Atomic and Optical Physics I, 02 ResonanceII: Larmor Procession Note

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Fourier limit?

Lorentz model for an atom

见另一独立的

Larmor Procession

经典磁矩可以假想电流得到

\begin{align} \vec{\mu} = I \vec{S} \end{align}

其中 \(I\) 为环形电流, \(\vec{S}\) 为环形电流围成的面积. 若是带电 \(-e\) 电子, 则

\begin{align} \mu = \frac{-e}{2\pi r / v} \cdot \pi r^2 = \frac{-e}{2m} \cdot m v r = -\frac{e}{2m} L \end{align}

写成矢量形式, 并定义旋磁比 gyromagnetic ratio \(\gamma = -\frac{e}{2m}\)

\begin{align} \vec{\mu} = \gamma \vec{L} \end{align}

磁矩在磁场中受到的力矩为

\begin{align} \vec{\tau} = \vec{\mu} \times \vec{B} \end{align}

那么角动量的变化就是力矩

\begin{align} \dot{\vec{L}} = \vec{\mu}\times \vec{B} = -\gamma \vec{B} \times \vec{L} \end{align}

可知, \(\vec{L}\) 变化的方向与 \(\vec{B}\) 和 \(\vec{L}\) 都垂直, 且大小不变, 也就是说 在绕 \(B\) 进动. 而 \(-\gamma \vec{B}\) 是频率的量纲, 是进动的频率, Larmor 频率

\begin{align} \Omega_L = \frac{e}{2m}B \end{align}

定义 Bohr magneton 为轨道角动量为 \(L = - \hbar\) 时对应的磁矩

\begin{align} \mu_B = \gamma \cdot (-\hbar) =\frac{e\hbar}{2m_e} \approx 2\pi \times 1.4 \hbar \mathrm{MHz /G} \end{align}

但是电子自旋的 gyromagnetic ratio 是经典值的 \(g_s = -g_e = 2\) 倍. 即 如, 电子的 gyromagnetic ratio is

\begin{align} |\gamma_e| \approx 2 \times \frac{e}{2m_e} = 2\pi \times 2.8 \mathrm{MHz /G} \end{align}

如果自旋 \(z\) 分量的 量子数为 \(m_s\) , 那么对应的磁矩为

\begin{align} \mu_z = g_s\cdot \frac{-e}{2 m_e}\cdot m_s\hbar = - g_s \mu_B m_s \end{align}

总的来说, \(L = \hbar, S = \frac{1}{2}\hbar\) 的电子, 对应的磁矩都是 \(\mu_B\) , 但是磁矩的方 向与角动量的方向相反.

The angular precession frequency has an important physical meaning: It is the angular cyclotron frequency, the resonance frequency of an ionized plasma being under the influence of a static finite magnetic field, when we superimpose a high frequency electromagnetic field.

Rotating Coordinate Trans

\begin{align} \left.\frac{\mathrm{d}}{\mathrm{d}t}\right|_{\mathrm{rot}} = \left.\frac{\mathrm{d}}{\mathrm{d}t}\right|_{\mathrm{inertial}} - \vec{\Omega}\times \end{align}

Exp:

\begin{align} \dot{\mathrm{L}}_{\mathrm{rot}} = \dot{\mathrm{L}}_{\mathrm{inertial}} - \Omega\times \vec{L} = \gamma \vec{L} (\vec{B} + \frac{\vec{\Omega}}{\gamma}) \end{align}

If choose rotating frequency

\begin{align} \vec{\Omega} = \Omega_L = - \gamma \vec{B} \end{align}

then

\begin{align} \vec{B}_{\mathrm{rot}} = 0 \end{align}

so, you can transform away the effect of a magnetic field by going to rotating fram at the Larmor frequency.

Reference