I Trouble understanding the SI definition of 1 second

[pbuk]

As far as Noether’s theorem is concerned, it is a relatively advanced topic best covered in analytical mechanics through the Lagrangian formalism (or its corresponding symmetry statements in the Hamiltonian formalism).

Absolutely. Conservation of energy is a concept that is introduced early in Physics (generally at the beginning of high school), and it will take you a long way - 1 or 2 years into college.

You need to learn a lot of mathematics (calculus, ordinary differential equations, vector calculus, partial differential equations, Lagrangian mechanics) before you are ready to use Noether's theorem.

 

[PeterDonis]

I am not getting the exact meaning of "two hyperfine levels of the ground state"

As others have explained, It means that the state that is normally referred to as the "ground state" of the Cs-133 atom is actually two states with slightly different energies, because of the interaction between the single unpaired electron spin of the 6s electron and the spin of the nucleus. The difference between those two energies corresponds to a specific photon frequency.

Perhaps, all excited states are unstable

Yes, this is correct. Since the electromagnetic field is always present, any atom in an excited state can interact with the electromagnetic field to emit a photon and drop to a lower energy state. This is called "spontaneous emission" and does not require any external "trigger" event to happen.

also the fact that higher energy states are unstable may contribute to this happening.

The fact that excited states are unstable is the same thing as interaction with the electromagnetic field happening. See above.

it seems that electrons falling to lower state is just a matter of scientific observations

It has certainly been observed, but you seem to be implying that we don't have a good theoretical understanding of it. That is not correct; the detailed theory of spontaneous emission was worked out decades ago.

like principle of energy conservation is simply what has been scientifically observed and not something that we can derive in Physics unlike principle of conservation of momentum that we can derive from Newton's second law.

First, I think you mean Newton's third law here, not his second; that's the one that is equivalent to momentum conservation in Newtonian physics.

Second, as others have commented, energy conservation can be derived; the derivation uses Noether's theorem. So does the modern derivation of conservation of momentum; Newton's third law in the modern viewpoint is viewed as a consequence of conservation of momentum, not the other way around.

Also, the "energy" in the conservation law derived from Noether's theorem is not the only kind of conserved "energy" in physics. See further comments below.

Beyond observations of electron in an atom through scientific experiments and may be a mathematical model derived scientifically to model electron transition, its impossible to explain why it happens.

This is not correct. See above.

Noerther's theorem that you quoted appears to talk of scenarios with conservative forces

No, it applies, as @Orodruin said, whenever there is time translation symmetry.

Why the authors chose to not explain using Noether's theorem is not clear to me.

Because they are talking about a different kind of "conservation of energy" from the one that Noether's theorem is about. The kind of "conservation of energy" that Halliday and Resnick are saying has been confirmed by many experiments deals with locally measured forms of energy, work done by one system on another, etc.; in relativity, this law is expressed as the divergence of the stress-energy tensor being zero. The conserved "energy" appearing in Noether's theorem when there is time translation symmetry is not the same thing; in relativity, it appears as a conserved quantity associated with a timelike Killing vector field in a region of spacetime, and might include components like "potential energy" in a gravitational field that are not locally measurable.

 

[PeterDonis]

Physics books lie.

I don't think this is a useful description in general, but in this particular case, I think it's incorrect regardless of one's attitude towards the general question involved (discussion of which is probably off topic for this thread). The "conservation of energy" Halliday and Resnick are talking about is not the kind embodied in Noether's theorem; it's the kind embodied in the divergence of the stress-energy tensor being zero. See my post #32 just now in response to the OP.

 

[Orodruin]

I don't think this is a useful description in general, but in this particular case, I think it's incorrect regardless of one's attitude towards the general question involved (discussion of which is probably off topic for this thread). The "conservation of energy" Halliday and Resnick are talking about is not the kind embodied in Noether's theorem; it's the kind embodied in the divergence of the stress-energy tensor being zero. See my post #32 just now in response to the OP.

