# Classical Mechanics ## Single Particle For a single particle in $\mathbb{R}^3$, we have the total [[Energy|energy]]: $ E = \frac{1}{2} m \|\dot{\pmb{q}}\|^2 + V(\pmb{q}) $ through the introduction of the [[momentum]] via the identification $\pmb{p} = m \dot{\pmb{q}}$, we rewrite $E$ as a function of $q$ and $p$ as: $ H(\pmb{p},\pmb{q}) = \frac{\|\pmb{p}\|^2}{2m} + V(\pmb{q}) $ # Quantum Mechanics The Hamiltonian is the [[operator]] corresponding to the total [[energy]] of the system. The [[eigenvalue]] of the Hamiltonian represent the possible results of measurement of the system's total energy.