Читать книгу Essential Concepts in MRI - Yang Xia - Страница 35

A nuclear spin in quantum mechanical description is represented by a spin angular momentum operator I, which can be written in the usual Cartesian coordinate system as a dimensionless quantity

(3.1)

where Ix, Iy, and Iz are the spin operators representing the x, y, z components of the spin operator I.

The magnetic moment µ is proportional to its spin angular momentum,

(3.2)

where γ is a proportionality constant (called the gyromagnetic ratio), different for different nuclear species. This equation is identical to that in the classical description [Eq. (2.1)], except the spin I is now an operator.

A single nucleus in an external magnetic field (B₀ = B₀k) experiences the nuclear Zeeman interaction¹ with the field. The evolution of a spin system ψ is governed by the time-dependent Schrödinger equation,

(3.3)

where ℋ is a Hamiltonian. (This equation plays a similar role as Eq. (2.3) in classical treatment of NMR in Chapter 2, where the Newton’s second law was used.) If ℋ is considered time-independent, the evolution of the spin system can be derived from the above equation as

(3.4)

where U(t) is the evolution operator. This equation effectively separates the time-independent part |ψ(t₀)> from the time-dependent part U(t).

Since the Hamiltonian operator for the case of B₀ = B₀k is given by the Zeeman Hamiltonian, we can write down the operator ℋ as

(3.5)

Note that only the Iz component is present in the last part of Eq. (3.5), which is due to the properties of the dot product (cf. Appendix A1.1) since B₀ = B₀k.

As in the classical description where I is the spin angular momentum (a vector) and the half-integer or integer values of I are called spin quantum number I, the spin operator Iz has m possible values (the eigenvalues), ranging from −I, −I + 1, …, I, where m is the azimuthal quantum number.

Therefore, the evolution operator U(t) can be written as

(3.6)

where the second step considers the fact that ω₀ = γB₀ and θ = ω₀t. U(t) is hence just a rotation operator [recall that exp^(iθ) = cosθ + i sinθ, also in Appendix A1.1], which corresponds to a rotation of the spin state |ψ> about the z axis with an angular frequency ω₀, known as the Larmor precession frequency:

(3.7)

This equation is identical to the equation that we have derived in the classical description [Eq. (2.5)].