Completing the square for integration of e^( )

[javaman1989]

Homework Statement

I'm working on problem 2.22 from Griffith's Intro. to Quantum Mechanics (a free particle problem). I am stuck on the final integral from part b. Part a of the problem is normalizing:
A*e^-a*x2 which I did. Part b wants the general, time-dependent wave function.

Homework Equations

Griffith says that integrals of the form
∫e^-(a*x2+b*x) can be solved by "completing the square." Griffith gives the example of defining y to be (a)^(1/2)*(x+(b/(2a)) and using that definition to convert (a*x²+b*x) to y² - (b²/(4*a)) I presume we are supposed to substitute in the converted expression into the integral and work from there.

The Attempt at a Solution

I have the following integral:
∫e^{-k2/(4*a)+i*(k*x-(h*k2)/(2*m)*t)}
I presume using the "completing the square" method is supposed to make this integral work out; but I don't see how this will help. If I do convert this into something like:
e^{y + (something about a, h, and m)},
how do I integrate with y?

Thanks,
Vance

 

[HallsofIvy]

The exponential, e^{-k^2(y^2- b^2/4a}) can be separated as e^{-k^2y^2}e^{k^2b^2/4} and then the constant, e^{k^2b^2/4a} can be taken out of the integral, leaving \int e^{-k^2y^2}dy. That integral cannot be done in terms of 'elementary' functions (polynomials, fractions, trig functions, exponential, logarithms). That integral is defined to be the "Error function" or "erf(x)".

 

[javaman1989]

Thank-you so much. The missing part was the change of variables.
Vance