# Maxwell's equations, Orthogonality, electric and magnetic fields in EM

1. Feb 17, 2008

Maxwell's equations give that the electric and magnetic fields in E-M radiation are orthogonal. This is a classic equation, but can it be related to the orthogonality of, for example, the momentum and position operators which lead to non-commutivity?

2. Feb 17, 2008

### Marco_84

what do u mean about "orthogonality" of q and p?????

3. Feb 17, 2008

### malawi_glenn

non-commutivity and orthonogality of states are not the same thing!

Non- commutivity means that you get different results when measuring the quantities A then B of state S, from when doing B then A of state S.

Ortonogality is about how states are related to each other. See the physical states as vectors in Hilbert space. A state n can be a superposition of several eigenstates to an operator O. Another state m can be a superposition of other eigenstates to the operator O. The different eigenstates o to O are mutual orthogonal. Compare with the cartesian unit(basis) vectors x,y,z. Every vector can be written as a superposition of x y z, for example A = 9x-5y+1z. The basis vectors are mutually orthogonal, but not every cartiesian vector are mutually orthogonal.

But to answer your real question, there is no connection I would say.

4. Feb 17, 2008

oops

First, I apologize for the haste in which I posed my question, and give my thanks to malawi_glenn for the deserved slap-on-the-wrist. Strictly, of course, as justly pointed out, non-commutivity is not justification for claiming any sort of orthogonality.(Especially to Marco_84: To see what was going on in my head: if we define ACD(X) as "ability to completely determine the observable X", then ACD (momentum) = 1-ACD(position), thereby fitting one sort of orthogonality, but contrary to the more appropriate definition of orthogonality in Hilbert space.) So, if I don't try everyone's patience too much, let me ask more simply: is it possible to measure the magnetic field and the electric field of a particle simultaneously? (I suspect, from malawi_glenn's closing comment that it is.)
Thanks again.

Last edited: Feb 17, 2008
5. Feb 17, 2008

### malawi_glenn

The magnetic field and E-field are generated by the same mechanism, so I would say yes, you can measure them simultaneously. But this was just by plain reason, I cant give you a theoretical derivation of this (not at this moment).

6. Feb 17, 2008

### belliott4488

nomadreid - I think you are confusing orthogonality of vectors in our 3-D position space with the idea of orthogonality in Hilbert space (although I'm not sure I understand your use of "orthogonal" with respect to incompatible observables). I could talk about the x and y displacement vectors that give position of a particle in space, and they are orthogonal but are certainly not incompatible in the sense of not being simultaneously observable. Eigenstates of position in the x direction would be orthogonal to eigenstates in the y direction, but that's just because they are linearly independent, not be cause the 3-space vectors are orthogonal.

7. Feb 17, 2008

OK, Thanks

OK, thanks for these last two replies. Clear. I think my original confusion was partly caused by the experiments involving spin, whereby the measurement apparati (apparatuses?) for spin in one or the other direction means that one cannot measure for the spin in two directions at once.

8. Feb 18, 2008

### Marco_84

Ok i think i've got where is you're problem:

I think ure reading Stern_Gerlach right???

What this experiment proves is the algebraic relation beetween the angular momentum operators:

$$[L_{\mu},L_{\nu}]=i\epsilon_{\mu\nu\lambda}L_{\lambda}$$

This expression suggest us that we cannot simoultanesly diagonalize all of them at the same time. In other words they do not commute.
A good "experimental way" to see this is rotating a box by 90° along two differebt axis and then switch the operations.

Then.

You're question about E and B fields require the tools of second quantization since they are dynamic variables with infinetly degrees of freedom.

Usually the canonical quantization of the radiation field is made upon the 4-vector potential $$A_{\mu}$$... see Itzykson_Zuber "Quantum field theory" chapter 3.

Ciao

Marco

9. Feb 18, 2008

Itzykson_Zuber "Quantum field theory"

Marco: thanks for the reference. Unfortunately, not living where I can easily get hold of this book from a library or even a local bookstore, I would have to order it (and pay, and wait, for its overseas delivery, not to mention other hassles and expenses involved). Hence: before I will go to the trouble to buy it, I will look online for an explanation of the points you made in the last two sentences (maybe you have a recommendation for a link?), but if I can't find it: would you say it is worth buying it? (I read that some people found it dated.) (This is a real question, not a rhetorical one.) Consider that my background is in mathematics, not physics (i.e., I do not know what sorts of assumptions the book makes about the reader.)

10. Feb 18, 2008

### peter0302

Electricity and Magnetism ARE commuting variables, though - their relationship is not, as I understand, governed by the uncertainty principle, and therefore the analogy would not apply.

The use of "state vectors" is simply a mathematically convenient way of speaking about states. Orthogonal states merely means mutually exlcusive base states. The orthogonality of the EM field, however, does have a real space aspect. The Hilbert space representing the states of a system has no direct physical analogue.

11. Feb 18, 2008

### malawi_glenn

yes, the magnetic moment and electric moment of a nucleus for instance are commuting operators.

