Harmonic Oscillations
Tags: #physics #harmonic #oscillationsEquation
$$\Psi(t)=\hat{\Psi}(t)e^{i(\omega t \pm \phi)}=\hat{\Psi}(t)\cos (\omega t \pm \phi) \\ \sum_{i} \hat{\Psi_{i}}\cos(\alpha_{i} \pm \omega t) =\hat{\Phi}\cos (\beta \pm \omega t) \\ \tan (\beta)=\frac{\sum_{i} \hat{\Psi_{i}} \sin (\alpha_{i})}{\sum_{i} \hat{\Psi_{i}} \cos (\alpha_{i})} \\ \hat{\Phi}^{2} = \sum_{i} \hat{\Psi_{i}^{2}} + 2 \sum_{j > i} \sum_{i} \hat{\Psi_{i}} \hat{\Psi_{j}} \cos (\alpha_{i} - \alpha_{j}) \\ \int x(t) dt=\frac{x(t)}{i \omega} \\ \frac{d^{n}(x(t))}{d t^{n}}=(i \omega)^{n} x(t)$$Latex Code
\Psi(t)=\hat{\Psi}(t)e^{i(\omega t \pm \phi)}=\hat{\Psi}(t)\cos (\omega t \pm \phi) \\
\sum_{i} \hat{\Psi_{i}}\cos(\alpha_{i} \pm \omega t) =\hat{\Phi}\cos (\beta \pm \omega t) \\
\tan (\beta)=\frac{\sum_{i} \hat{\Psi_{i}} \sin (\alpha_{i})}{\sum_{i} \hat{\Psi_{i}} \cos (\alpha_{i})} \\
\hat{\Phi}^{2} = \sum_{i} \hat{\Psi_{i}^{2}} + 2 \sum_{j > i} \sum_{i} \hat{\Psi_{i}} \hat{\Psi_{j}} \cos (\alpha_{i} - \alpha_{j}) \\
\int x(t) dt=\frac{x(t)}{i \omega} \\
\frac{d^{n}(x(t))}{d t^{n}}=(i \omega)^{n} x(t)
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Introduction
Equation
Latex Code
\Psi(t)=\hat{\Psi}(t)e^{i(\omega t \pm \phi)}=\hat{\Psi}(t)\cos (\omega t \pm \phi) \\
\sum_{i} \hat{\Psi_{i}}\cos(\alpha_{i} \pm \omega t) =\hat{\Phi}\cos (\beta \pm \omega t) \\
\tan (\beta)=\frac{\sum_{i} \hat{\Psi_{i}} \sin (\alpha_{i})}{\sum_{i} \hat{\Psi_{i}} \cos (\alpha_{i})} \\
\hat{\Phi}^{2} = \sum_{i} \hat{\Psi_{i}^{2}} + 2 \sum_{j > i} \sum_{i} \hat{\Psi_{i}} \hat{\Psi_{j}} \cos (\alpha_{i} - \alpha_{j}) \\
\int x(t) dt=\frac{x(t)}{i \omega} \\
\frac{d^{n}(x(t))}{d t^{n}}=(i \omega)^{n} x(t)
Explanation
Latex code for the harmonic oscillations. I will briefly introduce the notations in this formulation.
: Amplitude
- Superposition of more harmonic oscillations with the same frequency
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