# Talk:S-matrix

WikiProject Physics (Rated B-class, Mid-importance)
This article is within the scope of WikiProject Physics, a collaborative effort to improve the coverage of Physics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
B  This article has been rated as B-Class on the project's quality scale.
Mid  This article has been rated as Mid-importance on the project's importance scale.

## 'Classical' S-Matrix

It would be very helpful if there was a discussion of scattering parameters from the point of view of a microwave or RF engineer. I would be willing to provide such an explanation if others think it would be useful. Mradway 21:17, 25 Mar 2005 (UTC)

I would be interested in hearing what you have to say about the relation between classical field theory scattering operators and the quantum field theory version. Bas Michielsen
I think the point of view of a microwave/RF engineer would be very useful, and I encourage Mradway to provide this explanation. 05:48, 16 June 2006 130.188.8.11

Wow, my comment is essentially already here: We need a discussion of the scattering matrix formalism; there's really no particular need to have this topic described in a manner narrowly centered on quantum mechanics (and even more narrowly, particle physics). I understand it's even used in quite "classical" electrical transmission lines, and reading Siegman's book "Lasers", I realize that this earlier usage must be resp. have been widespread enough for him to omit the discussion e.g. of concatenation or free space evolution of radiation. So please go ahead, Mradway, your contribution is exactly what I came looking for! -- anonymous, July 10th 2007

Hey you guys should check out s-parameters where this topic is described in great length. 128.138.200.94 (talk) 21:15, 15 January 2010 (UTC)

## Wick's Theorem

Would it be useful to move the stuff about Wick's theorem to another page? --HappyCamper 16:05, 4 September 2006 (UTC)

I agree with this. Wick's theorem is far more general than something that is used to evaluate S-matrix elements. PhysicsBob 09:19, 26 July 2007 (UTC)
I copied the section there (it was a redirect here) as a start on that article. Now it alos needs some work and this section can be trimmed. RJFJR 13:48, 2 November 2007 (UTC)

## Reorganization of Article

The article at present seems to read like a page of mathematics rather than an encyclopedia entry. The LSZ reduction section especially reads like a mathematical proof. Perhaps someone who is competent in these matters could insert more explanation about the underlying ideas and applications of the S-matrix. Unfortunately, I don't know much about this kind of thing. PhysicsBob 16:19, 26 July 2007 (UTC)

The LSZ reduction formula has its own article; is there any reason why it needs a separate section here? --Starwed 19:25, 29 October 2007 (UTC)
I feel there is no reason why the section on the LSZ reduction and Wick's theorem should remain in the main article, since they are techniques used to evaluate S-matrix elements, rather than being part of the defintion of the S-matrix. Perhaps a 'see also' section would be more appropriate? PhysicsBob (talk) 06:45, 11 June 2008 (UTC)

## Properties

Towards the end of "mathematican definition" there is a list of properties that the S matrix "must" have. There is no mathematical need for these, as I understand it, but they are rather connected to (admittedly very reasonable) physical properties. like: unitarity <--> conservation of probability does not change vacuum <-- energy conservation <--> time independence of lagrangian S|k> = |k> <--> stable (non-decaying) particle momentum conservation <--> space independence of lagrangian most of these are related to Noethers theorem, which could be linked to. In fact, most of these are properties of the lagrangian rather than properties of the S-matrix imo, although they are tightly connected. Not really sure how to organise this article... Art.cascade (talk) 16:50, 25 November 2011 (UTC)

## "Heisenberg" picture

In the paragraph "Use of S-matrices", it seems to me that is is not the "Heisenberg" picture, but the "interaction" picture, isn't it ? — Preceding unsigned comment added by 31.33.159.210 (talk) 13:03, 26 July 2012 (UTC)

## Definition

Weinberg describes things seemingly very differently. The in-states and the out-states inhabit the same Hilbert space. The scattering matrix is then the set of inner products of suitable basis vectors (well defined free multi-particle like Lorentz transformation properties) between them. These states are eigenstates of the full Hamiltonian. Then there is the S operator. This is defined such that it's matrix elements between free particle states equal the S-matrix elements. I.e. <bfree|S|afree> = Sba = <bout|ain>. [This presupposes a suitable subdivision of the Hamiltonian in free and interacting parts.] Is this the same thing as described in this article? YohanN7 (talk) 14:07, 28 January 2013 (UTC)

