Gaussian integral for Peskin Schroeder page 14 propagator

[silverwhale]

Hi Everybody,

I am trying to do the calculation of Peskin Schroeder page 14, namely the first block of equations. The author moves from:

$$U(t) = \frac{1}{2 \pi^3} \int d^3p e^{-i(p^2/2m)t} e^{ip \cdot (x-x_0)}.$$
to
$$U(t) = (\frac{m}{2 \pi i t})^{3/2} e^{im(x-x_0)^2/2t}.$$
I guess the way to go is to do a spherical integration of a Gaussian. But I can't really advance through the calculation. I am stuck at this point:

$$\int \int \int p^2 sin \phi dp d\theta d\phi e^{-i (\frac{p^2}{2m}) t} e^{ip [\sin \phi \cos \theta (x-x0) + \sin \phi \sin \theta (y-y_0) + \cos \phi (z -z_0)]}.$$

Trying to get rid of the theta integral I get this function:
$$\int_0^\pi e^{ip sin \phi \cos(\theta) (x-x_0)} d\theta \equiv \int_0^\pi e^{i m \cos(\theta)} d\theta$$
which I do not know how to integrate.

Am I on the right track? Any hint is welcome!

 

[stevendaryl]

Hi Everybody,

I am trying to do the calculation of Peskin Schroeder page 14, namely the first block of equations. The author moves from:

$$U(t) = \frac{1}{2 \pi^3} \int d^3p e^{-i(p^2/2m)t} e^{ip \cdot (x-x_0)}.$$
to
$$U(t) = (\frac{m}{2 \pi i t})^{3/2} e^{im(x-x_0)^2/2t}.$$
I guess the way to go is to do a spherical integration of a Gaussian. But I can't really advance through the calculation. I am stuck at this point:

First, complete the square:

$\frac{p^2}{2m} t - p \cdot (x-x_0) = \frac{t}{2m} [(p - \frac{m}{t} (x-x_0))^2 - (\frac{m (x-x_0)}{t})^2]$

Now, change variables to $u = p - \frac{m}{t} (x-x_0)$.

 

[MathematicalPhysicist]

It seems in PF there are a lot of experts on the easy questions... :-D
https://www.physicsforums.com/threads/a-proof-of-kaku-8-18.945100/

 

[silverwhale]

Sometimes easy seems to be hard.. ;-)