Well, I was making the statement a bit tongue-in-cheek and general. That physics textbooks do not generally tell the whole truth is to be expected since many times the intended audience is not prepared for that.

 

[hutchphd]

With apologies

 

[sophiecentaur]

Why would an electron in caesium-133 undergo a transition of hyperfine states from a higher to a lower energy state, so as to emit energy?

As far as I understand, this is all about resonance. When in the presence of EM radiation of the right frequency, an atom will absorb and then emit that frequency. This is much the same as with a quartz crystal (but not in the quantum domain). Most transitions are not particularly well defined; in general, the f in the hf is not easily reproduced accurately enough. The particular transition that's chosen for the standard is easy to use (practical reasons) and to reproduce accurately. This is a general principle for choosing any standard for a unit. The atoms can be kept in an environment that gives a consistent value of energy change between the two chosen states. There are many possible choices for standards; this particular one has been chosen.

 

[PeterDonis]

As far as I understand, this is all about resonance. When in the presence of EM radiation of the right frequency, an atom will absorb and then emit that frequency.

More precisely, the cesium clocks that are used as time standards to define the SI second at places like the NIST have cesium atoms that are put in a cavity that contains EM radiation, and then allowed to fall back out of the cavity to see if they emit EM radiation. If they emit EM radiation, it means they must have absorbed it while they were in the cavity; and if the frequency of the EM radiation in the cavity is tuned just right, it will maximize the emission of EM radiation by the cesium atoms when they fall out of the cavity--i.e., the EM radiation in the cavity is in resonance with the desired hyperfine transition frequency that defines the SI second.

More here:

https://www.nist.gov/news-events/news/1999/12/nist-f1-cesium-fountain-clock

 

[haruspex]

This is wrong. First, conservation of energy is related to the invariance of physical laws under time translation (Noether's theorem).

Does that make it wrong to say the conservation law is based on experiment rather than theory? Noether's theorem says it is equivalent to the assumption that the laws are invariant under time translation, not that either is necessarily true.

 

[PeterDonis]

Does that make it wrong to say the conservation law is based on experiment rather than theory? Noether's theorem says it is equivalent to the assumption that the laws are invariant under time translation, not that either is necessarily true.

Noether's theorem, as a mathematical theorem, is of course based on the assumption of time translation symmetry, since that is a premise of the theorem.

But it can be tested by experiment whether or not a particular physical system actually has time translation symmetry, so it can be tested by experiment whether Noether's theorem, the mathematical theorem, is actually true of our physical world: just test to see whether the "energy" the theorem defines is in fact conserved if, and only if, the physical system being tested has time translation symmetry. (Note that the definition of "energy" that is used in the theorem, in itself does not require time translation symmetry; only the proof that it is conserved does.)

All that said, I think most experimental tests of "conservation of energy" are actually not testing for the Noether's theorem version of that, but are testing for something different. See the last part of post #32.

 

[Orodruin]

Noether's theorem, as a mathematical theorem, is of course based on the assumption of time translation symmetry, since that is a premise of the theorem.

No. Just if the conserved quantity is energy. Noether’s theorem generally relates conservation laws to continuous symmetries, whether they are rotations, translations, time invariance, field transformations or something else.

 

[PeterDonis]

Just if the conserved quantity is energy.

Yes, I meant my post to only apply to that case.

 

[profbuxton]

Greetings,
From my reading of the above explanation, am I right in understanding that the particular atom and frequencies are so chosen because they closely match our previous definition of a second and are extremely stable or do we now have to completely "recalibrate" our second to match the new definition.
I quess the same applies to our "newer" definitions of the speed of light etc.

 

[PeterDonis]

the particular atom and frequencies are so chosen because they closely match our previous definition of a second

Only if you think that the number 9192631770 counts as "closely matching". AFAIK the particular atomic transition used was chosen because it was the easiest to measure to a very high degree of accuracy, not because it was any better at matching the old definition of the second.

do we now have to completely "recalibrate" our second to match the new definition.

