A classical wave as sum of many coherent quantum wavelets?

[lightarrow]

Merzbacher - Quantum mechanics, second edition, chapter 1 page 4,writes:

"A classically observable wave will result only if
the elementary wavelets representing the individual quanta add
coherently. "

Is this analysis correct?

The context is the following:

We may read (1.3) [ë = h/p] the other way around and infer that any
wave phenomenon also has associated with it a particle, or quantum, of
momentum p = h/ë. Hence, if a macroscopic wave is to carry an
appreciable amount of momentum, as a classical electromagnetic or an
elastic wave may, there must be associated with the wave an enormous
number of quanta, each contributing a very small momentum.
A classically observable wave will result only if the elementary
wavelets representipg the individual quanta add coherently.
For example, the waves of the electromagnetic field are accompanied by
quanta (photons) for which the relation E = hv holds. Since photons
have no mass, their energy and momentum are related by E = cp. It
follows that (1.3) is valid for photons as well as for material
particles. At macroscopic wavelengths, corresponding to radio
frequency, a very large number of photons is required to build up a
field of macroscopically discernible intensity. Yet, such a field can
be described in classical terms only if the photons can act
coherently. This requirement, which will be discussed in detail in
Chapter 22, leads to the peculiar conclusion that a state of exactly n
photons cannot represent a classical field, even if n is arbitrarily
large.

 

[Bill_K]

Yes, Merzbacher has an unusually good discussion of the transition between quantum and classical, continued later in Chaps 15 and 22. Now I know why they made me buy Merzbacher!

 

[Naty1]

"
"A classically observable wave will result only if
the elementary wavelets representing the individual quanta add coherently. "

I take this to mean the real part of complex [real and imaginary] component waves add. Is this a modern perspective??

Your quote sounds ok to me...other views...

Albert Messiah, QUANTUM MECHANICS,1958:

, the idea of quantum field theory is that quantum fields are the basic ingredients of the universe, and particles are just bundles of energy and momentum of the fields... The corpuscle view of matter reflects localized grains of energy and momentum in contrast to the picture of a continuous distribution of energy and momentum.

so 'localized' would be coherent.

I still remember coming across this in college many years ago:

[Halliday and Resnick page 1015-1021: A boundary condition, where the wave equation of a particle or a vibrating string is zero, say fastened at both ends, quantizes the wavelength. Localizing, or bounding, a particle in space leads to energy quantization. Once the wave matter duality is accepted, setting boundary conditions to zero and formulating a plane wave in the form e to the iz immediately leads to sinz plus icosz and via trigonomic identiy 2SinzCosz which is a basic form of the Schrodinger wave equation.]

Messiah:

Any wave psi (x) is a superposition of plane waves exp (ikx)….To measure a particle location in a region delta x, the various plane waves forming psi(x) have constructive interference within that interval and destructive interference outside the interval.

So if you are starting out, I would urge you to remember that the deBroglie wavelength [of a matter particle] follows these conditions!

 

[Naty1]

A related way to think about such wave components is explained here:

In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines (or complex exponentials). The study of Fourier series is a branch of Fourier analysis.

http://en.wikipedia.org/wiki/Fourier_decomposition

this stuff relates back to my prior post exp [ikx] that is, ε^ikx]...

the relationship be between trig functions and 'e' still amazes...

 

[lightarrow]

[ë = h/p]

Clearly this was: λ = h/p

 

[lightarrow]

Yes, Merzbacher has an unusually good discussion of the transition between quantum and classical, continued later in Chaps 15 and 22. Now I know why they made me buy Merzbacher!

Thank you Bill, this encourages me to keep studying the book

 

[lightarrow]

A related way to think about such wave components is explained here:

http://en.wikipedia.org/wiki/Fourier_decomposition

this stuff relates back to my prior post exp [ikx] that is, ε^ikx]...

the relationship be between trig functions and 'e' still amazes...

Yes, but I'm not sure this is exactly what we are discussing here. That phrase of Merzbacher's seems to imply, to me, that we can decompose the wave function of a macroscopic system, made of many particles, as the sum of every particle's wave function, instead of being the wave function of the system as a whole. But I can have understood incorrectly...

 

[Naty1]

Hey, lightarrow, I reread the OP...maybe you are right!..

but the descriptions seem complementary in any case...Althought the presentation of stuff in these forums is often not in a logical learning sequence, one sure gets a lot of different perspectives...and that is valuable.

Bill_K:
that BETTER be a good book[!] because I just bought a used copy at Amazon based on your recommendation...

I have been meaning to get another to compare with my Albert Messiah text and
Merzbacher's seemed a reasonable choice...