Integrals With An Imaginary Linear Term in The Argument of The Exponent
The integral
is proportional to the Fourier transform of the gaussian where is the conjugate variable of .
By again completing the square we see that the Fourier transform of a gaussian is also a gaussian, but in the conjugate variable. The larger is, the narrower the gaussian in and the wider the gaussian in . This is a demonstration of the uncertainty principle.
Read more about this topic: Common Integrals In Quantum Field Theory, Variations On A Simple Gaussian Integral
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