Is Quamtum Mechanics Necessary for Understanding Magnetic Resonance?

In Chapter 18 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss magnetic resonance imaging. Like many authors, we derive an expression for the magnetization of the tissue using quantum mechanics. Must we use quantum theory? In an article published in Concepts in Magnetic Resonance, Lars Hanson asks the same question: “Is Quantum Mechanics Necessary for Understanding Magnetic Resonance?” (Volume 32, Pages 329–340, 2008). The abstract is given below.

Educational material introducing magnetic resonance typically contains sections on the underlying principles. Unfortunately the explanations given are often unnecessarily complicated or even wrong. Magnetic resonance is often presented as a phenomenon that necessitates a quantum mechanical explanation whereas it really is a classical effect, i.e. a consequence of the common sense expressed in classical mechanics. This insight is not new, but there have been few attempts to challenge common misleading explanations, so authors and educators are inadvertently keeping myths alive. As a result, new students’ first encounters with magnetic resonance are often obscured by explanations that make the subject difficult to understand. Typical problems are addressed and alternative intuitive explanations are provided.

How would IPMB have to be changed to remove quantum mechanics from the analysis of MRI? Quantum ideas first appear in the last paragraph of Section 18.2 about the source of the magnetic moment, where we introduce the idea that the z-component of the nuclear spin in a magnetic field is quantized and can take on values that are integral multiples of the reduced Planck’s constant, ℏ. Delete that paragraph.

Section 18.3 is entirely based on quantum mechanics. To find the average value of the z-component of the spin, we sum over all quantum states, weighted by the Boltzmann factor. The end result is an expression for the magnetization as a function of the magnetic field. We could, alternatively, do this calculation classically. Below is a revised Section 18.3 that uses only classical mechanics and classical thermodynamics.

18.3 The Magnetization

The MR [magnetic resonance] image depends on the magnetization of the tissue. The magnetization of a sample, M, is the average magnetic moment per unit volume. In the absence of an external magnetic field to align the nuclear spins, the magnetization is zero. As an external magnetic field B is applied, the spins tend to align in spite of their thermal motion, and the magnetization increases, proportional at first to the external field. If the external field is strong enough, all of the nuclear magnetic moments are aligned, and the magnetization reaches its saturation value.

We can calculate how the magnetization depends on B. Consider a collection of spins of a nuclear species in an external magnetic field. This might be the hydrogen nuclei (protons) in a sample. The spins do not interact with each other but are in thermal equilibrium with the surroundings, which are at temperature T. We do not consider the mechanism by which they reach thermal equilibrium. Since the magnetization is the average magnetic moment per unit volume, it is the number of spins per unit volume, N, times the average magnetic moment of each spin: M=Nμ>, where μ is the magnetic moment of a single spin.

To obtain the average value of the z component of the magnetic moment, we must average over all spin directions, weighted by the probability that the z component of the magnetic moment is in that direction. Since the spins are in thermal equilibrium with the surroundings, the probability is proportional to the Boltzmann factor of Chap. 3, e^–(U/k_BT) = e^{μBcosθ/k_BT}, where k_B is the Boltzmann constant. The denominator in Eq. 18.8 normalizes the probability:

The factor of sinθ arises when calculating the solid angle in spherical coordinates (see Appendix L). At room temperature μB/(k_BT) ≪ 1 (see Problem 4), and it is possible to make the approximation e^x ≈ 1 + x. The integral in the numerator then has two terms:

The first integral vanishes. The second is equal to 2/3 (hint: use the substitution u = cosθ). The denominator is

The first integral is 2; the second vanishes. Therefore we obtain

The z component of M is

which is proportional to the applied field.

The last place quantum mechanics is mentioned is in Section 18.6 about relaxation times. The second paragraph, starting “One way to analyze the effect…”, can be deleted with little loss of meaning; it is almost an aside.

So, to summarize, if you want to modify Chapter 18 of IPMB to eliminate any reference to quantum mechanics, then 1) delete the last paragraph of Section 18.2, 2) replace Section 18.3 with the modified text given above, and 3) delete the second paragraph in Section 18.6. Then, no quantum mechanics appears, and Planck’s constant is absent. Everything is classical, just the way I like it.