Doubly Rotating Coordinates: Wave Functions in Magnetic Resonance Problems

Author

Sunghyun Kim, University of Central FloridaFollow

Keywords

Spin, Magnetic Resonance, Rotation, Quantum Mechanics

Abstract

The nuclear spin response to a rotating field H has been theoretically investigated from the 1930s to the 1950s. Building upon Majorana's probability theory, the behavior of spin 1/2 is well-illustrated in the joint review by Rabi, Ramsey, andSchwinger, and their spin wave function ψ is succinctly restated by Gottfried: ψ(t) = e^-iI_zωt/ℏe^{-i[I_z(ω₀-ω)+I_xω₁]t/ℏ}ψ(0).

However, the complexity involved in evaluating the wave function ψ in terms of probability amplitudes C_m attributed to the noncommutative nature of spin operators [I_x, I_z] ≠0, hinders the application of this well-established theory to spins with arbitrary values I > 1/2. In a recent study by Hall and Klemm, a conjectural form of the spin wave function was suggested.

Here, we present an alternative formulation of the wave function ψ by controlling doubly rotating coordinates: ψ(t) = e^-iI_zωt/ℏ e^-iI_yθ/ℏ e^-iI_zΩt/ℏ e^iI_yθ/ℏ ψ(0). This formulation facilitates the computation of general state transitions from an initial state ψ(0)=∑_mC_m(0)ψ_m(0) to ψ(t)=∑_m'C_m'(t)ψ_m'(t). Moreover, by assuming an analogous form of the total electron spin J to that of the nucleus I, we can explore hyperfine structures in atoms and/or molecules traversing in the magnetic field H in terms of the nuclear-electronic spin interaction (I·J).

Through this approach, we not only formulate wave functions more effectively but also bridge quantum mechanics and algebraic perspectives.

Completion Date

2024

Semester

Spring

Committee Chair

Klemm, Richard

Degree

Doctor of Philosophy (Ph.D.)

College

College of Sciences

Department

Physics

Degree Program

Condensed Matter Theory, Physics

Format

application/pdf

Identifier

DP0028299

URL

https://purls.library.ucf.edu/go/DP0028299

Language

English

Rights

In copyright

Release Date

May 2024

Length of Campus-only Access

None

Access Status

Doctoral Dissertation (Open Access)

Campus Location

Orlando (Main) Campus

STARS Citation

Kim, Sunghyun, "Doubly Rotating Coordinates: Wave Functions in Magnetic Resonance Problems" (2024). Graduate Thesis and Dissertation 2023-2024. 130.
https://stars.library.ucf.edu/etd2023/130

Accessibility Status

Meets minimum standards for ETDs/HUTs