Relationship between Larmor precession and energy eigenstates

[docnet]

TL;DR Summary

How is Larmor precession related to the energy eigenstates?

Quantum mechanically, a spin 1/2 particle in a uniform magnetic field has two energy eigenstates $\ket{up}$ and $\ket{down}$ and rotational degrees of freedom (distinct from the energy eigenstates) about the axis of the magnetic field. this can be derived from the Pauli matrix commonly written $\sigma_z$. the energy difference is $\hbar\omega$ and the precession frequency is $\omega$, which is directly related to the energy of the photon emitted by the spin during quantum state transition.

In Virtual Photons in Magnetic Resonance, Larmor precession is described as a series of exchanges of virtual photons with nonzero angular momenta and zero energy. These exchanges conserve the angular momentum of the spin and the field during Larmor precession, while no energy exchange happens between the field and the spin.

What is the connection between Larmor frequency $\omega$ and the energy difference between the energy eigenstates $\hbar\omega$? It seems like the Larmor frequency and the energy difference are related because the faster the spin precesses about the magnetic field, the bigger is the energy difference of the spin's energy eigenstates. In the meanwhile, the Larmor frequency is only detectable if there is an input of energy into the spin (like radio frequency radiation) so they seem unrelated. I do not know whether there are experiments to prove that the spin evolves under a uniform magnetic field in the fashion of the Larmor precession, in the absence of the energy input and subsequent emission?

what connects the two seemingly disparate processes, other than the real photon of energy $\hbar\omega$?

 

[EigenState137]

Once again: spin is quantized, it has no classical analog, it does not precess.

The Larmor frequency is properly develop in terms of the magnetic moment, not the spin. See Larmor precession

 

[hutchphd]

The Larmor frequency is properly develop in terms of the magnetic moment

But the Larmor frequency is based on the experimental value of the "gyromagnetic ratio" and so the quantum coupling and the Larmor precession give the same result. Of course the Quantum picture is the more comprehensive but much of the physics works out classically.

 

[EigenState137]

But the Larmor frequency is based on the experimental value of the "gyromagnetic ratio" and so the quantum coupling and the Larmor precession give the same result. Of course the Quantum picture is the more comprehensive but much of the physics works out classically.

Reasonable for the most simplistic concepts of magnetic resonance spectroscopy. Not reasonable for more complex concepts and realistic experimental applications.

Spin is an inherently quantum phenomenon. Why not get it right for the outset?

 

[hutchphd]

Spin is an inherently quantum phenomenon. Why not get it right for the outset?

For the same reason we do not use QM to calculate the trajectories of artillery shells. Cannon balls are also inherently quantum phenomena. I am unaware of any parts of magnetic resonance which actually require quantum mechanical interpretation to avoid palpable error.
Personally I find the quantum mechanical treatment easier to understand, but I have invested perhaps a decade of effort to get to that point.

 

[EigenState137]

Greetings,

I find it rather ironic that an "A" level post in the Quantum Physics forum is answered with a hand-waving response of just treat it classically and that that response is lauded. Nor do I see such a response being consistent with the explicitly stated mission of these forums "to learn and discuss science as it is currently generally understood and practiced by the professional scientific community". Emphasis added.

Cannon balls are also inherently quantum phenomena. I am unaware of any parts of magnetic resonance which actually require quantum mechanical interpretation to avoid palpable error.

Are you seriously equating an intrinsic angular momentum with a cannon ball?

I fail to see how the limitations of your personal awareness of the field of magnetic resonance spectroscopy is relevant. A quantum treatment is necessary for a fundamental understanding of even the most basic spin-spin coupling effects within simple 1-dimensional NMR spectra. Yes, one can exploit rather easily applied, established rules to those splittings. But that is a far cry from real understanding, and that lack of understanding certainly restricts the new knowledge than might be extracted from an experimental investigation.

