Common Integrals in Quantum Field Theory - Variations On A Simple Gaussian Integral - Integrals With An Imaginary Linear Term in The Argument of The Exponent

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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