Time-independent Schrödinger Equation

[soul]

Hi everyone,
I have been studying Quantum mechanics course for one month and our subject for now is Time-independent Schrödinger Equation. What I couldn't figure out is whether $$\Psi(x,\,0) = \Psi(x)$$, since $$\Psi(x,\,0)$$ doesn't contain any time dependence and $$\Psi(x)$$ as well. Can someone explain me that that expression is true.

 

[olgranpappy]

your LaTeX isn't showing up for me... but it looks like you are asking whether psi(x,0) is equal to psi(x).

In which case, what do you mean by psi(x,0) and psi(x)?

 

[soul]

I am new in quantum and there could be some lack of terminology in my question. I mean that Schr. Eq. at t = 0 which is shown as psi(x,0) and wave function independent of time psi(x) are the same in sqaure well and in some other cases??

 

[Avodyne]

In general, no. Consider a potential like the square well that has only bound-state solutions. Then there is a discrete set of allowed energies, the energy eigenvalues, $E_n$, and corresponding eigenfunctions, $\psi_n(x)$, $n=1,2,\ldots$ ; these are the solutions of the time-independent Schrödinger equation. Then, the most general solution of the time dependent Schrödinger equation is
$$\psi(x,t)=\sum_{n=1}^\infty c_n e^{-iE_nt/\hbar}\psi_n(x),$$
where the $c_n$'s are arbitrary coefficients.

EDIT: something seems wrong with the TeX processing on the new server ...

 

[Dr Transport]

to fix your LaTEX issues, you need to close with [/itex] or [/tex]...

 

[Avodyne]

to fix your LaTEX issues, you need to close with [/itex] or [/tex]...

I did close with [/tex] and [/itex], but the slashes disappeared after uploading. This is a problem with the new server; see https://www.physicsforums.com/showthread.php?p=1922963