3.1.1 Logan textbook, page 30

problem number 143

From page 30, David J Logan textbook, applied PDE textbook.

Schrodinger PDE with zero potential (Logan p. 30)

Solve $$I\hslash f_t=-\frac{{\hslash }^2}{2m}{f}_{xx}$$ With boundary conditions $$\begin{array}{rl}f(0,t)& =0\\ f(L,0)& =0\end{array}$$

Figure 3.1: PDE specification

Mathematica ✓

ClearAll["Global`*"]; 
pde =  I*h*D[f[x, t], t] == -((h^2*D[f[x, t], {x, 2}])/(2*m)); 
bc  = {f[0, t] == 0, f[L, t] == 0}; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[{pde, bc}, f[x, t], {x, t}, Assumptions -> L > 0], 60*10]]; 
sol =  sol /. K[1] -> n;
 

$$\left\{\{f(x,t)\to \underset{n=1}{\sum ^{\mathrm{\infty }}}{e}^{-\frac{ih{n}^2{\pi }^{2}t}{2L^{2}m}}{c}_n\sin \left(\frac{n\pi x}{L}\right)\}\right\}$$

Maple ✓

restart; 
interface(showassumed=0); 
pde :=I*h*diff(f(x,t),t)=-h^2/(2*m)*diff(f(x,t),x$2); 
bc:=f(0,t)=0,f(L,t)=0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve([pde,bc],f(x,t)) assuming L>0),output='realtime'));
 

$$f(x,t)=\stackrel{\mathrm{\infty }}{\sum _{n=1}}\mathit{\text{\_F1}}\left(n\right){\mathrm{e}}^{-\frac{i{\pi }^{2}h{n}^{2}t}{2L^{2}m}}\sin \left(\frac{\pi nx}{L}\right)$$

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