Commit 022b2e08 by Michael Wimmer

### split figure of classical particle into two parts

parent 4e6e1be4
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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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