Some questions about quantum mechanics in NMR

[Roo2]

Hello,

I've been working in an NMR lab for two years and am trying to gain a deeper understanding of the phenomenon. I started my work right out of freshman year before even taking organic chemistry or electromagnetism so my initial knowledge was practically zero and my projects were assigned accordingly. As I spent more time in lab, however, I found it increasingly important to understand the physics of the process; it is now wholly insufficient for me to just know how to process and analyze the data. I've been studying NMR physics for most of the summer and am now comfortable with the vector model, signal processing, etc. However, I'm completely stuck on product operator formalism, the quantum mechanical description which I am told is imperative to understanding high-dimensional NMR experiments. I've therefore come here to ask for some help. My main source of information has been the Keeler NMR lectures, which lucidly present all of the other material. The chapter which discusses product operators is http://www-keeler.ch.cam.ac.uk/lectures/understanding/chapter_6.pdf" . I'll go through all of my questions in the order that they arise from this pdf; if anybody could answer any of them I'd appreciate it immensely.1. Page 2:

In quantum mechanics operators represent observable quantities, such an
energy, angular momentum and magnetization. For a single spin-half, the xy-
and z-components of the magnetization are represented by the spin angular
momentum operators Ix , Iy and Iz respectively. Thus at any time the state of
the spin system, in quantum mechanics the density operator, σ, can be
represented as a sum of different amounts of these three operators
σ(t) = a(t)Ix + b(t)Iy + c(t)Iz

Question: The operators must be operating on waves, and in the individual case, the operators are operating on a nucleus with associated wave W. If each nucleus in the sample is located at a slightly different position, won't it have a slightly different wave function due to the different potential term in the Hamiltonian? If so, and σ(t) represents the total magnetization of the system, how can it be abstracted to the weighted sum of Ix, Iy, and Iz, when each operator is operating on a different wave?

2. page 2

The density operator at time t, σ(t), is computed from that at time 0, σ(0) ,
using the following relationship
σ (t) = exp(−iHt)σexp(iHt)

I imagine the reason for this equation is complex so I can't expect anyone on here to tell me why this is so, but could anyone point me to a lecture/paper/online reference to the subject? The equation reminds me of an expectation value for an operator, but where Ψ*Ψ is some sort of hamiltonian equation rather than a wavefunction, and the whole thing is unnormalized and unintegrated. Where does this come from? Similarly, where can I find the derivation of the Hamiltonian as ωI?

3. page 3

Suppose that an x-pulse, of duration tp, is applied to equilibrium
magnetization. In this situation H = ω1Ix and σ(0) = Iz; the equation to be
solved is

σ(t) = exp(-iω1tpIx)Izexp(iω1tpIx)

Such equations involving angular momentum operators are common in quantum mechanics and the solution to them are already all know. The identity required here to solve this equation is

exp(-iBIx)Izexp(iBIx) = cosBIz - sinBIy

This is interpreted as a rotation of Iz by an angle B around the x axis.

Why can this be interpreted as such? What are the grounds for this interpretation? What I see in "cosBIz - sinBIy" is a scalar quantity with no indication of direction; the sine of one value subtracted from the cosine of another. Where is the rotation coming from?

4. page 6

I1z represents the z-magnetization of spin 1, and I 2z likewise for spin 2. I1x represents x-magnetization on spin 1. As spin 1 and 2 are coupled, the spectrum consists of two doublets and the operator I1x can be further identified with the two lines of the spin-1 doublet. In the language of product operators I1x is said to represent in-phase magnetization of spin 1; the description in-phase means that the two lines of the spin 1 doublet have the same sign and lineshape.

Two questions here: Why can the operator I1x be identified with the spin 1 doublet lines? I have no idea what that sentence even means. How can the (spin operator/component of a magnetization vector) of a nucleus "be identified with" a peak in a spectrum?

Also, what is in-phase magnetization? I get that it produces two lines in a doublet with the same size and lineshape, but what is it, in terms of the classical/vector description of magnetism? This question precludes me from actually understand any further information in the material; I can do the transformation equations until I'm blue in the face but I don't actually get what they mean.

5. page 8

The overall result is that anti-phase magnetization of spin 1 has been
transferred into anti-phase magnetization of spin 2. Such a process is called
coherence transfer and is exceptionally important in multiple-pulse NMR.

Again, I don't understand what it means to convert antiphase magnetization of one spin to another. Since this is the basis of all of the high-dimensional NMR experiments, I'm completely stuck.If anyone could help me understand any of this I'd greatly appreciate it. I'm feeling much like I did near the end of my quantum mechanics P-chem course; the math is doable but the physical interpretation of the results is over my head. Thanks!

 

[eljose79]

Thank you for reaching out for help with your understanding of NMR physics. As a fellow scientist, I understand the importance of having a deep understanding of the phenomena we study. I will try my best to answer your questions and provide some resources for further reading.

1. The operators in quantum mechanics do not necessarily operate on waves, but rather on states. In the case of NMR, these states represent the spin states of the nuclei in the sample. Each operator operates on the same wave function, but the resulting state (represented by σ(t)) is a weighted sum of the different spin states. This is because the sample contains multiple nuclei, each with their own spin state.

2. The equation for computing σ(t) is derived from the principles of quantum mechanics, specifically the time evolution of a quantum system. You can find a derivation of this equation in most quantum mechanics textbooks or online resources. As for the Hamiltonian being represented by ωI, this is because the Hamiltonian of a spin system is proportional to the magnetic field (ω) and the spin operator (I).

3. The identity exp(-iBIx)Izexp(iBIx) = cosBIz - sinBIy represents a rotation of the Iz state by an angle B around the x-axis. This is because the operators Ix, Iy, and Iz represent the x, y, and z components of the spin angular momentum, respectively. By applying the x-pulse, we are essentially rotating the spin state around the x-axis.

4. The operator I1x can be identified with the spin 1 doublet lines because it represents the in-phase magnetization of spin 1. In-phase magnetization means that the two lines of the spin 1 doublet have the same sign and lineshape, as you mentioned. In terms of classical magnetism, this means that the spins of the nuclei are aligned in the same direction, resulting in a stronger signal. The same applies for the antiphase magnetization of spin 1, which can be identified with the other peak in the spectrum. This is important for understanding how multiple-pulse NMR experiments work.

5. Coherence transfer is the process of transferring the antiphase magnetization of one spin to another. In the example given in the text, the antiphase magnetization of spin 1 is transferred to spin 2, resulting in a stronger signal for spin 2. This is