Doppler Effect: Minimum Recessional Velocity for Its Effect

[jay.yoon314]

Consider a stationary observer and an emitter of light radiation that may be either receding or approaching at a variety of nonrelativistic velocities, with the velocity of the emitter having only a radial component and no transverse component.

I was wondering whether the Doppler Effect, specifically for light, is observed for any recessional or approaching purely radial velocity, no matter how small the magnitude of that velocity. For example, if the emitter is receding at 0.00000001 m/s, will some tiny red shift exist?

If it did, how would the following paradox be resolved? Consider a sample of gas with N hydrogen atoms, where N is very large, and the absolute temperature T of the gas is nonzero so that the atoms are all moving in a multitude of directions. Let one of these N atoms be in its first excited state and all N-1 others be in their ground state, with the N hydrogen atoms as a whole comprising an isolated system. Let these hydrogen atoms also be spread out over a large enough space that the mean free path for a collision is long enough for its probability to be close to zero for the time-scale of this thought experiment.

That hydrogen atom in its excited state now drops down to its ground state, emitting a photon of fixed frequency. Since all of the other N-1 hydrogen atoms are moving with respect to this one, they have a nonzero velocity, and hence also a nonzero radial velocity with respect to the hydrogen atom that emitted the photon. If the Doppler Effect is exhibited for arbitrarily small radial velocities, then the frequency shift and consequent energy shift of the emitted photon will result in none of these N-1 hydrogen atoms being able to absorb this photon even in principle, since quantization of electron energy levels defines an exact transition energy requirement.

This implies that photons that are emitted by an atom of a particular element cannot be absorbed by another atom of that same element, and that emitted light essentially becomes "tired, dead, or unreactive," in that emitted photons are degraded and must forever hence propagate through space, unable to ever be absorbed again. One can fairly surmise that a reductio ad absurdum has been reached.

Therefore, assuming that there is no minimum radial recessional or approaching velocity required for the Doppler Effect (for light/radiation) to come into effect leads to the absurdity just mentioned; hence, there is such a minimum "threshold" velocity, which then implies that Doppler shifts are quantized.

Is it possible that physicists haven't noticed that Doppler shifts could be quantized since the phenomenon has only been measured with recessional or approaching radial velocities that are "too large?"

Thanks,
Jay

 

[Bobbywhy]

jay.yoon314,

As for your question, “For example, if the emitter is receding at 0.00000001 m/s, will some tiny red shift exist?"

Yes, the Doppler shift will exist owing to our trust in the laws. But whether or not we can detect such a tiny difference in frequency/wavelength depends on the characteristics of our detector. Owing to limitations in the resolving power such fine spectral resolution may not be possible.

When we measure radiation from a galaxy or a quasar we are not detecting any individual photon’s energy. The energy we detect is an uncorrelated continuum. It is the average of many emitters that we use to determining the redshift of that emitting object.

"Redshift quantization" has also been called redshift periodicity, redshift discretization, preferred redshifts, and redshift-magnitude bands. Are you suggesting we resurrect the original ideas of Tifft, Cocke, Croasdale, et al? If yes, then you are up against lots of mainstream astrophysicists,astronomers, and cosmologists who will question the validity of the idea.

 

[harrylin]

[..] I was wondering whether the Doppler Effect, specifically for light, is observed for any recessional or approaching purely radial velocity, no matter how small the magnitude of that velocity. For example, if the emitter is receding at 0.00000001 m/s, will some tiny red shift exist?

There is of course a difference between the existence of something and measuring it; but in principle there is no limit.

[..] quantization of electron energy levels defines an exact transition energy requirement.

I'm not sure if that is correct; in any case, also the receiving atoms are not exactly in rest and so on, so that the resonance peak is not infinitely small. See for example Mössbauer absorption experiments and a related discussion:
- http://prola.aps.org/abstract/PR/v129/i6/p2371_1
- http://en.wikipedia.org/wiki/Mössbauer_spectroscopy

 

[jay.yoon314]

jay.yoon314,

As for your question, “For example, if the emitter is receding at 0.00000001 m/s, will some tiny red shift exist?"

