Generalized Schrödinger equation

[Maximise24]

This equation (see attachment) appears in one of Prof. Susskinds's lectures on Quantum Mechanics: in trying to differentiate the coefficients of the eigenvectors of a wave function with respect to time, an exponential e^(-iEt) is introduced for alpha.

I can see that d/dt e^(-iEt) = -iE e^(-iEt), but why is the second part e^(-iEt) not in the top equation in the attachment? Is it disregarded because it's just a number?

Thanks for any help provided!

 

[Maximise24]

Attachment seems to have got lost.

 

[dextercioby]

But the exponential is there, disguised under the form of 'alpha' in the rhs.

 

[Maximise24]

OK, so [itex]\alpha[/itex]_j(0)e^-iEt has simply been conflated into [itex]\alpha[/itex]_j? Can you just do that since e^-iEt is not a constant?
Thanks!

 

[dextercioby]

The only variable is time. e^{-iEt} in units with hbar=1 gathers the time dependence of alpha.

 

[Jilang]

OK, so [itex]\alpha[/itex]_j(0)e^-iEt has simply been conflated into [itex]\alpha[/itex]_j? Can you just do that since e^-iEt is not a constant?
Thanks!

Aj is defined on the second line of your picture. It doesn't look like a constant to me.

 

[Maximise24]

OK, thanks guys.