Harmonic Oscillations
Tags: #physics #harmonic #oscillationsEquation
Latex Code
1 2 3 4 5 6 | \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
1 2 3 4 5 6 | \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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