Taken from Introduction to Quantum mechanics, second edition, by David Griﬃths, page 47. This is the same as the above problem but has an extra \(V(x) f(x,t)\) terms where \(V(x)\) is the inﬁnite square well potential deﬁned by \(V(x)=0\) if \(0\leq x \leq a\) and \(V(x)=\infty \) otherwise.

Solve for \(f(x,t)\) \[ I \hslash f_t = - \frac {\hslash ^2}{2 m} f_{xx} + V(x) f(x,t) \] With initial conditions \(f(x,0) = A x (a-x)\) for \(0\leq x \leq a\) and zero otherwise.

Mathematica ✓

\[\left \{\left \{f(x,y,t)\to \sqrt {2} e^{-\frac {5 i \pi ^2 h t}{m}} \left (\sin (\pi x) \sin (3 \pi y)+\sin (2 \pi x) \sin (\pi y) e^{\frac {5 i \pi ^2 h t}{2 m}}\right )\right \}\right \}\]

Maple ✓

Failed to convert to latex

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