2.3.5 David Griffiths, page 47

problem number 87

Taken from Introduction to Quantum mechanics, second edition, by David Griffiths, page 47.

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

Figure 2.20:PDE specification


\[\left \{\left \{f(x,t)\to \frac {\int _{-\infty }^{\infty }-\frac {A \exp \left (-\frac {i K[1] (2 a m-2 x m+h t K[1])}{2 m}\right ) \left (a K[1]+e^{i a K[1]} (a K[1]+2 i)-2 i\right )}{K[1]^3}dK[1]}{2 \pi }\right \}\right \}\]


\[f \left (x , t\right ) = -\frac {\left (a \left (\int _{-\infty }^{\infty }\frac {{\mathrm e}^{-\frac {i \left (\frac {h s t}{2}+\left (a -x \right ) m \right ) s}{m}}}{s^{2}}d s \right )+a \left (\int _{-\infty }^{\infty }\frac {{\mathrm e}^{-\frac {i h s^{2} t}{2 m}+i s x}}{s^{2}}d s \right )-2 i \left (\int _{-\infty }^{\infty }\frac {{\mathrm e}^{-\frac {i \left (\frac {h s t}{2}+\left (a -x \right ) m \right ) s}{m}}}{s^{3}}d s \right )+2 i \left (\int _{-\infty }^{\infty }\frac {{\mathrm e}^{-\frac {i h s^{2} t}{2 m}+i s x}}{s^{3}}d s \right )\right ) A}{2 \pi }\]