Yes, the Doppler shift will exist owing to our trust in the laws. But whether or not we can detect such a tiny difference in frequency/wavelength depends on the characteristics of our detector. Owing to limitations in the resolving power such fine spectral resolution may not be possible.

When we measure radiation from a galaxy or a quasar we are not detecting any individual photon’s energy. The energy we detect is an uncorrelated continuum. It is the average of many emitters that we use to determining the redshift of that emitting object.

"Redshift quantization" has also been called redshift periodicity, redshift discretization, preferred redshifts, and redshift-magnitude bands. Are you suggesting we resurrect the original ideas of Tifft, Cocke, Croasdale, et al? If yes, then you are up against lots of mainstream astrophysicists,astronomers, and cosmologists who will question the validity of the idea.

I'm vaguely familiar with the redshift-magnitude band theory that was proposed. Although both theirs and my ideas can be described as "redshift quantization," what they are specifically discussing are very, very different.

The ideas of Tifft, Cocke, Croasdale, et al. that you mentioned are, I believe, along the lines of saying that, i.e. galaxy recessional velocities appear to be clustered around regularly spaced intervals of velocity, with the intermediate intervals between these peaks being sparser in the number of galaxies. In other words, Hubble's law is not strictly linear, but has periodic oscillations above and below that linear relationship not unlike, perhaps, the "short-run" business cycles of expansion and contraction that accompany the overall "long-run" increase in GDP.

This is not at all what I am referring to at all. The ideas of Tifft et al claim that there is a redshift quantization in the z-values of galaxies receding from us, but my point isn't at all in reference to galaxies, astronomical observations of galaxies, or even any specific context in which Doppler shifts/red shifts are observed. I am instead asking whether the Doppler shift as a general phenomena in and of itself, could be quantized.

 

[Dale]

quantization of electron energy levels defines an exact transition energy requirement.

This is not correct. The uncertainty principle plays a role here making the energy requirement slightly uncertain. The uncertainty in the energy is related to the uncertainty in the time, transitions which happen very quickly have a very large uncertainty in the energy. This leads to a broad intrinsic natural linewidth.

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

 

[sophiecentaur]

There is a phenomenon called the Mossbauer effect in which a movement of a thin steel plate at only a few mm per second is enough for the Doppler effect to be observed.

I did the experiment at Uni in 1965, only a few years after it was published. It involved a gamma source and a Geiger Muller detector. The inelastic scattering of photons by the steel atoms has a very narrow band resonance - enough to detect the minute change in frequency when one plate moves slowly away from or towards another plate.

The same effect can also be used to show the relativistic time dilation due to differences in g between two floors in a building, I believe - but we didn't do that one.

 

[jay.yoon314]

This is not correct. The uncertainty principle plays a role here making the energy requirement slightly uncertain. The uncertainty in the energy is related to the uncertainty in the time, transitions which happen very quickly have a very large uncertainty in the energy. This leads to a broad intrinsic natural linewidth.

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

I see, I see. I was aware of line broadening but I didn't realize that the uncertainty principle was one of the causes of it. That really clears things up.

One follow up question: In the Wikipedia article on spectral lines, it says the following:

Natural broadening: The uncertainty principle relates the lifetime of an excited state (due to the spontaneous radiative decay or the Auger process) with the uncertainty of its energy. This broadening effect results in an unshifted Lorentzian profile. The natural broadening can be experimentally altered only to the extent that decay rates can be artificially suppressed or enhanced.[1]

Since the above description of linewidth broadening due to the uncertainty principle is discussed in the context of the lifetime of an excited state, it seems to be a phenomenon that would pertain to emission lines.

Does this same phenomena of linewidth broadening due to the uncertainty principle specifically occur with absorption lines as well as emission lines? I know that there are ways in which absorption lines are broadened, but since the absorption of a photon and the consequent transition from a lower electron energy state to a higher one doesn't seem to be a spontaneous process, as opposed to a transition in the other direction from a higher energy state to a lower one which clearly is a spontaneous process.

