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

In summary, the non-commutativity of the momentum and position operators means that they give different results when measuring the quantities A then B of state S. This experiment proves that the algebraic relation between the angular momentum operators implies that they do not commute.
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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?

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?

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

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.

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.

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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 can't give you a theoretical derivation of this (not at this moment).

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.

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.

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.

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

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.)

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.

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

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.)

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 I am not that smart for those kind of books :)

regards
Marco

"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.

minor correction

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

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.

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 don't know... but I am 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

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.

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

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?

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.

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

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peter0302 said:
I've never seen a simple or intuitive axiomizaiton of QM.

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

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.

peter0302 said:
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.

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

Since QM is born from experimentally evidence this approach I am 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...) which ------->(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 physics system S from the initial time 0 to the final time t is realized with a symmetry 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 which 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 thanks to prof Onofri and Destri for their book.

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

reilly said:
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

Obviously we can modelize an H Atom at different levels...approximations.
Lets assume Non relativistic speed of electron and proton mass infinite:

1) I find the proper repr of the hilbert space... counting also the Spin, and reminding that the potential is a central one i would say that a good one is :$$L^{2}(S^{2})\otimes L^{2}(\mathbb{R})\otimes \mathbb{C}^2$$.
So when solving the dynamic later will be easier to find the spherical harmonics and laguerr's polynomial, if i remeber right.

2)what are we search for? the energy of the ground state and the other excited??, Our Observable would be H and its eigenvalues have to be compared with the Spectroscopy ones.

3)the quantum numbers usually are referred as (n,l,m,ms)... At this stage of approx the energy depend only on n ---> E=E(n)~$$1/n^2$$

4)Since H does not depend on time we can factor out the time by the solution and solve the eigenvalue proble H|psi>=E|psi>.
standing waves...

Only one particle-----> no problem with statistics...

what else... many more obviously...

regards
marco

If you have really 'axiomized' QM than those axioms should be the basis for answering such questions a: Why does the Schrodinger EQ have kinetic and potential energy terms? Where does the Hamiltonian come from? What are the appropriate boundary conditions?

More challenging: how do your axioms deal with a quantized E&M field interacting with a classical current. That is, how do your axioms give you the tools to do a simple QFT problem?

Regards,
Reilly Atkinson

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reilly said:
If you have really 'axiomized' QM than those axioms should be the basis for answering such questions a: Why does the Schrodinger EQ have kinetic and potential energy terms? Where does the Hamiltonian come from? What are the appropriate boundary conditions?

Im not the founder of QM...
i think that Schrodinger eq has K+V term cause a physical system (electron this time) it is suppose to travel in space and time so K represents the "movement energy", it is bounded so we modelize with a potential V which represents a "configurational energy".

Obvyously the problem is always to treat V in more than two body problems... :)

But this concepts are classical not only QM, the canonical formalism its not a new tool for modern physicist H=K+V is an old formula.

The problem arise in QM where (p,q)------>(P,Q) the fundamental observables become a pair of non-commuting operators... and fortunately, always the founders, suggested us how to work; I am referring to things like the switching from poisson's brackets to commutators.
The BC depends upon the problem.

reilly said:
More challenging: how do your axioms deal with a quantized E&M field interacting with a classical current. That is, how do your axioms give you the tools to do a simple QFT problem?

Regards,
Reilly Atkinson

Thats right! the problem of coupling QM partcles to Classical fields is solved quantizing also the classical relativistic fields. And in the relativistic scheme were paticles can be crated and annhilated N is not anymore a good quantum number in fact [H,N]!=0.
The tools of second quantization in any case are usefull also in many other branches of physics as many-body theories at low enrgies wher c<<<1 and we don't need anymore postulates on the vacuum state or something else...
My post was about QM non QFT...

regards
marco

## 1. What are Maxwell's equations?

Maxwell's equations are a set of four mathematical equations that describe the behavior of electric and magnetic fields in electromagnetism. They were first formulated by James Clerk Maxwell in the 1860s and are fundamental to our understanding of electricity and magnetism.

## 2. What is the concept of orthogonality in relation to Maxwell's equations?

Orthogonality refers to the perpendicular relationship between electric and magnetic fields in Maxwell's equations. In other words, the electric field and magnetic field are at right angles to each other, and this relationship is crucial for understanding how they interact with each other.

## 3. How do electric and magnetic fields interact in electromagnetic waves?

According to Maxwell's equations, changes in electric fields will create magnetic fields, and changes in magnetic fields will create electric fields. This continuous exchange between the two fields is what allows electromagnetic waves to travel through space, carrying energy and information.

## 4. What are the implications of Maxwell's equations for technology?

Maxwell's equations are the foundation of modern technology, as they provide a mathematical understanding of how electricity and magnetism work. They have led to the development of technologies such as radio, television, and cell phones, and continue to be crucial in fields such as telecommunications, electronics, and power generation.

## 5. How do Maxwell's equations relate to the behavior of light?

Maxwell's equations also describe the behavior of light as an electromagnetic wave. This means that light is made up of oscillating electric and magnetic fields that travel through space at the speed of light. Understanding Maxwell's equations is essential for understanding the properties and behavior of light.

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