Proton wave function in a water molecule: Breakdown of degeneration caused by interactions with the magnetic field of a Magnetic Resonance Imaging device

TL;DR

This paper models the proton wave function in water molecules, demonstrating how magnetic fields used in MRI lift degeneracy in energy levels through an analytical approach.

Contribution

It introduces a simple analytical model of water molecule protons, showing how MRI magnetic fields break degeneracy in energy levels.

Findings

Energy levels are degenerate without magnetic field.

Magnetic field lifts degeneracy and shifts energy levels.

Numerical results align with typical MRI magnetic field strengths.

Abstract

The concept of a Magnetic Resonance Imaging (MRI) device is based on the emission of radio waves produced by the protons of the hydrogen atoms in water molecules when placed in a constant magnetic field after they interact with a pulsed radio frequency (RF) current. When the RF field is turned on, the protons are brought to a spin excited state. When the RF field is turned off, the MRI sensors are able to detect the energy released as the protons realign their spins with the magnetic field. In this work we provide a simple model to describe the basic physical mechanism responsible for the operation of MRI devices. We model the water molecule in terms of a central force problem, where the protons move around the (unstructured) doubly negatively charged oxygen atom. First, we employ an analytical treatment to obtain the system's wave function as well as its energy levels, which we show are degenerate. Next, the energy levels from the water molecule are studied in the presence of a uniform external magnetic field. As a result, they get shifted and the degeneration is lifted. We provide numerical results for a magnetic field strength commonly used in MRI devices.