00. Classical Particles and Waves

Properties of Point Particles

Consider Point Particle
An idealised object that, at any given instant time, is located at a point in space

1. Position of Particle governed by Newton’s Second Law

2. If particle has constant $m$ and acted on by force $\mathbf{F}$, then position vector $\mathbf{x}(t)$ satisfies

$$m\frac{d^2\mathbf{x}}{d{t}^2}=\mathbf{F}$$

3. Total Energy of Particle is sum of kinetic energy and potential energy

$$E=\frac{1}{2}m|\dot{\mathbf{x}}{|}^2+V=\frac{1}{2}m{\left|\frac{d\mathbf{x}}{dt}\right|}^2+V$$

where it is acted by a conservative force $\mathbf{F}=-\nabla V$ where $V=V(\mathbf{x})$ is the potential

4. Total Energy is Conserved

$$\frac{dE}{dt}=m\frac{d\mathbf{x}}{dt}\cdot \frac{d^2\mathbf{x}}{d{t}^2}+\nabla V\cdot \frac{d\mathbf{x}}{dt}=\dot{\mathbf{x}}\left(m\frac{d^2\mathbf{x}}{d{t}^2}-\mathbf{F}\right)=\dot{\mathbf{x}}\cdot 0=0$$

5. Kinetic Energy relation to Momentum

$$E_{\text{kinetic}}=\frac{|\mathbf{p}{|}^2}{2m}$$

where $\mathbf{p}=m\dot{\mathbf{x}}$ is the momentum of the particle

Classical Wave Equation

$$\frac{1}{v^2}\frac{{\partial }^2\varphi }{\partial t^2}={\nabla }^2\varphi$$

where $v$ is the constant speed of wave

Note that it governs e.g. propagation of sound or light

Complex Plane Wave

$$\varphi (\mathbf{x},t)=A\exp [i(\mathbf{k}\cdot \mathbf{x}-wt)]$$

where
$k$ is the constant wave vector
$\omega$ is the angular frequency
$A$ is the (complex) amplitude

Substitution into Classical Wave Equation gives

$$v=\frac{w}{|\mathbf{k}|}$$