hand, will appear to some others as a distraction* for the lack of utility to the question at hand. Sorry, a good scientist should also be helpful.

I cannot think of any other word but "trolling" for what you are doing.

I am sure the OP will be delighted this helpful nugget found its way into the conversation.All this from: "How can we entangle 2 arbitrary particles?"

Now hold it for a minute. You specifically asked me for exactly that information. Giving it to you is now dismissed as not being helpful?

I will say that I plan to step out of this conversation until the OP returns, at which time I will comment further if I have anything USEFUL to add.

How very convenient for you.

Congratulations!

*plonk*

Jazz

* I would call it splitting hairs as I see no meaningful disagreement.

Then, clearly, you have not understood the disagreement.

Thank you all for your responses.

So, could be the entanglement the more nature way to be for quantum particles?

DrChinese
Gold Member
So, could be the entanglement the more nature way to be for quantum particles?

Atoms and molecules naturally exist in configurations in which there is a lot of entanglement. Which is to say there is a lot of superposition of states. Those are not always useful for entanglement experiments, however. The effort to separate the entangled particles can cause them to cease acting as a combined system and instead act as separate ones.

That should be taught on my quantum mechanics course...

When we use Bell's basis? Which is his usefulness? We use different basis each case of entanglement?

I ask these questions in order to perform a more important one.

I have notions of the mathematics involving (entangled state is one which is not a product state of two qubits, etc) the idea of entanglement. However, still can not figure out how to, let us say, pick a particle (which type of particle we can use as qubit? protons? electrons? ununumbiums!?!?) A in one laboratory of Max Planck and a particle B in Boston and then entangle them. How can we entangle 2 arbitrary particles?

In principle, you can use any two quantum systems. For them to become entangled they need to take part in a physical interaction whose outcome depends on the state of both systems. Interactions are local, even in quantum theory, which means you need to bring the systems together for them to interact. (Exceptions arise if you have photons involved, which can transport entanglement between lightlike related events).

Because the two systems interact, the state of each system after the interaction will depend on the states of both system before the interaction. So if you have an incoming product state ##\psi_A \times \phi_B## let's assume for the moment that the outgoing state is also a product state ##\psi'_A \times \phi'_B##. Both outgoing factor states are supposed to depend on both incoming states. That means if we change the incoming state of just one of the systems by adding ##\Lambda_B## and respect the linearity in->out map we get the outgoing state ##(\psi'_A+\Lambda^{'(1)}_B)\times(\phi'_B+\Lambda^{'(2)}_B)##. This no longer necessarily factors into two systems at A and B as B appears in both factors. That means we generally have to expect the outcome of an interaction between two systems to be entangled, with the unentangled outcome being the exception.

When we talk about entanglement between two distant particles or systems we usually mean something slightly stronger than just entanglement, namely two maximally entangled systems. Entanglement is maximised if there there are four orthonormal vectors ##\phi_A##, ##\psi_A##, ##\phi_B##, ##\psi_B## so that we can write the state of the combined system as ##\phi_A \phi_B + \psi_A \psi_B##. Generating such a maximally entangled pair is clearly more difficult than just generating an entangled pair. We need carefully prepared input states and a suitable interaction to make sure that the result is maximally entangled. Finding such interactions for a given pair of systems is very much an art and there are no general rules I am aware of. Symmetries of the interaction usually help to determine what incoming states result in maximally entangled outgoing states without having to know all the details of the interaction. Often it is much simpler to not entangle two systems that you already have but to product two new particles in such a way that they must be entangled. Pair production for example can generate spin entangled photon pairs relatively easily.

I hope this clarifies some of the concepts involved. As you can see from the discussion here, entanglement is not an easy topic and it all too easily leads to confusion and misconceptions are common even among physicists.

Cheers,

Jazz

Thank you all for your responses.

So, could be the entanglement the more nature way to be for quantum particles?

In fact, in the space of all quantum states, entangled states are much more common than product states. It is even so that product states are a set of measure zero. Even if you admit some uncertainty and a cutoff for what you count as "approximately non-entangled" the number of entangled states quickly dominates the non-entangled ones if the dimension of the state space is increased. So yes, entanglement is normal. The lack of entanglement is the very rare exception. But be careful: maximally entangled states are just as rare as disentangled ones.

Cheers,

Jazz

That was very clear, thank you.

Do you know the answer of my last post?