No. The new second is the same period of time as the old second, to within the accuracy of the two measurements involved (a fraction of the Earth's tropical year--old definition--vs. a particular number of cycles of a given frequency of radiation--new definition). The new definition was chosen because (a) it can be measured more accurately so it can serve as a better time standard, and (b) it allows the meter to be defined in terms of the second by fixing the value of the speed of light in SI units, reducing the number of independent measurement standards that need to be maintained. (Since the new SI second definition was adopted, other units have also been redefined in terms of it--I believe that since the redefinition of the kilogram in terms of a fixed value for Planck's constant and the SI second, there are no other independent standards remaining.)

 

[Orodruin]

Only if you think that the number 9192631770 counts as "closely matching". AFAIK the particular atomic transition used was chosen because it was the easiest to measure to a very high degree of accuracy, not because it was any better at matching the old definition of the second.

I’m not sure this was the question. Yes, the atom was choosen because it can be measured precisely and easily, but the frequency of radiation of the transition (1/9192631770 Hz) was chosen such that the new definition matches the previous one within experimental error.

 

[PeterDonis]

the frequency of radiation of the transition (1/9192631770 Hz) was chosen such that the new definition matches the previous one within experimental error.

Yes, agreed; but choosing another atom would not make it any harder to match the previous definition. It would just mean a different many-digit number would appear in the new definition.

 

[DrClaude]

More precisely, the cesium clocks that are used as time standards to define the SI second at places like the NIST have cesium atoms that are put in a cavity that contains EM radiation, and then allowed to fall back out of the cavity to see if they emit EM radiation. If they emit EM radiation, it means they must have absorbed it while they were in the cavity; and if the frequency of the EM radiation in the cavity is tuned just right, it will maximize the emission of EM radiation by the cesium atoms when they fall out of the cavity--i.e., the EM radiation in the cavity is in resonance with the desired hyperfine transition frequency that defines the SI second.

This is actually incorrect. Emission is never measured, it is rather the number of atoms that have made the transition.

 

[DrClaude]

For what it is worth, the days of caesium being the reference for the definition of the second are counted (pun intended). It is no longer the atom allowing the highest precision. The future standard will probably be based on an ion instead of a neutral atom, using much higher optical frequencies (rather than microwave as in the case of caesium).

 

[sophiecentaur]

This is actually incorrect. Emission is never measured, it is rather the number of atoms that have made the transition.

That sounds to me exactly like an absorption measurement. Apart from using moreu precise conditions, it is much the same as optical and microwave spectrographic astronomy which identifies the presence of resonant structures. In a time standard cell there are plenty of the right atoms present and the pressure, temperature and levels of EM are adjusted to standard values.

 

[PeterDonis]

This is actually incorrect. Emission is never measured, it is rather the number of atoms that have made the transition.

The number of atoms that have made the transition is measured by the intensity of EM radiation emitted from those atoms and hitting the detector, correct?

 

[DrClaude]

The number of atoms that have made the transition is measured by the intensity of EM radiation emitted from those atoms and hitting the detector, correct?

No, it is absorption spectroscopy. They shine a laser to make a transition from the $F=4$ state to an excited state, and measure the amount of absorbed light.

I am not experimentalist, but I think that measuring emission would be very difficult. Spontaneous emission will be isotrope, so you would only collect a small fraction of the light, and then there is also the question of the timing of the emission. Also, that would require detecting microwaves, much more difficult than the IR (852 nm) light used for absorption spectroscopy.

 

[PeterDonis]

it is absorption spectroscopy. They shine a laser to make a transition from the state to an excited state, and measure the amount of absorbed light.

In the NIST article I linked to in post #37, it is described this way:

"As the atoms interact with the microwave signal—depending on the frequency of that signal—their atomic states might or might not be altered. The entire round trip for the ball of atoms takes about a second. At the finish point, another laser is directed at the cesium atoms. Only those whose atomic states are altered by the microwave cavity are induced to emit light (known as fluorescence). The photons (tiny packets of light) emitted in fluorescence are measured by a detector."

