Understanding Quantum Wave Packets: How to Derive the Schrodinger Equation

[silimay]

Sorry... my message got posted by mistake before I started typing. Here is what I was going to say:

I'm having a problem just understanding something from my quantum book. They're deriving something to do with a wave packet with the Schrödinger equation, and they have the equation of a wave packet at time t = 0:

$$\psi(x,0) = \int_{- \infty}^{+ \infty} dk A(k) e^{i(kx)$$

where $$A(k) = e^{- \alpha (k - k_o)^2 / 2}$$

They change variables to $$q' = k - k_o$$ and then they get

$$\psi(x,0) = e^{i k_o x} e^{- x^2 / {2 \alpha}} \int_{- \infty}^{+ \infty} dq' e^{- \alpha {q'}^2 / 2}$$

I don't understand how they got that (specifically, the $$e^{-x^2 / {2 \alpha}}$$ term).

 

[Dick]

They did the substitution and then completed the square in q' in the integral and brought the non-square stuff outside the integral. Finally they shifted q'->q'-ix/alpha.

 

[silimay]

Thanks so much for your help =) =) =) I understand it now.