Any more complex experiments clearly do demand a more profound understanding of the quantum phenomena. A list of examples would be long. A few examples will suffice:
1D NMR in a liquid crystal solvent
2D NMR
3D NMR
CIDNP and CIDEP
ENDOR
Cross-relaxation and cross-polarization techniques
ODMR
Hyperfine structure interactions
...

I suppose my point is what is the objective of this forum in attempting to address questions posed? If it is to help achieve understanding of the scientific fundamentals, then I fail to see how cutting corners and propagating basic misconceptions is fruitful.ES

 

[WernerQH]

What is the connection between Larmor frequency $\omega$ and the energy difference between the energy eigenstates $\hbar\omega$?

It is important to distinguish between the gyro-motion of the proton as a whole (controlled by its charge) and the precession of its spin (controlled by the magnetic moment). In a homogeneous magnetic field charged particles occupy Landau levels, which are uniformly spaced in energy like the states of an harmonic oscillator. Transitions of one level to the next are accompanied by the emission or absorption of a photon with a frequency equal to the gyro- or cyclotron-frequency $eB / 2 \pi m$. This is analogous to the transitions between the spin states, but these occur at a different frequency, $\gamma eB / 2 \pi m$, with the gyro-magnetic ratio $\gamma$ (2.79 for the proton). And there are only $2s + 1$ spin states as opposed to the infinite number of Landau levels. Nuclei with spin $s = 0$ don´t show up in NMR at all.

I suspect what confused you is the idea of a static magnetc field as a "bath" or "condensate" of virtual photons. This is just a figure of speech. It is not meaningful to describe the motion of a charged particle in a static magnetic field as an exchange of infinitely many, infinitely small "quanta". As is said somewhere in Landau / Lifshitz: "static fields are always classical".

 

[hutchphd]

Are you seriously equating an intrinsic angular momentum with a cannon ball?

Nor did I. I am simply pointing out that to a man with a hammer everything looks like a nail.

 

[docnet]

But the Larmor frequency is based on the experimental value of the "gyromagnetic ratio" and so the quantum coupling and the Larmor precession give the same result. Of course the Quantum picture is the more comprehensive but much of the physics works out classically.

@ hutchphd thank you so much for your reply. Could the gyromagnetic ratio be a representation of something more fundamental (not thinking of anything in particular) that could explain, for example, why some nuclei have positive and others have negative gyromagnetic ratios (a reason for clockwise and anti-clockwise spin/magnetic moment precession)?

It is important to distinguish between the gyro-motion of the proton as a whole (controlled by its charge) and the precession of its spin (controlled by the magnetic moment). In a homogeneous magnetic field charged particles occupy Landau levels, which are uniformly spaced in energy like the states of an harmonic oscillator. Transitions of one level to the next are accompanied by the emission or absorption of a photon with a frequency equal to the gyro- or cyclotron-frequency $eB / 2 \pi m$. This is analogous to the transitions between the spin states, but these occur at a different frequency, $\gamma eB / 2 \pi m$, with the gyro-magnetic ratio $\gamma$ (2.79 for the proton). And there are only $2s + 1$ spin states as opposed to the infinite number of Landau levels. Nuclei with spin $s = 0$ don´t show up in NMR at all.

I suspect what confused you is the idea of a static magnetc field as a "bath" or "condensate" of virtual photons. This is just a figure of speech. It is not meaningful to describe the motion of a charged particle in a static magnetic field as an exchange of infinitely many, infinitely small "quanta". As is said somewhere in Landau / Lifshitz: "static fields are always classical".

Thank you so much for your reply. For some reason the term Landau Levels sounds familiar. Is it a spectrum of discrete energy levels that correspond to quantized values of orbital angular momenta (hence quantized cyclotron-frequency)? If so, that was a topic in Professor Susskind's advanced QM lectures.

Yes, the QFT language confused me and I'm glad to hear that static magnetic fields can be thought of classically!