Emphasis mine.

How does this match up with your description in what I quoted from you above?

 

[DrClaude]

In the NIST article I linked to in post #37, it is described this way:

"As the atoms interact with the microwave signal—depending on the frequency of that signal—their atomic states might or might not be altered. The entire round trip for the ball of atoms takes about a second. At the finish point, another laser is directed at the cesium atoms. Only those whose atomic states are altered by the microwave cavity are induced to emit light (known as fluorescence). The photons (tiny packets of light) emitted in fluorescence are measured by a detector."

Emphasis mine.

How does this match up with your description in what I quoted from you above?

I'm sorry, I was mistaken about the detection method. It is not absorption spectroscopy, but fluorescence spectroscopy, meaning that rather than measure how much of the laser light is absorbed, they measure how much fluorescence the atoms emit when irradiated with that laser light. But it is still probing with a laser, not simply looking at emission of atoms on the $F=4 \rightarrow F=3$ transition.

One difficulty of measuring directly emission I forgot to mention above is that the hyperfine transition has a long lifetime, so the probability of actually measuring spontaneous emission is very small.

To summarize how this all work, without going into all the details, is that cesium atoms are somehow prepared in the $F=3$ state, then they go through a microwave cavity twice, and then the number of atoms that have made the transition to $F=4$ is measured by shining IR laser light to excite the atoms to an excited electronic state, and the amount of fluorescence from these excited atoms gives a measurement of how many atoms made the transition $F=3 \rightarrow F=4$, and thus how close the microwave generator is to the actual frequency of cesium. Maximizing the signal is what keeps the clock ticking at the correct rate.

 

[PeterDonis]

they measure how much fluorescence the atoms emit when irradiated with that laser light.

Yes.

But it is still probing with a laser, not simply looking at emission of atoms on the transition.

I'm not sure I see a big difference here. The emission is stimulated emission, true, not spontaneous emission, but it's still the same transition.

One difficulty of measuring directly emission I forgot to mention above is that the hyperfine transition has a long lifetime, so the probability of actually measuring spontaneous emission is very small.

Yes, that's why they use the probe laser for stimulated emission. But, as above, it's still the same transition causing the emission either way.

 

[DrClaude]

I'm not sure I see a big difference here. The emission is stimulated emission, true, not spontaneous emission, but it's still the same transition.

It is not stimulated emission, but spontaneous emission. The atoms absorb the laser light and the fluorescence (by spontaneous emission) is measured.

Yes, that's why they use the probe laser for stimulated emission. But, as above, it's still the same transition causing the emission either way.

Unless I misunderstood what you wrote previously, my entire point is that it is not the same transition. In the microwave cavity, atoms get excited on the transition $F=3 \rightarrow F=4$, but the number of atoms making the transition is not measured by emission on the reversed hyperfine transition $F=4 \rightarrow F=3$, which is long-lived (it is a forbidden transition) and in the microwave part of the spectrum.

Instead, the atoms in $F=4$ are optically pumped on the D2-line to $F'=5$ and the decay back (fluorescence, by spontaneous emission) to $F=4$ is measured. The lifetime is much shorter, detecting IR light is much easier, and fluorescence is localized, making it easier to measure the emitted light.

 

[PeterDonis]

the atoms in $F=4$ are optically pumped on the D2-line to $F'=5$ and the decay back (fluorescence, by spontaneous emission) to $F=4$ is measured.

Ah, ok. The NIST page doesn't make this clear.

 

[DrClaude]

Ah, ok. The NIST page doesn't make this clear.

A better description can be found in this Physics Today article:
https://physicstoday.scitation.org/doi/10.1063/1.1366066

 

[PeterDonis]

A better description can be found in this Physics Today article:
https://physicstoday.scitation.org/doi/10.1063/1.1366066

Thanks for the reference!