At any rate,

${\displaystyle \left\langle F,\beta |I,\alpha \right\rangle ,}$

(from article) doesn't make sense unless the states inhabit the same Hilbert space. YohanN7 (talk) 14:19, 28 January 2013 (UTC)

There is indeed an incoherence: if we say that S takes us from ${\displaystyle {\mathcal {H}}_{IN}}$ to ${\displaystyle {\mathcal {H}}_{OUT}}$ then ${\displaystyle \left\langle I,\beta |S|I,\alpha \right\rangle }$ makes no sense.

## S-matrix in one dimension

I am Dibyendu Bala,student of Tata Institute of Fundamental Research,want to extend this page in the following way.I want to add S-matrix in one dimensional quantum mechanics for a plane wave function(definite momentum state).Initially I will prefer to give the definition of S-matrix in case of short range potential barrier.Then relate transmission coefficient and reflection coefficient in terms of matrix element of S-matrix.I am also interested to show that the unitary matrix property is related to the conservation of probability current.Finally I will also include the effect of symmetry of the Hamiltonian on the S-matrix and the one dimensional version of optical theorem.

I like the new section. The problem is that there are errors. The k to the left and right of the origin aren't identical. The potential V(x) should be given explicitly, and inline LaTeX shouldn't be used for inline equations. Please fix them. Then, also, expressions for the Sij wouldn't be out of place. Moreover, make sure to include inline citations. (I suspect the added reference is there for this purpose. Also, use the citation template for references (as for the other references.) YohanN7 (talk) 01:50, 26 October 2014 (UTC)
You need to note that this is (I guess) not the S-matrix in the context of quantum field theory, but rather from ordinary quantum mechanics. Citations are still lacking. YohanN7 (talk) 13:03, 27 October 2014 (UTC)
Yes,in this new section S-matrix is in ordinary quantum mechanics.Dibyendum (talk) 16:40, 27 October 2014 (UTC)
Thanku YohanN7 for correcting this new section.Dibyendum (talk) 16:40, 27 October 2014 (UTC)
You are welcome. (Please sign your posts and indent them properly. Type four tildes →~ to sign.)
You need to make it clear that this is ordinary QM in the article. Also, since this is an "easy" example, it should probably be one of the first sections. (Don't forget inline citations.) YohanN7 (talk) 14:09, 27 October 2014 (UTC)
The k to the left and right of the origin aren't identical. When I wrote that I was under the impression that a Step potential was to be treated. My bad, so forget that. YohanN7 (talk) 07:55, 2 December 2014 (UTC)

### Question about S-matrix in 1D quantum mechanics

Is it really correct to say that the reflection and transmission coefficients specify the S-matrix? Since the transmission coefficient is known once the refelction coefficient is given, this would mean that the S-matrix is specifies by one real number. But one needs four real numbers to specify a 2 by 2 unitary matrix. Is there some more information in the S-matrix?

If the problem is time-reversal symmetric, does this mean that the amount of information in the matrix is smaller? Is this related to the Optical Theorem?

Finally, why is it that the S-matrix in quantum mechanics is two-dimensional? Shouldn't the S-matrix connect the whole Hilbert space of incoming waves to the whole Hilbert space of outgoing states? In which case wouldn't one expect the S-matrix to be infinite dimensional? — Preceding unsigned comment added by 158.144.32.127 (talk) 16:42, 27 November 2014 (UTC)

