Complicated integral

Question

The question has to do with the Schrodinger equation in one dimension and using the propagator. I have gotten an integral but it has to be solved to get the final solution which I can't get. The integral is:

ψ(x,t) = ∫[-∞,∞] of (dx'/sqrt(4πDt))*exp[(-|x-x'|^2)/4Dt]*A*exp[-β(x'-b)^2]

where D=(ih)/(2m)

please help if you can for I'll be really grateful! :)

Answer 1

The integral is symmetric about x = 0, so you can double the integral from 0 to ∞.  Let α = 1/4Dt.

 

 

exp[-(x-x')²/4Dt]exp[-β(x' - b)²] = exp[-α(x² - 2xx' + x'²)]exp[-β(x'² - 2bx' + b²]

 

= exp[-αx² + 2αxx' - αx'² - βx'²+ 2βbx' - βb²] = exp[-(α + β)x'² + 2(αx + βb)x' -(αx² + βb²)]

 

Complete the square: -(α + β)x'² + 2(αx +βb)x' = -(α + β)[x'² - 2(αx + βb)x'/(α + β)]

 

= -(α + β)[x' - (αx + βb)/(α + β)]² + (αx + βb)²/(α + β)

 

= -(α + β)[x' - (αx + βb)/(α + β)]² - (αx + βb)²

 

 

Use this last expression and a published definite integral for the exponential to get a final result.  Good luck!