 

[hutchphd]

One of the reasons the classical description of dynamic for magnetic resonance is that one is dealing with a large ensemble of spins at relatively high temperatures (compared to the energy of the spin "flipping". My point was to use the simplest analysis that works for what you need.
In general, the measurement of the magnetic properties of particles with spin is very much an active area of research. In fact there is recently considerable excitement over apparent discrepancy in the measured value (relative to theory) for the muon gyromagnetic ratio:

https://en.wikipedia.org/wiki/Muon_g-2

No one understands it...might be interesting,,, we'll see. I know little more about this!

 

[PeterDonis]

Nor do I see such a response being consistent with the explicitly stated mission of these forums "to learn and discuss science as it is currently generally understood and practiced by the professional scientific community".

Can you give some references regarding the current practice of the professional scientific community in analyzing scenarios like those under discussion in this thread?

 

[WernerQH]

Is it a spectrum of discrete energy levels that correspond to quantized values of orbital angular momenta (hence quantized cyclotron-frequency)?

That's right. It's actually a harmonic oscillator in two dimensions. Besides the familiar $n$ of the harmonic oscillator you need another quantum number $l$. Motion along the field is not quantized. (Because of the degeneracy of the Landau levels there is considerable freedom in the choice of gauge; the wave functions that you find in the literature can look quite different from those in section 111 of vol III of Landau / Lifshitz.)

 

[PeroK]

I suppose my point is what is the objective of this forum in attempting to address questions posed? If it is to help achieve understanding of the scientific fundamentals, then I fail to see how cutting corners and propagating basic misconceptions is fruitful.ES

One weakness in that argument is that a pure mathematician might describe all the mathematics that you as a physicist use as "hand waving" and "cutting corners".

And someone who studies fundamental logic and the foundations of mathematics might say that about regular pure mathematics.

 

[EigenState137]

Greetings

Can you give some references regarding the current practice of the professional scientific community in analyzing scenarios like those under discussion in this thread?

The literature is voluminous to say the least, much of it is applied to specific objectives such as the elucidation of the structure of organic or biological molecules and polymers, and very likely most of it would not available to many readers of this thread. Instead, allow me to walk through a very simple NMR spectrum, utilizing references to Wikipedia (yes, I know) and citations therein, that are readily available to all. I believe this will serve to illustrate my basic point: quantum mechanics is required to understand the phenomenon, although not necessarily to apply it effectively to a specific experiment, for example the assignment of a particular molecular structure. However, the capacity to utilize the technique without fundamental understanding is possible only because others have made the effort to understand the fundamental physics, and those efforts have been successful to the extent that general, qualitative rules have been established that are applicable by non-specialists.

Consider the $_{}^{1}\textrm{H}$-NMR spectrum of ethanol, $\mathrm{{CH_{3}CH_{2}OH}}$ in dilute, isotropic solution as illustrated below. Spectral assignments are given within the illustrated spectrum.

Two features are immediately obvious. First, the $\mathrm{CH_{2}}$ protons appear as a quartet of transitions while the $\mathrm{CH_{3}}$ protons appear as a triplet of transitions. This is the result of spin-spin interactions as will be described in greater detail below. Second, the two kinds of protons appear to be shifted from one another, this is the so-called chemical shift chemical shift that results from the nuclear spin being “shielded” from the applied Zeeman field by the bonding electrons within the molecules. These shifts are of the order of parts per million (ppm) of the applied field and need not concern us for the current discussion.

The spin-spin couplings that give rise to the triplet and quartet transitions are an example of $\mathit{j}$-coupling. For two nuclear spins, $I_{i}$ and $I_{k}$, within the same diamagnetic molecule the coupling is given by the following Hamiltonian:
$$H=2\pi I_{i}\cdot J_{ik}\cdot I_{k}$$
where $J_{ik}$ is the $\mathit{j}$-coupling tensor which under the specified experimental conditions of a dilute, isotropic solution reduces to a scalar coupling constant characterized both by magnitude and by sign. Under the experimental conditions specified, the sign of the coupling constant is not important.

