2.3.2 In a square, zero potential

problem number 84

With initial and boundary conditions. In a square, zero potential.

Solve for \(f(x,y,t)\) \[ I f_t = - \frac {\hbar ^2}{2 m} \nabla ^2 f(x,y) \] With boundary conditions \begin {align*} f(0,y,t) &= 0\\ f(1,y,t) &=0 \\ f(x,1,t) &=0 \\ f(x,0,t) &=0 \end {align*}

And initial conditions \(f(x,y,0)=\sqrt {2} \left ( \sin (2\pi x) \sin (\pi y) + \sin (\pi x) \sin (2 \pi y) \right )\)

pict
Figure 2.17:PDE specification

Mathematica


\[\left \{\left \{f(x,y,t)\to 2 \sqrt {2} \sin (\pi x) \sin (\pi y) e^{-\frac {5 i \pi ^2 \text {hBar}^2 t}{2 m}} (\cos (\pi x)+\cos (\pi y))\right \}\right \}\]

Maple


\[f \left (x , y , t\right ) = \sqrt {2}\, \left (\sin \left (\pi x \right ) \sin \left (2 \pi y \right )+\sin \left (\pi y \right ) \sin \left (2 \pi x \right )\right ) {\mathrm e}^{-\frac {5 i \pi ^{2} \mathit {hBar}^{2} t}{2 m}}\]

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