Quantum Mechanical Description of NMR: From Wave Function to Bloch Equation

This report outlines steps in deriving the Bloch Equation, beginning by applying the principles of relativistic and non-relativistic quantum mechanics to simple physical models of the proton, and deriving the two-component Schrödinger Equation, which describes the interaction of a single proton with externally applied magnetic fields. The off-diagonal entries of the Bloch equation matrix are derived by taking the time derivative of the expectation values of the solutions (spinors) of the Schrödinger Equation that describe the dynamics of the collection (ensemble) of protons in each voxel. Derivations of the Boltzmann distribution, which determines the thermal equilibrium magnetization, and of the mechanisms of magnetization relaxation, are also described to complete the entries of the Bloch Equation. The mathematical foundations of proton intrinsic spin and magnetic moment from relativistic and non-relativistic quantum mechanics will be described, and an intuitive understanding of the proton gyromagnetic ratio, and of the physical origin of the Nuclear Magneton fundamental constant, is obtained from non-relativistic quantum mechanics. The paragraphs below briefly describe these topics, which will be elaborated in the talk.