In more complex experiments, such as anisotropic solutions (liquid crystals for example), or in solid state experiments, $J_{ik}$ no longer reduces to a scalar and a more in-depth analysis is necessary.

In the vast majority of cases, an NMR spectrum is taken in the presence of an homogeneous, static external magnetic field, $\mathrm{B_{0}}$, that is applied, by convention, in the $z$-direction. That Zeeman field serves to lift the degeneracies of the spin states allowing observation of a spectrum. I am not aware of a satisfactory treatment of the Zeeman effect upon spin systems that is not quantum mechanical.

I hope that the above is sufficient to address the question asked. By no means do I suggest that every specific application of magnetic resonance spectroscopy demands a full quantum mechanical analysis. I do suggest that such analyses benefit dramatically from those earlier researchers who did undertake such full quantum mechanical treatments to enable the more qualitative analyses.

Lastly to perhaps clarify my initial point within this thread. Spin is quantized and to treat it otherwise is a misconception, admittedly a rather common one. I see no reason why such a misconception should not be pointed out at this forum. Would an equivalent misconception regarding cosmic expansion go unchallenged in the Cosmology forum? A misconception regarding gravitation in the General Relativity forum? I suspect not, nor should they.

If you will excuse me, some nails await.ES

 

[hutchphd]

.

Lastly to perhaps clarify my initial point within this thread. Spin is quantized and to treat it otherwise is a misconception, admittedly a rather common one.

Of course it is and no one here would claim otherwise.
The article you provided was interesting and I learned a few things so thank you. The nuclear magnetic interaction (j-coupling) is mediated by the exchange interactions of the bond electrons and this is absolutely a Quantum Mechanical Phenomenon. No one would argue that the fine structure of atoms can be otherwise successfully addressed.
This has nothing to do with the original question about Larmor precession of the nuclear magnetic moment however. My point to the OP was that in this circumstance, the classical description of nuclear magnetic resonance phenomena in a classical Magnetic Field worked out fine because the gyromagnetic ratio was essentially a fitted parameter and population averages are involved. I won't reiterate.
Your point is perhaps valid if a little overstated. For instance you did not use Quantum Electrodynamics to describe the magnetic field. Why would we teach such a common misconception?? Small hammers some times work out fine.

 

[docnet]

NMR is taught at the undergraduate level organic chemistry level

Greetings

The literature is voluminous to say the least, much of it is applied to specific objectives such as the elucidation of the structure of organic or biological molecules and polymers, and very likely most of it would not available to many readers of this thread. Instead, allow me to walk through a very simple NMR spectrum, utilizing references to Wikipedia (yes, I know) and citations therein, that are readily available to all. I believe this will serve to illustrate my basic point: quantum mechanics is required to understand the phenomenon, although not necessarily to apply it effectively to a specific experiment, for example the assignment of a particular molecular structure. However, the capacity to utilize the technique without fundamental understanding is possible only because others have made the effort to understand the fundamental physics, and those efforts have been successful to the extent that general, qualitative rules have been established that are applicable by non-specialists.

Consider the $_{}^{1}\textrm{H}$-NMR spectrum of ethanol, $\mathrm{{CH_{3}CH_{2}OH}}$ in dilute, isotropic solution as illustrated below. Spectral assignments are given within the illustrated spectrum.

View attachment 286615

Two features are immediately obvious. First, the $\mathrm{CH_{2}}$ protons appear as a quartet of transitions while the $\mathrm{CH_{3}}$ protons appear as a triplet of transitions. This is the result of spin-spin interactions as will be described in greater detail below. Second, the two kinds of protons appear to be shifted from one another, this is the so-called chemical shift chemical shift that results from the nuclear spin being “shielded” from the applied Zeeman field by the bonding electrons within the molecules. These shifts are of the order of parts per million (ppm) of the applied field and need not concern us for the current discussion.

