Читать книгу Essential Concepts in MRI - Yang Xia - Страница 34

Although the visualization of a vector M moving under the direction of a B₁ pulse is useful for the understanding of the simplest cases in NMR and MRI, a deep understanding of magnetic resonance [1–6] requires the aid of quantum mechanics, where the essential information of the nuclear systems is contained in the complex wave functions (labeled with Greek letters Ψ, Φ, φ, ψ). These wave functions can be described by the use of a vectoral term called a ket and written as |φ>. For each |φ>, one further defines a conjugated vector of a different nature, called a bra and written as <φ|. (The terms bra and ket come from truncations of the word bracket.)

In modern physics, the energies (E ) and wave functions (ψ) for a molecular or atomic system can be investigated by the use of the Schrödinger equation (ℋ ψ = Eψ), where the operator ℋ is called the Hamiltonian and commonly contains the differential operator ∇2. (A spin term is usually neglected for the computation of atomic and molecular orbitals because its influence, in terms of energy shift, is negligibly small in the absence of a magnetic field.) A similar quantum mechanical equation can describe the nuclear spins where the Hamiltonian contains the spin angular momentum operator. In NMR, the stable states of quantum mechanics systems are the eigenfunctions of H. Hence, to calculate NMR spectra we must find the eigenvalues of H.

In Chapter 2, the classical description of NMR, spin angular momentum is visualized as a spinning sphere that carries a charge (Figure 2.3b). In a quantum mechanical description of NMR, spin angular momentum is a quantum mechanical quantity without a classical analog; spin angular momentum is determined by the internal nuclear structure of the spin system. A classical limit is only approached in the case of orbital angular momentum and in the limit of large quantum numbers. Appendix 2 has some background introduction in quantum mechanics. This chapter presents the quantum mechanical description of the fundamental NMR concepts.