There is something funny abut this talk page section. When it is opened in the editor, there is more text than is visible in the browser. Parts are commented away. This may have to do with that you forgot (repeatedly) to sign your posts. Try to fix this. (And log in!)
As for your questions, I think S = S(k, V) explains some. YohanN7 (talk) 09:13, 1 December 2014 (UTC)
Still not ok. Are you logged in? Also, remove the subst:Unsigned|1=Dibyendum|2=08:47, 1 December 2014 (UTC) Autosigned by SineBot stuff.
You are creating a mess YohanN7 (talk) 09:27, 1 December 2014 (UTC)
Hint: Use the preview before you save your edits - and don't reorder posts by me and you. YohanN7 (talk) 09:30, 1 December 2014 (UTC)
Moved my reply to where it was originally. Still one unsigned post above. YohanN7 (talk) 10:05, 1 December 2014 (UTC)

When we have to calculate reflection and transmission coefficient,we have to take mod value of that ratio.Phase information is lost.Matrix elements of the S-matrix has phase.So matrix elements of S-matrix is not reflection coefficient and transmission coefficient.Sorry for this.Thank you for correcting me. As S-matrix elements are not reflection and transmission coefficient,so to specify S-matrix we have to specify four real number. To specify a 2 by 2 unitary symmetric matrix we need 3 real parameter.whereas to specify unitary matrix we need 4 real parameter. So it does not mean that the information is smaller in S-matrix ,it only means that one information is related to other by one more constraint(symmetric constrained). In optical theorem the departure from free particle S-matrix is described by two complex function.So it has 4 real parameter.But there is a real relation.So the number of parameters is 3.

Here the S-matrix is given for a particular value of wave vector.That is why the S-matrix is 2 by 2.As there is infinite number of wave vector, the S-matrix is in general infinite dimensional block diagonal matrix.Each block is 2 by 2,correspond to particular value of wave vector.Dibyendum (talk) 09:44, 1 December 2014 (UTC)

## Conventions of the S-matrix in one-dimensional quantum mechanics section

It looks like the odd conventions of this section, namely S→symplectic metric, instead of the identity, for the free theory (T=0) may well confuse the reader who has just read that S→I in the preceding sections.

Specifically, assuming a universal time-dependence by a phase exp(−iωt) in ψ, A describes an in right-mover and C an out-right mover; while D an in left-mover and B an out left-mover. So, interchanging B with C in ψout while leaving ψin alone, has the effect of interchanging the two rows of the S-matrix written at present, so that the free case is, indeed, the identity, as advertised above this section.

In this way, the optical theorem could present less deformed, indeed, bizarre, and the missing further relation for the imaginary part of r would be apparent. Cuzkatzimhut (talk) 02:03, 2 December 2014 (UTC)

But this convention is used in Eugen Merzbacher quantum mechanics book.Dibyendum (talk) 09:13, 2 December 2014 (UTC)
(It's common practice to indent replies.) This is the perfect example of where you need to put in a literature reference as a footnote. I'll do this one. YohanN7 (talk) 09:34, 2 December 2014 (UTC)
Done. Feel free to fill out with page number or anything. The ref-tag creates a popup, and the template harvnb connects to the reference list and places the actual popup text under "Notes". It takes a while to get used to the system, but it works. YohanN7 (talk) 09:45, 2 December 2014 (UTC)

OK, for future reference by the inquiring reader, but not for the article, I record that S22 = 2ir* (1+2it)/(1−2it*), as things stand defined at present. Cuzkatzimhut (talk) 16:24, 2 December 2014 (UTC)

## New section closer to top?

I think the new section should be moved closer to the top since it is relatively accessible. Some reorganization is required for this. But I don't know exactly what that reorganization would be. Just an idea. YohanN7 (talk) 08:06, 2 December 2014 (UTC)

Done. Looks ok? YohanN7 (talk) 12:24, 2 December 2014 (UTC) Yes,its looking ok.Dibyendum (talk) 12:27, 2 December 2014 (UTC)

## Intro

Sanity-check for the intro to the new section appreciated. I think the "continuous matrix" can be formalized rigorously, but that would be gross overkill to do here. YohanN7 (talk) 15:26, 2 December 2014 (UTC)

## Assessment comment

The comment(s) below were originally left at Talk:S-matrix/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

 I have just one suggestion: replace the word "void" by it's most commonly used equivalent "vacuum". thierry

Last edited at 01:50, 1 January 2012 (UTC). Substituted at 05:09, 30 April 2016 (UTC)