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


  • Cristian Heber Zepeda Fernández CONACyT, FCFM-BUAP
  • Jorge Luis Aguilar Cuevas
  • Eduardo Moreno Barbosa



Magnetic resonance imaging, water molecule, energy levels, quantum mechanics


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.


Care at Mayo Clinic.

Web MD.

Medical News Today

J. A. Fessler. Optimization Methods for Magnetic Resonance Image Reconstruction: Key Models and Optimization Algorithms, IEEE Signal Processing Magazine, 37 (2020) 33,

F. Knoll et al., Deep Learning Methods for Parallel Magnetic Resonance Image Reconstruction. arXiv:1904.01112v1[eess.SP].

X. Qu, et al., Magnetic resonance image reconstruction from undersampled measurements using a patch-based nonlocal operator. Medical Image Analysis 18 (2013) 843,

B. Wen, et al., Transform Learning for Magnetic Resonance Image Reconstruction: From Model-Based Learning to Building Neural Networks, IEEE Signal Processing Magazine 37 (2020) 41,

Mayo Clinic,

National Institute of Biomedical Imaging and Bioengineering,

computed-tomography-ct. For patients,

Medical News Today,

Harvard Health Publishing Harvard Medical School,

U.S. Food and drugs Administration.

S.D. Serai, et al., Components of a magnetic resonance imaging system and their relationship to safety and image quality. Pediatr. Radiol. 51 (2021) 716,

J. Zhang, G. Shim, S. M. de Toledo, and E. I. Azzam, The Translationally Controlled Tumor Protein and the Cellular Response to Ionizing Radiation-Induced DNA Damage, in TCTP/tpt1 - Remodeling Signaling from Stem Cell to Disease. Results and Problems in Cell Differentiation, edited by A. Telerman and R. Amson (Springer, Cham, 2017), 12.

R. A. Pooley, Fundamental Physics of MR Imaging, The AAPM/RSNA physics tutorial for residents 25 (2005) 1087,

A. Pai, R. Shetty, Y. S. Chowdhury, Magnetic Resonance Imaging Physics. (StatPearls2021)

V. S. Vassiliou et al., Magnetic resonance imaging: Physics basics for the cardiologist. JRSM Cardiovascular Dis. 7 (2018) 1,

J.H. Sanders and K. C. Turberfield, The Magnetic Moment of the Proton. I. The Value in Nuclear Magnetons. Proc. R. Soc. Lond. 272 (1963) 79,

Britannica, The Editors of Encyclopaedia. Magnetic resonance. Encyclopedia Britannica, 25 Jan. 2011, Accessed 14 September 2021.

R. McWeeny and K. A. Ohno, A quantum-mechanical study of eater molecule, Proc. R. Soc. Lond. A 255 (1960) 367,

J. Chem, Multiconfiguration Wavefunctions for the Water Molecule, J. Chem. Phys. 55 (1971) 1720,

A. Sprague Coolidge, A Quantum Mechanics Treatment of the Water Molecule, Phys. Rev. 42 (1932) 189.

S. Aung, Thesis of Doctor of Philosophy. Part I Approximate Hartree-Fock Wavefunctions One-electron Properties and Electronic Structure of the W ater Molecule. Part II PerturbationVariational Calculation of the Nuclear Spin-Spin Isotropic Coupling Constant in HD California Institute of Technology Pasadena California (1969).

B. Clark et al., Computing the energy of a water molecule using MultiDeterminants: A simple, efficient algorithm. J. Chem.

Phys. 135 (2011) 244105,

M. Johansson, Introduction to Electronic Structure Theory. CSC/PRACE Spring School in Coputational Chemistry 2018.

S. S. Zumdahl water Encyclopedia Britannica, (2021).




How to Cite

C. H. Zepeda Fernández, J. L. Aguilar Cuevas, and E. Moreno Barbosa, “Proton wave function in a water molecule: Breakdown of degeneration caused by interactions with the magnetic field of a Magnetic Resonance Imaging device”, Rev. Mex. Fís., vol. 68, no. 3 May-Jun, pp. 031101 1–, May 2022.