12. Feb 18, 2008

### Marco_84

Also many friends at my UNI think that is dated and they prefer something else...
but i think that It is one of the most rigourus mathematically speaking...
An if i got right you are mathematician!!

Im currently studying those topics and using 4/5 books.
Obviously Peskin.. then Lebellac is a really good one for connections with statistical mechanics...The bible (Weinberg) but i think it does not go so deep with the calculus, maybe im not that smart for those kind of books :)

regards
Marco

13. Feb 19, 2008

"Devil's circles"

Thanks to peter0301, malawi_glenn, and Marco_84 for the answers!
Marco, I will follow up your recommendation and look into ordering those books. Yes, my academic specialty is in mathematics, and as this is in pure (as opposed to applied), maths/math (depending on your geographical position), various philosophical considerations involving Model Theory have brought me to looking into Quantum Theory; I have a number of books from different angles. I know that QM grew up as a potpourri of "well, this works, so use it"; nonetheless, except for the standard axiomatisations of Hilbert space, Measure Theory, and Projection Logics and a listing of about 20 "irreducible constants", I have not come across an axiomatisation of QM which is as compact as I feel is out there somewhere. For example, looking for a characterization of non-commuting variable pairs, I come across the explanation that two observables are incompatible if their associated Hamiltonians are not simultaneously diagonalizable, or more generally in terms of their spectral measures and associated Borel sets etc. However, since there seems to be no other criteria for assigning the details of the Hamiltonians, the characterization seems to be begging the question. p and q are incompatible because P and Q are non-commuting because p and q are incompatible…… I am obviously missing something that allows everyone to tell me that E and M are compatible (unless they just tell me that it is because the Hamiltonians commute, which again would be begging the question). Hence I must read further. But I am always grateful for guiding hints.

14. Feb 19, 2008

minor correction

oops, I meant "Hermitian operators" instead of "Hamiltonians" in that last diatribe, sorry. But that does not change my point.

15. Feb 19, 2008

### Marco_84

To answer this kinds of questions you should start reading an axiomatisation of QM and the problem of mesaure (not math) in physics in generally. I have a good book, but is in italian :(
It start explaining what is the difference between an experiment of first kind and of the second kind... introducing then the postulates and stuff...
I dont know... but im pretty sure that a lot of this work was done already by Von neumann and Gleason...

If you cannot find anything that you can appreciate i think i would try to translate it...

regards
marco

16. Feb 19, 2008

Grazie, Marco. Italian is a language that I read without difficulty (although I know longer use it actively, so speaking and writing it has dropped out of my capabilities, but I still read it). As far as the measure problem in general in Physics, I do have several references on that as well. (Most expositions are similar to Peeble's exposition, chapter 4 of his "Quantum Mechanics", although a somewhat different approach is taken by Nielsen and Chuang in their "Quantum Computation and Quantum Information".) All of these do put a few more steps, mainly working from Schrödinger's equation, in the "circolo vizioso" that I mentioned. So many steps, that one can lose sight of the vicious circle, but after I worked through these, the vicious circle seemed to still be there: there was always a step in which one was, essentially assuming compatibility or non-compatibility. I intend to work through them again to see if I can resolve the problem: that is, find some general criteria to find out, given any two observables without any hidden assumptions about their compatibility, to be able to test if they are compatible or not by any means besides experimental ones.

17. Feb 19, 2008

### per.sundqvist

You don't have to buy anything, it's here: http://people.seas.harvard.edu/~jones/ap216/lectures/lectures.html. EM-theory, QM and second-quantization (QED-theory). There, dE*dB>constant, is for example derived (similar to Heisenberg) and the Hamiltonian for the harmonic oscillator H=x^2+p^2 is an analogue to the EM-energy H~E^2+B^2, and operators could be thought of as: p~d/dx and B~dE/dx (not really but as a weak analogy it may help)

Per

18. Feb 19, 2008

Thanks. er...where?

Thanks very much, per.sundqvist. I have bookmarked it, and will be looking through it carefully. At a first glance, I do not see dE*dB>const anywhere -- the closest I see is in III-4b of "Representation of Photon States" where dE*dH = ......; in that and the preceding section ("Quantization of the E-M Field" that deals with the canonical quantization of the fields), B is only mentioned explicitly in a formulation of Maxwell's Eq's, and implicitly in H. Do you mean that I should carry through the process to come up with that, or is this inequality derived elsewhere in the notes?

19. Feb 19, 2008

### peter0302

I've never seen a simple or intuitive axiomizaiton of QM. I don't think there really is a "good one." Certainly you cannot boil it down to something as simple and intuitive as the 5 Euclidean postulates or Einstein's postulates of SR.

The closest I've seen is many-worlds, but that nonetheless assumes a priori the validity of the Schrodinger Equation.

20. Feb 19, 2008

### reilly

Sad to say, in general the quantum E and B fields do not commute -- ultimately this can be seen by expressing the fields in terms of creation and destruction operators. All this stuff is discussed in detail in Mandel and Wolf, Optical Coherence and Quantum Optics, and in many other books. A Google on E&M field commutation rules be useful, no doubt.Regards,Reilly Atkinson

Last edited: Feb 19, 2008
21. Feb 19, 2008

### George Jones

Staff Emeritus
Why should there be such a thing?

