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- Homework Statement
- Consider a free particle whose state at t=0 given by the gaussian wave packet. Find psi(x,t)

- Homework Equations
- Gaussian wave packet at t=0, More equations in attempt at solution

**EQ 1:**Ψ(x,0)= Ae

^{-x2/a2}

__A. Find Ψ(x,0)__So I normalized Ψ(x,0) by squaring the function, set it equal to 1 and getting an A

**I.**A=(2/π)

^{¼}(1/√a)

**Ψ(x,t)= 1/(√2π) ∫ ∅(k) e**

EQ:2

__B. To find Ψ(x,t)__EQ:2

^{i(kx-ωt)}dk --------->when ω=(ħk

^{2})/2m and integral from -∞ to +∞

**EQ 3:**∅(k)= 1/(√2π) ∫ Ψ(x,0) e

^{-ikx}dx -------> integral from -∞ to +∞, i is an imaginary number

Using eq3 to find ∅(k), I got

∅(k)=(2π)

^{1/4}(√a) e

^{-(ka)2/4}

Using Eq2, I got

**II.**Ψ(x,t)=(2π)

^{-3/4}(√a) ∫ e

^{-k2(a2/4 -iht/(2m)) +ikx}dk -------> integral from -∞ to +∞

But, in the formula to complete the square

∫ e

^{-(Ax2+Bx}dx= (π/A) e

^{B2/4A}

I dont know how to manipulate II in order to get exp

**-**(Ax

^{2}+Bx). That placement of the negative in II makes me think I either normalized it wrong in part a or I messed up somewhere and don't know where.