By the way, what really means "Entanglement can not be created locally"?

$$\left|\widetilde{\phi}\right> = \frac {1}{\sqrt{2}} ( \left|01\right> - \left|10\right>) = \frac {1}{\sqrt{2}} ( \left|\widehat{x}-\widehat{x}\right> - \left|-\widehat{x} \widehat{x}\right>)$$

What means the ##\widehat{x}, -\widehat{x}## and the ~ over phi?

PS.: This equations comes from the correlations in bell pairs topic, but I already see the ~ over other kets in other topics, must has a general meaning.

The inverse of SWAP operator is itself?

Well, we really have very simple methods to see quantum entanglement. Measuring devices are usually treated as classic objects, however, everything is quantum. A good device will be highly entangled with the particle being measured. Messages will also cause entanglement, as well as force.

vanhees71
Gold Member
2021 Award
We are both speaking in general terms. However your objections are themselves objectionable. :-)

1. There is no meaningful way to say a single particle is entangled. Entanglement is represented by a system of 2 or more particles.

No, I'd rather say that properties of quantum systems are entangled. As was already stated, the spin and position of one particle can be entangled. The paradigmatic example is the Stern-Gerlach experiment, where you sent a silver atom (electrically neutral) through an inhomogeneous magnetic field which is taylored such that silver atoms with spin up go preferredly in one and those with spin down in another direction.

This you'd describe in the following way: In non-relativistic physics we can write the state of the silver atom as the direct product of the orbital and spin part and superpositions thereof. So let ##|\Phi_{\vec{x}_j} \rangle## (##j \in \{1,2 \}##) describe two wave packets, peaking at two positions ##\vec{x}_j## with a spatial width small compared to the distance between these places. Then the state when the silver atom has run through the magnetic field is described by
$$|\Psi_{\text{SG}} \rangle=|\Phi_{\vec{x}_1},\sigma_z=+1/2 \rangle+|\Phi_{\vec{x}_2},\sigma_z=-1/2 \rangle.$$
Here, the position of the particle and its spin component in ##z## direction are entangled. When registering an atom around place ##\vec{x}_1## (##\vec{x}_2##) it has spin up (down) with (almost) 100% probability
3. Could you give me an example of this? Specifically, what does the "different states" criteria have to do with anything? And just to be clear about MY objection to your concept about entanglement of fermions versus bosons: there is no requirement that entangled particles be the same kind of particle at all.

I guess, what's meant with the example of indistinguishable particles is that in 3 (and more) dimension they necessarily must be described as bosons or fermions. Fermions (bosons) consist of totally antisymmetrized (symmetrized) ##N##-particle Fock states and superpositions thereof. Take two fermions in a Fock state. The two single-particle states must be necessarily different, because otherwise antisymmetrization implies that the vector vanishes. The Fock state is thus always a superposition of the form
$$|\Psi \rangle=|\psi_1,\psi_2 \rangle-|\psi_2,\psi_1 \rangle.$$
You can say, in a loose sense, the particles are entangled. It's however not so clear to me in which sense in this general case. It depends on the single-particle states chosen which properties (observables) are entangled.

Two bosons, of course, can be in the same state. In the extreme you even have a macroscopic number of particles in a single one-particle state (e.g., a Bose-Einstein condensate in a trap). Then (for two particles) you simply have a product state
$$\Psi \rangle=|\psi,\psi \rangle,$$
i.e., there is no "entanglement" as when you have a (in this case symmetrized) product of two different single-particle states.

vanhees71
Gold Member
2021 Award
As always, I stand ready to acknowledge the limits of my knowledge. Your argument for "correctness", on the other hand, will appear to some others as a distraction* for the lack of utility to the question at hand. Sorry, a good scientist should also be helpful.

-DrC

* I would call it splitting hairs as I see no meaningful disagreement.

No, it's not splitting hairs! Particularly in quantum theory it's of utmost important to state things as precisely as one can. The famous dictum by Feynman, "nobody understands quantum mechanics" has some truth in it, because we all struggle with precisely this necessity of rigor when talking about quantum phenomena. In my opinion, you cannot understand entanglement without a minimum of the mathematical foundations. It's impossible to explain quantum theory without math, because we simply don't have a precise enough language other than the mathematics. The more care one has to take to translate the mathematical expressions into plain English!

dlgoff, VantagePoint72 and bhobba
DrChinese
Gold Member
1. No, I'd rather say that properties of quantum systems are entangled. As was already stated, the spin and position of one particle can be entangled. The paradigmatic example is the Stern-Gerlach experiment...

2. I guess, what's meant with the example of indistinguishable particles is that in 3 (and more) dimension they necessarily must be described as bosons or fermions. Fermions (bosons) consist of totally antisymmetrized (symmetrized) ##N##-particle Fock states and superpositions thereof.
..

