-- Erwin Schrödinger in Letter to John Lighton Synge (1959)

At the atomic level, the up and down quantum states of a spin 1/2 nucleus carry a small energy difference allowing the lower energy state to be populated slightly more than the other (

E = - **μB** = - γ**IB**

where

E = - μ_{z}B_{z} = - γI_{z}B_{z} = - ωB_{z}

where the last term ω = 2πf = γI

It comes naturally that sending in an RF wave to a sample sitting in the high-field magnet will

A

To create a coherence, using a classical picture, we start from the polarized spins (i.e. I

More interesting, however, are spin-spin correlations that can be probed to provide structural information. A

The spins have X, Y and Z components. Fortunately, for solution-state NMR, the situation is much simpler due to the fact that fast random molecular rotations randomize and cancel any XY components of the through-bond interaction to the first-order approximation, leaving only a coupling along Z-axis, with an energy contribution of:

where I

Given the main energy term for

Therefore, J-coupling simply adds or subtracts πJ from the main energy term, leading to a splitting of J of the original resonance.

The two spin coupling Hamiltonian,

The extra energy affecting

ΔE = 2πJ_{1}I_{z}S_{z} + 2πJ_{2}T_{z}S_{z}

With I

-π(J_{1} + J_{2})S_{z}, π(J_{1} - J_{2})S_{z},
π(- J_{1} + J_{2})S_{z}, π(J_{1} + J_{2})S_{z}

with the chance of occurrence of 1 (++), 1 (+-), 1 (-+), and 1 (--), respectively. These states correspond to both spins up, one up and one down, and both down. The couplings lead to four separate resonances, a

It needs to be noted that the sign of

-2πJS_{z}, 0 , 2πJS_{z}

and correspondingly three lines for the

It is obvious now that in the example above, from the S-spin point of view, NMR cannot tell the difference between (1) two non-equivalent spins (I, T) coupling to the same other spin (S) if J

Continuing onto more neighboring (vicinal) equivalent I spins in the couplings, we arrive at Pascal's Triangle for general splitting patterns. The basis of these patterns is due to the fact that the small perturbations involved are additive and the quantum transition rule dictates that only single quantum transitions (with angular momentum change of 1) in the spin order are allowed (and observable).

How to create a useful spin-spin correlation? In the double rotating frames of J-coupled I and S spins, the initial Hamiltonian is simply

If we write

This is essentially an

Alternatively, we may not convert the operator back to I spin, but rather convert

Before the conversion back to I spin detection, we have the option to "

To encode the S spin chemical shifts (frequencies), we will revert back from the rotating frame to the normal lab frame of reference for S spin.

In another variation, at the end of the first coherence transfer period, we may choose not to flip I spin back to Z when dealing with

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