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Statements

Subject Item
dbr:Product_operator_formalism
rdfs:label
直積演算子 Product operator formalism
rdfs:comment
In NMR spectroscopy, the product operator formalism is a method used to determine the outcome of pulse sequences in a rigorous but straightforward way. With this method it is possible to predict how the bulk magnetization evolves with time under the action of pulses applied in different directions. It is a net improvement from the semi-classical vector model which is not able to predict many of the results in NMR spectroscopy and is a simplification of the complete density matrix formalism. 核磁気共鳴において直積演算子（プロダクト演算子、プロダクトオペレーター）法とは、複数のスピン系の状態を表す密度演算子を直積演算子で表すことで、スピン系の時間発展を記述する方法である。 一般的な量子統計力学による取り扱いでは密度行列を用いて状態を表さなければならないが、これは数個のスピン系でさえも大変複雑で、物理的イメージも分かりにくい。たとえば最も単純な2スピン系であっても、密度行列の行列要素は合計16個になる。そこでSørensenらによって考案されたのが直積演算子法である。スペクトルの強度を問題にしないならば、行列要素の値そのものは問題ではなく、どのように時間発展するか分かればよい。
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dbo:abstract
In NMR spectroscopy, the product operator formalism is a method used to determine the outcome of pulse sequences in a rigorous but straightforward way. With this method it is possible to predict how the bulk magnetization evolves with time under the action of pulses applied in different directions. It is a net improvement from the semi-classical vector model which is not able to predict many of the results in NMR spectroscopy and is a simplification of the complete density matrix formalism. In this model, for a single spin, four base operators exist: , , and which represent respectively polarization (population difference between the two spin states), single quantum coherence (magnetization on the xy plane) and the unit operator. Many other, non-classical operators exist for coupled systems. Using this approach, the evolution of the magnetization under free precession is represented by and corresponds to a rotation about the z-axis with a phase angle proportional to the chemical shift of the spin in question: Pulses about the x and y axis can be represented by and respectively; these allow to interconvert the magnetization between planes and ultimately to observe it at the end of a sequence. Since every spin will evolve differently depending on its shift, with this formalism it is possible to calculate exactly where the magnetization will end up and hence devise pulse sequences to measure the desired signal while excluding others. The product operator formalism is particularly useful in describing experiments in two-dimensions like COSY, HSQC and HMBC. 核磁気共鳴において直積演算子（プロダクト演算子、プロダクトオペレーター）法とは、複数のスピン系の状態を表す密度演算子を直積演算子で表すことで、スピン系の時間発展を記述する方法である。 一般的な量子統計力学による取り扱いでは密度行列を用いて状態を表さなければならないが、これは数個のスピン系でさえも大変複雑で、物理的イメージも分かりにくい。たとえば最も単純な2スピン系であっても、密度行列の行列要素は合計16個になる。そこでSørensenらによって考案されたのが直積演算子法である。スペクトルの強度を問題にしないならば、行列要素の値そのものは問題ではなく、どのように時間発展するか分かればよい。
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