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Some questions about quantum mechanics in NMR

  1. Jul 24, 2011 #1

    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" [Broken]. 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:
    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
    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

    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

    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

    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!
    Last edited by a moderator: May 5, 2017
  2. jcsd
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