Frequency oscillations and Planck's constant

[DiracPool]

This is a two part question and something I think I should know off the top of my head, but I don't, and I can't find a ready answer to it on at least a cursory search.

The first part has to do with defining a frequency of a wave in general. I can sit here and tap out a beat on a drum that executes 5 cycles (taps) in 2 seconds. Ostensibly, that appears to me to be a beat frequency of 2.5 cycles per second (cps), or Hertz. I can similarly imagine a water wave or electromagnetic wave beating out the same frequency. However, I've never heard of a fractional frequency as such. Do these exist? In my example of my tapping on a drum again, my "rough assessment" is that the frequency was 2.5 cps, but I'm probably off of that a little in each cycle, so the actual frequency of the oscillations are likely described by an irrational number. Yet again, I've never seen a fractional frequency described. Perhaps it has to do with only an integer number of cycles being able to fit on a unit circle? I don't know the answer to this, please help.

The second part of the the question relates the above to the Planck-Energy relationship, E=hf. I'm assuming that you find the energy of an electromagnetic (EM) wave by taking the integer value of it's frequency and then multiplying that by Planck's constant. Sounds fair enough. But why are we so confident that every EM wave has an exact integer value that we can use for the calculation?

I think a confusion, for me at least, arises here over the meaning of the concept of "quantization." My understanding of quantization is generally that it arises because standing wave patterns that describe particles must have integer values that fit into a 2π circumference. But then you also have Planck's constant that appears to quantize energy, momentum, etc. in terms of the numerical value "chunk" of that constant. Which of these two serves as the canonical "chunk" of quantization? Again, this relates back to the traveling wave example above. If the oscillation is not confined to some sort of potential well that constrains an integer value of it's frequency, then isn't there a relatively continuous distribution of non-integer frequencies possible for say, any given electromagnetic wave?

 

[mfb]

What do you mean with "fractional frequency"? A frequency does not have to be a multiple of 1 Hz - why should it, the definition of a second is arbitrary.

I'm assuming that you find the energy of an electromagnetic (EM) wave by taking the integer value of it's frequency and then multiplying that by Planck's constant.

You don't find any integer. You just find the frequency (as accurate as possible, if it is an actual measurement).

This has nothing to do with quantization. Quantization will tell you that just some specific frequencies can excite an atom, for example, but these atoms do not care about the units we use to express those frequencies.

 

[Vanadium 50]

If we defined a new unit of time, the di-second, equal to 2 seconds, your 2.5 cps magically becomes an integer: 5 cpd. Since the physics cannot depend on the units we measure it with, that means there is no restriction that frequencies be integral.

It's probably good to get Part 1 straight before going on to Part 2.

 

[DiracPool]

If we defined a new unit of time, the di-second, equal to 2 seconds, your 2.5 cps magically becomes an integer: 5 cpd.

A frequency does not have to be a multiple of 1 Hz - why should it, the definition of a second is arbitrary.

In relation to these responses, I guess what I may be asking, is that, can't we all agree on what one second of time is in each of our own proper time frames? And if we took that frame, and looked at all of the the EM radiation passing us, wouldn't there necessarily be frequencies of fractional values in relation to an agreed upon standard of what "one second" of time was?

that means there is no restriction that frequencies be integral.

This is kind of what I'm getting at.. I've never seen a characterization of an EM wave, or "photon," as having a fractional frequency in print, ever. They are always described by integer values. I'm sure there is just some basic thing I'm not getting here, but whatever it is, I'm not getting it, so that's what I'm trying to figure out

 

[mfb]

wouldn't there necessarily be frequencies of fractional values in relation to an agreed upon standard of what "one second" of time was?

Yes, sure. There is nothing special about integer values. That is the main message of posts 2 and 3.

Frequencies for photons are often large numbers - like 1.53*10¹³ Hz. That is not an integer, but if the number is not known to 14 digits you don't see the fractional part.

 

[DiracPool]

Frequencies for photons are often large numbers - like 1.53*1013 Hz. That is not an integer, but if the number is not known to 14 digits you don't see the fractional part.

Wow, that's really interesting. I didn't know that. I guess it wouldn't make a difference anyway in terms of the energy value if you were multiplying it by an irrational number such as Planck's constant, the value would be irrational regardless of whether it was an integer or not. I guess I was just thrown off because I've never seen fractional frequency values. Thanks for the insight.

 

[DiracPool]

A: Just as a follow up, can we then say that the electromagnetic spectrum is an essentially continuous distribution of wavelengths/frequencies? And when we talk about quantization, we are talking strictly about an energy that has to be broken up into discrete values of multiples of the Planck constant, namely that value multiplied by the frequency of the photon?

B: Is there any constraint on this continuous distribution of EM wavelengths? i.e., can we say something like that, at the very least, the value of these wavelengths must be quantized or "granularized" at the distance of the Planck length? I've read that there is no known theoretical limit to how small the wavelength of a photon can be, but there's an argument it can't get any shorter than the Planck length. Is this true? Is there some other barrier that would prevent a photon wavelength from getting even this small? I think the highest energy photons ever recorded are many order of magnitude longer than the Planck length..

 

[mfb]

A: Just as a follow up, can we then say that the electromagnetic spectrum is an essentially continuous distribution of wavelengths/frequencies?

Sure.

And when we talk about quantization, we are talking strictly about an energy that has to be broken up into discrete values of multiples of the Planck constant, namely that value multiplied by the frequency of the photon?

The Planck constant does not have units of energy. I'm not sure what you are asking.
If you have some specific (arbitrary) frequency only, then you cannot absorb arbitrary energy values, but only multiples of the energy of a photon, which is h*f.

B: Is there any constraint on this continuous distribution of EM wavelengths? i.e., can we say something like that, at the very least, the value of these wavelengths must be quantized or "granularized" at the distance of the Planck length?

Probably not. The Planck length could be a lower limit (=the Planck frequency could be an upper limit), at least we don't know what happens at higher frequencies.