I have a quick calculational question.
In Peskin and Schroeder, Chapter 2, they want to look at the amplitude for a particle to propagate between two arbitrary points, x and x0, in an arbitrary amount of time t given the Hamiltonian √p2c2+m2c4. To do this, we look at the inner product of the time-evolved particle that was at x0 with the x eigenket: ⟨→x|exp(−iˆHt/ℏ)|→x0⟩, correct?
If we insert two complete sets of momenta, the integral becomes (give or take factors of ℏ):
1(2π)3∫d3pexp(−it√p2c2+m2c4/ℏ)exp(i→p⋅(→x−→x0))
I'm not quite sure where to proceed from here... Peskin and Schroeder somehow reach:
12π2|x−x0|∫∞0dppsin(p|x−x0|)exp((−it√p2c2+m2c4/ℏ).
To get here, however, it seems that you would have to assume that p points in the same direction as x−x0 to get spherical symmetry or something. Why can we do this? Aren't we summing over all possible momenta?
Thanks!
Answer
Work with spherical coordinates in momentum space and choose z-axis in p-space along →x−→x0. Then →p⋅(→x−→x0)=p|x−x0|cosθ
Then your integral 1(2π)3∫d3pexp(−it√p2c2+m2c4/ℏ)exp(i→p⋅(→x−→x0))=1(2π)3∫∞0p2dp∫2π0dϕ∫1−1d(cosθ)exp(ip|x−x0|cosθ)exp(−it√p2c2+m2c4/ℏ)
Now do the ϕ and cosθ integrals to get 14π2i|x−x0|∫∞0pdp[exp(ip|x−x0|)−exp(−ip|x−x0|)]exp(−it√p2c2+m2c4/ℏ)
Recognize that the quantity in the square brackets is 2isin(p|x−x0|) to get your answer.
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