How did it happen!!!!!!

[Sreejith A Nair]

Can someone please tell me how
Y (x,t) became y (x)f (t).{I am using y for psy}

[The Confused Guy (O.o)?]

y(x,t) is a function of x and t so we can write it as a combination two funtion of each variable x and t

[Sreejith A Nair]

But it need not always be a product of the two.

[Dinesh PR]

Yes, you're right. But it's only a particular solution. If you add all possible solutions in this form, it's again mathematically a solution! And that's really the most general solution that incorporates all possible energies the electron can have!

[schrodinger]

As electrons are considered standing waves , we can do the above.

[Confusions allowed]

These pics are from Introduction to Quantum Mechanics, D. J. Griffiths, 2nd Ed. The first two explanations are physical, and the first was already discussed in the class. I'm not clear regarding the 3rd explanation, but 2nd seems promising. Also, the expectation value is a fancy term for average of a certain quantity in QM. The parts are as follows:
Part 1:

Part 2:


Part 3:

[Confusions allowed]

One thing that I missed, to explain what is Normalization, which is mentioned in the text. Well, we do know that :
The infinite integral ∫|Ψ(x, t)|^2 dx has to be 1, as the electron is somewhere. But, the found solution may not give you that. Therefore, we devise a new Ψ' = AΨ, where A is a complex constant. We do know that Ψ appears on both the sides of the wave function, so one gets the same Physics as before. This can be used to get the total probability of finding the electron to be 1, which is called as Normalization.

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