- #1
silverwhale
- 78
- 2
Hi Everybody,
I am trying to do the calculation of Peskin Schroeder page 14, namely the first block of equations. The author moves from:
<br /> U(t) = \frac{1}{2 \pi^3} \int d^3p e^{-i(p^2/2m)t} e^{ip \cdot (x-x_0)}.<br />
to
<br /> U(t) = (\frac{m}{2 \pi i t})^{3/2} e^{im(x-x_0)^2/2t}.<br />
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:
<br /> \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<br />
which I do not know how to integrate.
Am I on the right track? Any hint is welcome!
I am trying to do the calculation of Peskin Schroeder page 14, namely the first block of equations. The author moves from:
<br /> U(t) = \frac{1}{2 \pi^3} \int d^3p e^{-i(p^2/2m)t} e^{ip \cdot (x-x_0)}.<br />
to
<br /> U(t) = (\frac{m}{2 \pi i t})^{3/2} e^{im(x-x_0)^2/2t}.<br />
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:
<br /> \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<br />
which I do not know how to integrate.
Am I on the right track? Any hint is welcome!