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Diffstat (limited to 'source/know/concept/larmor-precession')
| -rw-r--r-- | source/know/concept/larmor-precession/index.md | 32 |
1 files changed, 16 insertions, 16 deletions
diff --git a/source/know/concept/larmor-precession/index.md b/source/know/concept/larmor-precession/index.md index 6b101e0..774af7b 100644 --- a/source/know/concept/larmor-precession/index.md +++ b/source/know/concept/larmor-precession/index.md @@ -10,18 +10,18 @@ layout: "concept" Consider a stationary spin-1/2 particle, placed in a [magnetic field](/know/concept/magnetic-field/) -with magnitude $B$ pointing in the $z$-direction. -In that case, its Hamiltonian $\hat{H}$ is given by: +with magnitude $$B$$ pointing in the $$z$$-direction. +In that case, its Hamiltonian $$\hat{H}$$ is given by: $$\begin{aligned} \hat{H} = - \gamma B \hat{S}_z = - \frac{\hbar}{2} \gamma B \hat{\sigma_z} \end{aligned}$$ -Where $\gamma = - q / m$ is the gyromagnetic ratio, -and $\hat{\sigma}_z$ is the Pauli spin matrix for the $z$-direction. -Since $\hat{H}$ is proportional to $\hat{\sigma}_z$, -they share eigenstates $\Ket{\downarrow}$ and $\Ket{\uparrow}$. -The respective eigenenergies $E_{\downarrow}$ and $E_{\uparrow}$ are as follows: +Where $$\gamma = - q / m$$ is the gyromagnetic ratio, +and $$\hat{\sigma}_z$$ is the Pauli spin matrix for the $$z$$-direction. +Since $$\hat{H}$$ is proportional to $$\hat{\sigma}_z$$, +they share eigenstates $$\Ket{\downarrow}$$ and $$\Ket{\uparrow}$$. +The respective eigenenergies $$E_{\downarrow}$$ and $$E_{\uparrow}$$ are as follows: $$\begin{aligned} E_{\downarrow} = \frac{\hbar}{2} \gamma B @@ -29,9 +29,9 @@ $$\begin{aligned} E_{\uparrow} = - \frac{\hbar}{2} \gamma B \end{aligned}$$ -Because $\hat{H}$ is time-independent, -the general time-dependent solution $\Ket{\chi(t)}$ is of the following form, -where $a$ and $b$ are constants, +Because $$\hat{H}$$ is time-independent, +the general time-dependent solution $$\Ket{\chi(t)}$$ is of the following form, +where $$a$$ and $$b$$ are constants, and the exponentials are "twiddle factors": $$\begin{aligned} @@ -40,9 +40,9 @@ $$\begin{aligned} \:+\: b \exp(- i E_{\uparrow} t / \hbar) \: \Ket{\uparrow} \end{aligned}$$ -For our purposes, we can safely assume that $a$ and $b$ are real, -and then say that there exists an angle $\theta$ -satisfying $a = \sin(\theta / 2)$ and $b = \cos(\theta / 2)$, such that: +For our purposes, we can safely assume that $$a$$ and $$b$$ are real, +and then say that there exists an angle $$\theta$$ +satisfying $$a = \sin(\theta / 2)$$ and $$b = \cos(\theta / 2)$$, such that: $$\begin{aligned} \Ket{\chi(t)} = \sin(\theta / 2) \exp(- i E_{\downarrow} t / \hbar) \: \Ket{\downarrow} @@ -50,7 +50,7 @@ $$\begin{aligned} \end{aligned}$$ Now, we find the expectation values of the spin operators -$\expval{\hat{S}_x}$, $\expval{\hat{S}_y}$, and $\expval{\hat{S}_z}$. +$$\expval{\hat{S}_x}$$, $$\expval{\hat{S}_y}$$, and $$\expval{\hat{S}_z}$$. The first is: $$\begin{aligned} @@ -86,8 +86,8 @@ $$\begin{aligned} \matrixel{\chi}{\hat{S}_z}{\chi} = \frac{\hbar}{2} \cos(\theta) \end{aligned}$$ -The result is that the spin axis is off by $\theta$ from the $z$-direction, -and is rotating (or **precessing**) around the $z$-axis at the **Larmor frequency** $\omega$: +The result is that the spin axis is off by $$\theta$$ from the $$z$$-direction, +and is rotating (or **precessing**) around the $$z$$-axis at the **Larmor frequency** $$\omega$$: $$\begin{aligned} \boxed{ |