Time dependence of a given wave equation

Question

I have a quick question. Particle in 1D box with length $L$ and wave equation $$ \psi(x,t=0)=Ax^3(L-x) $$ If I want to express time evolution do I just add time factor at end like this? $$ \psi(x,t)=x^3(L-x)e^{-iEt/\hbar} $$ Or am I suppose to use superposition of wave functions describing particle in a box? Then the time dependence would be $$ \psi(x,t)=\sum_{n=1}^\infty c_n\psi_n(x)e^{-iE_nt/\hbar}, $$ where $$ \psi_n(x)=\sqrt{\frac{2}{L}}\sin(n \pi x/L) $$ $$ c_n=\sqrt{\frac{2}{L}}A\int_0^L\sin(n\pi x/L)x^3(L-x). $$

Answer 1

For time-independent Hamiltonian, We know that $$|\Psi(t)\rangle=U(t)|\Psi(0)\rangle=e^{-i Ht/\hbar}|\Psi(0)\rangle$$

I'm using $|n\rangle$ to show the energy eigenbasis. $$|\Psi(0)\rangle=\sum_n|n\rangle\langle n|\Psi(0)\rangle=\sum_n\int |n\rangle\langle n|x\rangle\langle x|\Psi(0)\rangle dx $$ $$=\sum_n\left(\int \phi_n(x)\psi(x,0)dx \right)|n\rangle$$ $$|\Psi(t)\rangle=\sum_n\left(\int \phi_n(x)\psi(x,0)dx \right)e^{-i Ht/\hbar}|n\rangle=\sum_n\left(\int \phi_n(x)\psi(x,0)dx \right)e^{-i E_nt/\hbar}|n\rangle$$

$$\psi(x,t) = \langle x|\Psi(t)\rangle=\sum_n\left(\int \phi_n(x)\psi(x,0)dx \right)e^{-i E_nt/\hbar}\phi_n$$

where $$\phi_n=\langle x|n\rangle=\sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right)$$ That's cleared what you should use.