Sorry, I am going to stick with my original statements (as I replied to Jazzdude) and I will say you too are splitting hairs in a way that communicates no useful information to our readers. You can't quote a textbook with every answer in your desire to be "precise". There are exceptions to almost any general rule - including this one. :-)

1. As quoted below, quantum entanglement involves systems of 2 or more particles being represented by a single wave function which is not a product state - ie what I said originally. I am not going to quibble about whether position and spin are entangled in your example (they are correlated), but I would say I have never seen it described as entanglement in any article about SG. I guess by your thinking, the output of a PBS is also and example of position-polarization entanglement since the same principle is involved. I would point out that position/spin are commuting (your example). Whereas most Bell Inequalities - the usual gold standard for demonstrating entanglement - depend on there being entanglement on bases that do not commute (for example photon spin being entangled at both 0 degrees and 45 degrees). My point being not that your example is or is not entanglement, just saying it is not usually cited as such and you are going overboard insisting it should be brought into this discussion when a simple reading of the original OP question would explain why that makes no sense at all.

Wiki: "Quantum entanglement is a physical phenomenon that occurs when pairs or groups of particles are generated or interact in ways such that the quantum state of each particle cannot be described independently—instead, a quantum state may be given for the system as a whole."

In case you reject Wiki as a source, perhaps you can find something different (that supports your contentions) on Plato: http://plato.stanford.edu/entries/qt-entangle/

2. Entanglement has absolutely NOTHING fundamentally to do with entangled particle pairs being bosons or fermions - you can entangle either obviously. (Of course bosons must still act like bosons and fermions like fermions.) You will be jumping through hoops to "prove" your statement as meaningful in a manner consistent with traditional definitions of entanglement. Again, a technical treatise is inappropriate as an answer for a general question.

For example: In PDC you start with 1 spin 1 particle and end up with 2 entangled ones. With entanglement swapping you start with 4 photons and end up with 2 entangled. Nothing fundamental about symmetric or anti-symmetric states there! And the output pairs can be parallel or anti-parallel in both cases. I can cite examples of these if you need them. And I could cite plenty of discussions of entanglement that will make it clear that there are no special factors for entangling fermions as opposed to bosons past what I have said. Conceptually, each can be entangled on the same observables, and there are observables beyond the usual spin/momentum/etc they can be entangled on as well. And to support my comment to the OP about conservation in entanglement:

Wiki: "The decay events obey the various conservation laws, and as a result, the measurement outcomes of one daughter particle must be highly correlated with the measurement outcomes of the other daughter particle (so that the total momenta, angular momenta, energy, and so forth remains roughly the same before and after this process). For instance, a spin-zero particle could decay into a pair of spin-1/2 particles. Since the total spin before and after this decay must be zero (conservation of angular momentum), whenever the first particle is measured to be spin up on some axis, the other (when measured on the same axis) is always found to be spin down."

Nothing there implying fermions obey different rules than bosons (I did not mention this but you did), although it says conservation (which I mention and you skip) is relevant. So who is correct, my friend?
-------------------------------

I am not going to be drawn into a debate with another science advisor over the best way to present an idea or how to most accurately describe an idea. As far as I am concerned, there is no meaningful question here and no need for further comment in a thread called "How To Entangle Two Particles". Or if you feel as if the subject is worthy of more debate: start a new thread about "Single Particle Entanglement" or "Fermion vs Boson Entanglement" and we can discuss it there. BTW: Your posts #27 and #28 seem a bit out of character for you.

vanhees71
Gold Member
2021 Award
I stick to my point too. Entanglement describes quantum correlations between properties (observables), not necessarily between properties of two or more particles. Recently there was reported a very interesting experiment dubbed "quantum chesire cat"

http://www.tuwien.ac.at/en/news/news_detail/article/8921/

Also the original publication seems to be open access:

http://www.nature.com/ncomms/2014/140729/ncomms5492/full/ncomms5492.html

The relation between the symmetry properties under particle exchange (totally symmetric states for bosons and antisymmetric states for fermions) and entanglement is so obvious that I don't understand why you deny this relation so vehemently.

I also don't want to discuss didactics here, but I insist on the right of all posters to indicate errorneous statements and to discuss about them. I'm always glad when I'm being corrected, because then I learn something from it. In addition it's important to correct mistakes in a forum like this, not to confuse readers, particularly students starting to study a new subject. That's what science is all about, and it's a great achievement of this forum to be really scientific.