A Quantum Mechanical Description of the Zeeman Interaction

The classical Zeeman interaction between a magnetic moment and an applied magnetic field was defined above. In a quantum mechanical description the operators for the quantities need to be used, such as the nuclear spin I. With an externally applied magnetic field B the Zeeman Hamiltonian is given by 

The usual convention is to let the direction of the externally applied static magnetic field Bo define the z-direction in the laboratory frame. Then the Zeeman Hamiltonian can be written as 

Multinuclear Solid-State NMR of Inorganic Materials Bo Hnite 

The Zeeman Hamiltonian can be used to calculate the energy difference between the nuclear spin states. For a nucleus the energy eigenvalues can be obtained by taking the Hamiltonian operator given in Eq. 2.11 so that 

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