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I am trying to derive the TISE, but I am having many questions, and the textbook (Griffiths) does not give any adequate explanation and I have minimal access to my professor. My goal is to find ##\Psi (x,t)##. The book says the solution is

$$ \Psi (x,t) = \sum_{n=0}^{\infty} c_{n} \psi_{n}(x) exp(\frac {-iE_{n}t}{\hbar}) $$

So I start with the general SE (I am just taking this as a fact since the first page of Griffiths puts this equation and says its right)

$$ i \hbar \frac {\partial \Psi}{\partial t} = - \frac {\hbar^{2}}{2m} \frac {\partial^{2} \Psi}{\partial x^{2}} + V \Psi $$

I can separate ##\Psi (x,t) = \psi (x) f(t)## and substitute

$$ i \hbar \frac {\partial [\psi (x) f(t)]}{\partial t} = - \frac {\hbar^{2}}{2m} \frac {\partial^{2} [\psi (x) f(t)]}{\partial x^{2}} + V [\psi (x) f(t)]$$

Then divide by ##\psi (x) f(t)## in order to have the potential, ##V##, stand alone

$$ i \hbar \frac {1}{f} \frac {\partial [f(t)]}{\partial t} = - \frac {\hbar^{2}}{2m} \frac {1}{\psi (x)} \frac {\partial^{2} [\psi (x)]}{\partial x^{2}} + V $$

Now comes the first question. Griffiths says (without explanation)

$$ i \hbar \frac {1}{f(t)} \frac {\partial [f(t)]}{\partial t} = - \frac {\hbar^{2}}{2m} \frac {1}{\psi (x)} \frac {\partial^{2} [\psi (x)]}{\partial x^{2}} + V = E $$

Now, what exactly is ##E##? Is this "equality" actually just a definition, such that ##E## is just defined this way? Also, does ##E## stand for "energy"?

Anyways, they go on to find ##f(t)##, which with trivial integration is exponential, and ##f(t) = exp( \frac {-iE}{\hbar} t)##

So now I'm at this point, and I am not seeing how I will find ## \Psi (x,t) ##. I do know ## \Psi (x,t) = \psi (x) exp( \frac {-iE}{\hbar} t)## My guess is to try and substitute back into the SE

$$ i \hbar \frac {\partial [\psi (x) exp( \frac {-iE}{\hbar} t)]}{\partial t} = - \frac {\hbar^{2}}{2m} \frac {\partial^{2} [\psi (x) exp( \frac {-iE}{\hbar} t)]}{\partial x^{2}} + V [\psi (x) exp( \frac {-iE}{\hbar} t)] $$

Okay, well where the heck do I get this linear combination? This just got ugly, and the book doesn't show the steps.

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# Time-independent SE linear combination solution help

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