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1.3.5 Schrödinger

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Choosing units where Planck's constant $ \hbar=1$ and the mass $ m=1/2$ , the time-dependent Schrödinger equation clearly appears as a special type of wave / diffusion equation:

$\displaystyle i\frac{\partial\psi}{\partial t}=H\psi$ $\displaystyle \hspace{5mm}\textrm{with}\hspace{5mm}$ $\displaystyle H(x)=-\frac{\partial^2}{\partial x^2}+V(x)$ (1.3.5#eq.1)

In quantum mechanics, the Schrödinger equation is used to evolve the complex wave-function $ \vert\psi>=\psi(x,t)$ to describe the probability $ <\psi\vert\psi>=\int_\Omega \vert\psi\vert^2 dx$ of finding a particle in a given interval $ \Omega=[x_L;x_R]$ . Take the simplest example of a free particle modeled with a wave-packet in a periodic domain and assume a constant potential $ V(x)=0$ . The JBONE applet below shows the evolution of a low energy $ E=-<\psi\vert\partial_x^2\vert\psi>$ (long wavelength) particle that is initially known with a rather good accuracy in space (narrow Gaussian envelope):
JBONE applet:  press Start/Stop to run the simulation showing how a well localized low energy wave-packet spreads out in time, in agreement with Heisenberg's uncertainty principle in quantum mechanics.
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"; ?> the wave-function $ \Re e(\psi)$ (blue line) starts to oscillate, the probability density $ \vert\psi\vert^2$ spreads out (black line) and the position increasingly becomes uncertain... reproducing the famous uncertainty principle.



Numerical experiments: Heisenberg's uncertainty principle
  1. Reduce the ICWavelength down to 2 and relate an increase in the kinetic energy or the velocity of the wave-packet with a more accurate localization.
  2. Reload the applet with a lower kinetic energy ICWavelength=6 and reduce the accuracy of the initial position ICWidth=12 by spreading out the envelope: by spreading out the wave-function, you in effect reduced the uncertainty in the velocity.

SYLLABUS  Previous: 1.3.4 Wave-breaking  Up: 1.3 Prototype problems  Next: 1.4 Numerical discretization