PX156 - B4 - classical vs quantum

classical approach

consider a particle of mass, $m$ , in 1D, subjected to a force, $F (x, t)$
$v = \frac{d v}{d t}$ , $p = m v$ , $K E = \frac{1}{2} m v^{2}$
from newton's second law:

F = m a

potential energy function, $V (x)$

m \frac{d^{2} v}{d t^{2}} = - \frac{\partial V}{\partial x}

+ initial conditions

quantum approach

bohr's correspondence principle

"Bohr reasoned that any new theory had to do more than describe atomic phenomena correctly; it must be applicable to conventional phenomena, too, in such a way that it would reproduce the old physics" SRC

the wave equation

considering a transverse wave travelling in the $x$ -direction with a speed, $v$ (displacement is in $y$ ), the wave equation is:

\frac{\partial^{2} y}{\partial x^{2}} = \frac{1}{v^{2}} \frac{\partial^{2} y}{\partial t^{2}}

solutions $: y = Y (x \pm v t)$
general solutions:
$y = A \exp (i k x - i ω t)$
$y = A [\cos (k x - ω t) + i \sin (k x - ω t)]$