Saturday, 28 May 2016

quantum mechanics - Free particle propagation amplitude calculation


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/)|x0, correct?


If we insert two complete sets of momenta, the integral becomes (give or take factors of ):


1(2π)3d3pexp(itp2c2+m2c4/)exp(ip(xx0))


I'm not quite sure where to proceed from here... Peskin and Schroeder somehow reach:


12π2|xx0|0dppsin(p|xx0|)exp((itp2c2+m2c4/).


To get here, however, it seems that you would have to assume that p points in the same direction as xx0 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 xx0. Then p(xx0)=p|xx0|cosθ


Then your integral 1(2π)3d3pexp(itp2c2+m2c4/)exp(ip(xx0))=1(2π)30p2dp2π0dϕ11d(cosθ)exp(ip|xx0|cosθ)exp(itp2c2+m2c4/)


Now do the ϕ and cosθ integrals to get 14π2i|xx0|0pdp[exp(ip|xx0|)exp(ip|xx0|)]exp(itp2c2+m2c4/)


Recognize that the quantity in the square brackets is 2isin(p|xx0|) to get your answer.


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