论文标题
旋转波近似中任意自旋值的核磁共振
Nuclear Magnetic Resonance for Arbitrary Spin Values in the Rotating Wave Approximation
论文作者
论文摘要
为了探测核旋转$ s $从其量量子数量$ m $之一到另一个替代$ m'$的过渡，实验者将磁场$ {\ bm bm b} _0在某个特定方向上应用，例如沿$ \ hat $ \ hat {\ bm z} $，然后使用薄弱领域的$ rive $ rive unity IS $ {\ bm b}。角频$ω$，通常垂直于$ {\ bm b} _0 $，例如$ {\ bm b} _1（t）= b_1 \ hat {\ bm x} \ cos（ωt）$。在旋转波近似中，$ {\ bm b} _1（t）= b_1 [\ hat {\ bm x} \ cos（ωt）+\ hat+\ hat {\ bm y} \ sin（ωt）] $。尽管在每个量子力学教科书中为旋转$ \ frac {1} {2} $解决了此问题，但对于一般的旋转$ s $案例，它的一般解决方案仅出于国家之间过渡的总体概率而发布，但是在每个替代方案中找到核的概率的时间依赖性，以前尚未出版过。在这里，我们提出了一种基本方法来准确解决此问题，并为各种$ s $值的各种初始取代概率提供了各种取代的时间依赖性的数字。我们发现了一个新的结果：与$ s = \ frac {1} {2} $案例不同，如果在其中一个取代的一个最初的概率为1，而每个替代物的概率的时间依赖性为0和1的概率，则在0和1之间振荡，对于更高的旋转值之间的概率，则更多的概率的时间依赖于粒子的粒子，以便在每个粒子中找到粒子的替代品，而这些定期是在替代品中的替代性，这是在替代品中的。
In order to probe the transitions of a nuclear spin $s$ from one of its substate quantum numbers $m$ to another substate $m'$, the experimenter applies a magnetic field ${\bm B}_0$ in some particular direction, such along $\hat{\bm z}$, and then applies an weaker field ${\bm B}_1(t)$ that is oscillatory in time with the angular frequency $ω$, and is normally perpendicular to ${\bm B}_0$, such as ${\bm B}_1(t)=B_1\hat{\bm x}\cos(ωt)$. In the rotating wave approximation, ${\bm B}_1(t)=B_1[\hat{\bm x}\cos(ωt)+\hat{\bm y}\sin(ωt)]$. Although this problem is solved for spin $\frac{1}{2}$ in every quantum mechanics textbook, for the general spin $s$ case, its general solution has been published only for the overall probability of a transition between the states, but the time dependence of the probability of finding the nucleus in each of the substates has not previously been published. Here we present an elementary method to solve this problem exactly, and present figures for the time dependencies of the various substates states for a variety of initial substate probabilities for a variety of $s$ values. We found a new result: unlike the $s=\frac{1}{2}$ case, for which if the initial probability of finding the particle in one of the substates was 1, and the time dependence of the probabilities of each of the substates oscillates between 0 and 1, for higher spin values, the time dependencies of the probabilities finding the particle in each of its substates, which periodic, is considerably more complicated.
