Commit 022b2e08 authored by Michael Wimmer's avatar Michael Wimmer
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split figure of classical particle into two parts

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......@@ -18,13 +18,16 @@ title: The WKB approximation
## Classical intuition
![Classical particle](figures/classical_scat.svg)
The WKB approximation is used to describe a particle of mass $m$ in a one-dimensional (1D) potential with some initial velocity. To call some intuition, let us recall the classical case: consider a particle of energy $E$ moving in a potential $V(x)$. In this situation, wthe total energy of the particle is conserved, and can be at every point $x$ be separated into kinetic and potential energy as $E = E_\text{kin}(x) + V(x)$.
If $E > V(x)$, as shown in the upper plots, then the particle will always have an energy larger than the potential at any point $x$. Hence, there will always be some non-zero kinetic energy at every point $x$. As a consequence, the particle will continuously move in one direction (determined by the initial velocity). This happens irrespective of the potential shape: for example, it does not matter whether the potential is smooth or changes abruptly.
If $E > V(x)$, then the particle will always have an energy larger than the potential at any point $x$. Hence, there will always be some non-zero kinetic energy at every point $x$. As a consequence, the particle will continuously move in one direction (determined by the initial velocity). This happens irrespective of the potential shape: for example, it does not matter whether the potential is smooth or changes abruptly.
If $E < V(x)$, as shown in the lower plots, will reach a maximum position $x_0$ such that $E = V(x_0)$ and the kinetic energy becomes zero. Then, the particle will be reflected. Again, this will happen for any potential shape.
![Classical particle with energy smaller than potential energy](figures/classical_scat2.svg)
If $E < V(x)$, a classical particle will reach a maximum position $x_0$ such that $E = V(x_0)$ and the kinetic energy becomes zero. Then, the particle will be reflected. Again, this will happen for any potential shape.
![Classical particle with energy larger than potential energy](figures/classical_scat1.svg)
In summary, in classical mechanics the global motion of a particle in a 1D potential is only determined by the value of the total energy compared to the potential $V(x)$.
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