Lagrangian and QTF

Phase

Recall the time solution of the Schrodinger equation is given by,

\[ \phi(x,t) \sim \exp\left[-\frac{i}{\hbar}Et\right] \]

We call the \(E\) (energy of the photon), the phase factor. In classical particles the phase factor is \(S\):

\[ \phi(x,t) \sim \exp\left[ i S / \hbar\right] \]

where we note that the principle of least action \(\delta S / \delta x(t) = 0\) causes \(\partial \phi / \partial t = 0\).

Intuitively, the classical particle considers all trajectory that cost some total action \(S\). The observed trajectory is the one that gives a stationary action \(\delta S / \delta x(t) = 0\).