Dataset Information

The Product Operator Formalism: A Physical and Graphical Interpretation.

ABSTRACT: The product-operator formalism is the most commonly used tool for describing and designing multidimensional NMR experiments. In spite of its relative simplicity and sound theoretical underpinnings, however, students and practitioners often find it difficult to relate the mathematical manipulations to a physical picture. In an effort to address this pedagogical challenge, the present paper begins with a quantum-mechanical treatment of pure populations of scalar-coupled spin-pairs, rather than the equilibrium population of spin-pairs in different quantum states, which is the usual starting point for treatments based on the density matrix and product operators. In the context of pure populations, the product operators are shown to represent quantum correlations between the nuclei in individual molecules, and a new variation on the classical vector diagram is introduced to represent these correlations. The treatment is extended to mixed populations that begin at thermal equilibrium, and the density matrix is introduced as an efficient means of carrying out quantum calculations for a mixed population. Finally, it is shown that the operators for observable magnetization and correlations can be used as a basis set for the density matrix, providing the formal justification for the widely-used rules of the product-operator treatment. Throughout the discussion, the vector diagrams are used to help maintain a connection between the mathematics and the sometimes subtle physical principles. An electronic supplement created with the Mathematica computer program is used to provide additional mathematical details and the means to carry out further calculations.

SUBMITTER: Goldenberg DP

PROVIDER: S-EPMC3088292 | biostudies-literature | 2010 Mar

REPOSITORIES: biostudies-literature

ACCESS DATA

Publications

The Product Operator Formalism: A Physical and Graphical Interpretation.

Goldenberg David P DP

Concepts in magnetic resonance. Part A, Bridging education and research 20100301 2

The product-operator formalism is the most commonly used tool for describing and designing multidimensional NMR experiments. In spite of its relative simplicity and sound theoretical underpinnings, however, students and practitioners often find it difficult to relate the mathematical manipulations to a physical picture. In an effort to address this pedagogical challenge, the present paper begins with a quantum-mechanical treatment of pure populations of scalar-coupled spin-pairs, rather than the ...[more]

PMID: 21552432

Similar Datasets

Project description:OBJECTIVES:The purpose of this study was to analyze distortion product otoacoustic emission (DPOAE) level and signal to noise ratio in a group of infants from birth to 4 months of age to optimize prediction of hearing status. DPOAEs from infants with normal hearing (NH) and hearing loss (HL) were used to predict the presence of conductive HL (CHL), sensorineural HL (SNHL), and mixed HL (MHL). Wideband ambient absorbance was also measured and compared among the HL types. DESIGN:This is a prospective, longitudinal study of 279 infants with verified NH and HL, including conductive, sensorineural, and mixed types that were enrolled from a well-baby nursery and two neonatal intensive care units in Cincinnati, Ohio. At approximately 1 month of age, DPOAEs (1-8?kHz), wideband absorbance (0.25-8?kHz), and air and bone conduction diagnostic tone burst auditory brainstem response (0.5-4?kHz) thresholds were measured. Hearing status was verified at approximately 9 months of age with visual reinforcement audiometry (0.5-4?kHz). Auditory brainstem response air conduction thresholds were used to assign infants to an NH or HL group, and the efficacy of DPOAE data to classify ears as NH or HL was analyzed using receiver operating characteristic (ROC) curves. Two summary statistics of the ROC curve were calculated: the area under the ROC curve and the point of symmetry on the curve at which the sensitivity and specificity were equal. DPOAE level and signal to noise ratio cutoff values were defined at each frequency as the symmetry point on their respective ROC curve, and DPOAE results were combined across frequency in a multifrequency analysis to predict the presence of HL. RESULTS:Single-frequency test performance of DPOAEs was best at mid to high frequencies (3-8?kHz) with intermediate performance at 1.5 and 2?kHz and chance performance at 1?kHz. Infants with a conductive component to their HL (CHL and MHL combined) displayed significantly lower ambient absorbance values than the NH group. No differences in ambient absorbance were found between the NH and SNHL groups. Multifrequency analysis resulted in the best prediction of HL for the SNHL/MHL group with poorer sensitivity values when infants with CHL were included. CONCLUSIONS:Clinical interpretation of DPOAEs in infants can be improved by using age-appropriate normative ranges and optimized cutoff values. DPOAE interpretation is most predictive at higher F2 test frequencies in young infants (2-8?kHz) due to poor test performance at 1 to 1.5?kHz. Multifrequency rules can be used to improve sensitivity while balancing specificity. Last, a sensitive middle ear measure such as wideband absorbance should be included in the test battery to assess possibility of a conductive component to the HL.

Dataset Information

The Product Operator Formalism: A Physical and Graphical Interpretation.

Publications

The Product Operator Formalism: A Physical and Graphical Interpretation.

Similar Datasets

OmicsDI is part of the ELIXIR infrastructure

Tweets