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main.tex — Quantum Notes
\documentclass{article} \usepackage{amsmath} \title{On Harmonic Oscillators} \author{A. Physicist} \begin{document} \maketitle \section{Introduction} The Hamiltonian of the system is \begin{equation} H = \frac{p^2}{2m} + \frac{1}{2}m\omega^2 x^2 \end{equation} % energy levels follow below \end{document}

On Harmonic Oscillators

A. Physicist
November 2026

1  Introduction

The Hamiltonian of the system is
H = p²/2m + ½ mω²x²      (1)
from which the quantised energy levels En = ħω(n + ½) follow directly.

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iħ ∂ψ/∂t = Ĥψ ∇·E = ρ/ε₀ S = k_B ln Ω G_μν + Λg_μν = 8πG T_μν

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