# Classical fields and quantum fields

## Main Question or Discussion Point

We describe classical fields with electromagnetic WAVES. Why can quantum theory describe the electromagnetic field with harmonic oscillations? An electromagnetic wave has a certain direction where it moves with a certain velocity. And an oscillation just oscillates up and down. So, what are the analogies of these two things?

thanks for everyone who can give an answer

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vanesch
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Kruger said:
We describe classical fields with electromagnetic WAVES. Why can quantum theory describe the electromagnetic field with harmonic oscillations? An electromagnetic wave has a certain direction where it moves with a certain velocity. And an oscillation just oscillates up and down. So, what are the analogies of these two things?
If you solve for classical fields, you see that their solution space is build up from different modes (this is sometimes called "separation of variables" and you separate the spatial variables from the time variable), so you write your general field solution, satisfying boundary (but not initial) conditions:

E(r,t) = a1(t) E1(r) + a2(t) E2(r) +...

and it turns out that the solutions for a1(t)... are of the form A sin(w1.t) + B cos(w1.t).
Each term in this sum is called a field mode.

These constants (A and B) for each a(t) are fixed by the initial conditions and give you your solution E(r,t) that satisfies boundary and initial conditions.
So when looking at the dynamics in time, it is as if the final E-field is just a composition of different harmonic oscillators, described by a1(t), a2(t) ...
The field is then just seen as a kind of book keeping device that helps us keep track of all these different oscillators, but just specifying A and B of each of them is equivalent to this description.
So there is a 1-1 relationship between a set of oscillators of frequencies w1, w2, ... and a field solution E(r,t).

cheers,
Patrick.

scroll down to the https://www.physicsforums.com/journal.php?s=&action=view&journalid=13790&perpage=10&page=7 [Broken] -entry

regards
marlon

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Besides, think of a quantumfiels as if it were a mattress of which the springs are harmonic oscillators

marlon

reilly
Even though the "oscillator-wave" connection is tricky, it is as basic to quantum theory as is the Schrodinger Eq. The connection started wih Heisenberg's matrix mechanics of the oscillator, which merged with classical work relating infinite sytems of springs to wave systems -- usually done in lagrangian style --, yielded standard 2nd quantization in NR QM, with the appropriate unitary xforms between the oscillator and wave descriptions.

That being said, the topic is discussed in excruciating detail in almost any book that covers basic QFT and canonical quantization. (I think the discussion in Mandel and Wolf's book on Quantum Optics is the best, but check out Google. You'll hit lots of paydirt.)

Regards,
Reilly Atkinson

Kruger said:
We describe classical fields with electromagnetic WAVES. Why can quantum theory describe the electromagnetic field with harmonic oscillations? An electromagnetic wave has a certain direction where it moves with a certain velocity. And an oscillation just oscillates up and down. So, what are the analogies of these two things?

thanks for everyone who can give an answer

A classical wave can also be described like a collection of harmonic oscillators, an infinite one.

At each point of the transversal wave you have harmonic motion, site there a harmonic oscillator and ready!

Mhh, if there is only one em wave in space how do we then describe its motion with a harmonic oscillator?

Kruger said:
Mhh, if there is only one em wave in space how do we then describe its motion with a harmonic oscillator?
An EM-wave is just a fluctuation of the EM field. The dynamics is described in terms of the constituent harmonic oscillators. Just think of this : you jump on a trampoline with a very big surface. Once you have jumped, the trampoline surface will vibrate up and down and this vibration will propagate over the surface : that is the wave or the excitation of the trampoline.

Now, in QFT, the transition from a flat trampoline to an oscillating one demands some energy dE. Via E=mc², energy is the same as mass. So through this excitation, you have created a mass : ie a particle. That is what we mean when we say particles are excitations of fields in QFT

regards
marlon

reilly
Kruger -- It's all there in the books and Google, any physics dept. library, and public library, and so on. If you will take the trouble to read any text on the matter for a few hours, I'll guarantee that you will more than answer your questions. The ideas and connections involved are indeed primarily technical, and mathematical. But the best game in town for getting past the beancounting stuff, is immersion in the details --that's how you get understanding and an enhanced intuition. Nobody can do this for you. Many thousands of students have learned this stuff; so can you.
Regards,
Reilly Atkinson

Is there any book you could recommand on this topic?

Anthony Zee's QFT in a Nutshell.

But beware : only read the first few chapters. :)

marlon

Ok, I read some pdf. files and some Weinberg "Introduction to quantum field theory" lectures. If I have understood this then the basic prinziples is the use of walls:

Consider two parallel walls. Between this walls there can only be certain electromagnetic waves. The electric and magnetic field vanishes at the wall. On electromagnetic wave just forms one electromagnetic oscillation between the walls.

If we want to use this for free space we just say the distance of the two walls is "infiniti". Thus we can apply the same prinziple. Only difference is that the harmonic oscillations can have any wave numbers k.

Is that right? Can we just imagine this as a reason why the em-quantum field is described by a collection of independant harmonic oscillations?