The further we move from the realm of everyday experience, the further we move from the real of everyday experience

22. Feb 19, 2008

### peter0302

Because - I know I'm going to piss off some people, but - I see the debate between axiomatic approaches to QM and "shut up and calculate" approaches QM as _somewhat_ analogous to evolution vs. creationism. On the one hand, you have very simple rules that are applied objectively (if non-deterministically) over very long periods of time to create complexity. But the complexity is not beyond our understanding the fundamentals. On the other hand, you have rather blind acceptance of "the way things are" without asking for a more fundamental, deeper explanation. I'm sorry to say Copenhagenism falls into the latter category IMO and I find it extremely unsatisfying.

23. Feb 20, 2008

### Marco_84

What do you mean with "simple or intuitive"... I think that obvously QM is not intuitive...
so??
Well i think ill translate from a nice book the postulates of QM and see what people think.

regards
marco

24. Feb 20, 2008

### Marco_84

Since QM is born from experimentally evidence this approach im going to show is the best for me, it is an operational one.
We will call the system under exam S.

Experiment of FIRST and SECOND KIND:
It an experiment of FIRST KIND we are sure that making a mesurement on one or more observables and performing the same operation after a while we get the same entries.
Say A, B,C are our observable and a,b,c the value we mesaure and the arrow (----->) means making the experiment and mesauring.

1)A,B,C----->a,b,c

after a while

2)A,B,C----->a,b,c

(Later we will call a,b,c eigenvalues).
Another thing; in this experiment we have a miximal information in the sense that
$$\triangle A,B,C=0$$.

Instead in a SECOND KIND experiment one ore more observables are mesaured with operations that imply a partial/total loss of information on the value they assume after the observation. (In this contest observation=mesurament=making exp).

DEFINITION: COMPLETE ENSEMBLE OF COMPATIBLE OBSERVABLES:

Generally speaking we say that 2 observable A and B of system S are compatible if a FIRST KIND mesurament on A do not imply indetermination on B, and viceversa. We say that $$\triangle A$$ and $$\triangle B$$ are statistically NON-correllated.
Example (A=polarization of a photon; B=Its momentum). An important thing is that the observables are INDEPENDENT, in the sense that they are not functions of each others.
We can of course add another one, say C and see how the observation will be.
WE affirm that an experimental set-up is complete when we have the max information on S and adding more observables and maikng the experiment of FIRST KIND we lose some knowledge. This definition is not absolute since new experimental datas can suggest us the introduction of new Observables (see the electron spin).
So we have what follow:
For each experimental set-up of S prepared in the pure state X there exist at least one set of compatible observables (A,B,C...) wich ------->(a,b,c...) always.

Sorry i forgot to tell what is a pure state X:
For every $$\epsilon >0$$ there exist X such that $$\triangle A<\epsilon$$ for every observable A.
Note that this definition is still good in a classical picture. Just to tell there exist also the statistical mixture in contrast to pure states.

This is was a short intro to what I was teelling you last time.

POSTULATES:

1) The pure states of a system S are in a one to one mapping wit the rays of a complex and separable Hilbert space Hs. So it is a projective Representattion and the space is something like H/C*.

2)Every observable is represented by a linear self-adjoint operator that acts on Hs.

3)If $$|\psi>$$ is a vector of Hs representatives of the ray correspondent to the pure state X at the moment of the mesuration of the observable A, the probability to obtain a value in the borelian subset b is :

$$Pr_{A}(b|x)=\frac{<\psi|\widehat{E}_{A}(b)|\psi>}{<\psi|\psi>}$$

where $$\widehat{E}_{A}(b)$$ is the spectral family of the operator $$\widehat{A}$$.

4)The undisturbed temporal evolution of a physic system S from the initial time 0 to the final time t is realized with a simmetry transformation T(t,0) acting on Hs.

From wigner theorem we know that this Trasformation in implemented via a unitary or Anti-Unitary operator U(t,0); in this case is unitray since T(0,0)=1.

5)Th e Observables of a system of N identical paticles correspond to self-adjoints operators that commute with every permutation P wich belong to the symmetric group SN. The Hilbert space become:

$$H^{(N)}_{\varsigma}=P_{\varsigma}H^{(N)}$$

where $$\varsigma$$ is +1 for bosons, -1 for fermions and $$P_{\varsigma}$$ is the total (anti)symmetrizer:

$$P_{\varsigma}=\frac{1}{N!}\sum\varsigma^{|P|}P$$

This include the Pauli exclusion principle...

I hope that helped a little bit.

regards
marco.

PS.
Many thanx to prof Onofri and Destri for their book.

25. Feb 22, 2008

### reilly

Marco_84. So you got axioms. That's all well and good, but what next? How do you go from your axioms to a description of Hydrogen energy levels? We, are, after all, talking physics
Regards, Reilly Atkinson