There seems to be no analogous concept of a "lifetime" of a ground state, as the lifetime of a ground state without any external energy input is, as far as I know, indefinitely long; i.e. an electron in a ground state cannot really "decay" spontaneously (since it is already at its lowest energy state), and it cannot be "promoted to a higher energy state" spontaneously (since that process is by definition nonspontaneous; it doesn't happen without an external contribution of energy). So the "lifetime" of a ground state in absence of external energy inputs seems to be infinite, meaning that there is no uncertainty in its lifetime; i.e. if you know something under a specified set of conditions will never happen, then that knowledge is "absolutely certain," as "never" seems to be an "absolute" statement. Or is that this absorption process can occur spontaneously under the uncertainty principle also, with a nonzero probability?

Essentially, my question can be phrased more succinctly as:

Do the mechanisms by which emission line broadening occur, and in particular emission line broadening as a result of the uncertainty principle, operate in analogous or identical ways in the broadening of absorption lines also?

 

[Dale]

Hmm, that is an interesting question that I didn't think of. I should start by saying that I am just making an educated guess here, so please check it with other sources.

The equations of QM are time-reverse symmetric. Since absorption and excitation to a higher energy state is the time-reverse of emission and relaxation to the ground state the line width should be the same.

Again, this is just a guess.

 

[mfb]

The equations of QM are time-reverse symmetric. Since absorption and excitation to a higher energy state is the time-reverse of emission and relaxation to the ground state the line width should be the same.

Again, this is just a guess.

And it is a good guess.
Absorption and (stimulated) emission follow the same rules with the same transition probabilities/amplitudes (with appropriate prefactors to compare them). Therefore, you have to consider the lifetime of both the excited and the ground state, or two excited states for other transitions.

 

[jay.yoon314]

And it is a good guess.
Absorption and (stimulated) emission follow the same rules with the same transition probabilities/amplitudes (with appropriate prefactors to compare them). Therefore, you have to consider the lifetime of both the excited and the ground state, or two excited states for other transitions.

I see. The connection between absorption and stimulated emission certainly makes sense, and the uncertainty principle resolves the paradox in my original post as well as the analogy between emission lines and absorption lines.

But what about quantization and the Heisenberg uncertainty principle considered in and of themselves, and their possible connection/apparent contradictions?

It seems quite strange that that two fundamental realities of quantum mechanics, namely the Heisenberg uncertainty principle, and the quantization principle are both realities at very small scales. What I mean is that it doesn't seem necessary, from an aesthetic point of view, to have both the uncertainty principle and quantization. For instance, if the nature of matter at very small scales was only uncertain, and not quantized, or vice versa, in which it was quantized, but "certain," that kind of universe seems somewhat plausible to me, as at least you either have, in the former, fuzziness without the specifically allowed values, and in the latter, complete harmony and certainty. I "feel" that the coexistence of uncertainty and quantization is quite similar to the wave-particle duality principle, which is quite counterintuitive; with the uncertainty perhaps being a dimension of the "wave" nature of matter and the quantization being a dimension of the "particle" nature.

I'm speculating that the two can be reconciled by thinking of it as the following: even if the the world is uncertain at the very small scale, the uncertainty always is centered about some average value so that these average values can be taken to be the "allowed" quantized values of whatever physical quantity one is looking at.

Can either the uncertainty principle be a consequence of the fact that physical quantities (such as energy) are quantized, or vice versa in that quantization of physical quantities (some of them) is a consequence of their uncertainties? Also, if it were indeed the case that one or the other were more "fundamental" in a sense, which one do you think would be so?

In my opinion, uncertainty would be the more fundamental principle, because the uncertainty relation seems to exist between various pairs of variables such as energy and time, and also momentum and position, whereas quantization is also really important, but there's a greater chance that it could an "emergent phenomenon," especially in light of its interpretation as the "average value" that the uncertainties settle at.

What are your thoughts?
Thanks. (If what I am posting is off-topic or not in line with the rules of the forum in a subtle/more obvious way, feel free to tell me.)

 

[Dale]

Neither the uncertainty principle nor quantization are added to QM ad-hoc nor for aesthetic reasons. They both fall naturally out of the formulation of the state as a vector in a Hilbert space and the observables as operators on that space. I.e. the fundamental principles are the state vector and the observable operator. Both uncertainty and quantization are derived from that.