The spin-spin couplings that give rise to the triplet and quartet transitions are an example of $\mathit{j}$-coupling. For two nuclear spins, $I_{i}$ and $I_{k}$, within the same diamagnetic molecule the coupling is given by the following Hamiltonian:
$$H=2\pi I_{i}\cdot J_{ik}\cdot I_{k}$$
where $J_{ik}$ is the $\mathit{j}$-coupling tensor which under the specified experimental conditions of a dilute, isotropic solution reduces to a scalar coupling constant characterized both by magnitude and by sign. Under the experimental conditions specified, the sign of the coupling constant is not important.

In more complex experiments, such as anisotropic solutions (liquid crystals for example), or in solid state experiments, $J_{ik}$ no longer reduces to a scalar and a more in-depth analysis is necessary.

In the vast majority of cases, an NMR spectrum is taken in the presence of an homogeneous, static external magnetic field, $\mathrm{B_{0}}$, that is applied, by convention, in the $z$-direction. That Zeeman field serves to lift the degeneracies of the spin states allowing observation of a spectrum. I am not aware of a satisfactory treatment of the Zeeman effect upon spin systems that is not quantum mechanical.

I hope that the above is sufficient to address the question asked. By no means do I suggest that every specific application of magnetic resonance spectroscopy demands a full quantum mechanical analysis. I do suggest that such analyses benefit dramatically from those earlier researchers who did undertake such full quantum mechanical treatments to enable the more qualitative analyses.

Lastly to perhaps clarify my initial point within this thread. Spin is quantized and to treat it otherwise is a misconception, admittedly a rather common one. I see no reason why such a misconception should not be pointed out at this forum. Would an equivalent misconception regarding cosmic expansion go unchallenged in the Cosmology forum? A misconception regarding gravitation in the General Relativity forum? I suspect not, nor should they.

If you will excuse me, some nails await.ES

Although the physicists and mathematicians in this thread may be impressed by your narrative, I have repeatedly seen that kind of narrative in my sophomore organic chemistry textbook (your posts are understandable by a typical sophomore-level chemistry student). I am fairly certain that working-level NMR theory uses a rudimentary level of quantum mechanics. I know that the post-doc NMR theorists in my laboratory (who are very talented and knowledgeable NMR theorists who graduated from top U.S universities mind you) do not ever discuss QM principles like the uncertainty principle, entanglement, and superposition states (and I never heard them use QFT to explain anything NMR related). I do not believe the post-docs in my group would know how to explain every aspect of the basic spectra you posted there, without resorting to the NMR theorist's unbreakable habit of ensemble averaging. This is because act of ensemble averaging makes a semi-classical version of quantum mechanics that is practical for doing NMR.

FYI, I am a graduate student researcher in a lab that specializes in solid state NMR experiments (which are more theory-heavy than liquid state NMR because as you mentioned, the interactions are anisotropic and require magic-angle spinning to average them out). I know a few talented and knowledgeable post doc NMR theorists in my group. The post docs are very passionate and intelligent NMR theorists who have only taken QM at the graduate level quantum chemistry level. They could talk about advanced techniques in NMR all day and I would barely understand what they are saying because NMR theory is specialized and so far from conventional quantum mechanics.

I am not trying to be mean, but I'm guessing your native language is not english. Reading your repeated the statement "once again, spin is quantized" I cannot understand which part of my question you are trying to address, or how it is a valid response to my question. I am well aware that spins have integer and half-integer quantum numbers. I never said I was confused about that. The posts from the mentors like @hutchphd, @PeroK, and @WernerQH have all been relevant to my question and I beg them to continue answering my confused questions in my future posts. But I would like to kindly ask @EigenState137 to please stop derailing my threads. If I wanted a hand-wavy explanation about NMR spectroscopy, I would ask the post-doc in my lab who is currently working on new solid-state NMR theory and has a degree from MIT, and who is aware of the size of the hammer he wields. I would not ask @EigenState137 (no offense).

edit: I realized that my response sounded angrier than I was feeling. It was not my intention to be rude. I just wanted to explore my curiosity in QM and not in an applied theory like NMR.