Harmonic Oscillations

Tags: #physics #harmonic #oscillations

Equation

Ψ(t)=Ψ^(t)ei(ωt±ϕ)=Ψ^(t)cos(ωt±ϕ)iΨi^cos(αi±ωt)=Φ^cos(β±ωt)tan(β)=iΨi^sin(αi)iΨi^cos(αi)Φ^2=iΨi2^+2j>iiΨi^Ψj^cos(αiαj)x(t)dt=x(t)iωdn(x(t))dtn=(iω)nx(t)

Latex Code

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                     \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

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\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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