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I am trying to obtain a representation of the momentum-space wavefunction $<p'|\alpha>$

Its position space wavefunction is given as $$ <x'|\alpha> = N \exp [-(a+ib)x'^2 +(c+id)x'] $$ where $N,a,b,c,d$ are all real.

I have tried to calculate the following integral using Gaussian integral, or Fourier transform method but both were not easy because of the complex numbers in the exponent.

$$ <p'|\alpha> = \int dx' <p'|x'><x'|\alpha> = \int dx' {1 \over \sqrt{(2 \pi \hbar)}} \exp[-i{p'x' \over \hbar}] N \text{exp} [-(a+ib)x'^2 +(c+id)x'] dx'$$

where I have used $$<p'|x'> = {1 \over \sqrt{(2 \pi \hbar)}} \exp[-i{p'x' \over \hbar}].$$

Can someone give me a hint about calculating indefinite integral with complex exponent?

Qmechanic
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raccoon
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    Hint 1: Try completing the square in the exponential. Does the resulting function seem familiar? Hint 2: It is just a Gaussian integral https://en.wikipedia.org/wiki/Gaussian_integral ! – Gabriel Ybarra Marcaida Mar 24 '24 at 13:36
  • Some of the formulas for gaussian integration continue to work regardless of whether the parameters are real or complex, and regardless of whether you are integrating along the real line ($-\infty$ to $\infty$), or along a line in the real direction but displaced off the real axis ($-\infty + i\Delta$ to $+\infty + i\Delta$). This means you can often play fast and loose with your variable substitutions and your limits of integration and still get the right answer. – hft Mar 24 '24 at 17:08
  • Does this answer your question? Gaussian integral with imaginary coefficients and Wick rotation – hft Mar 24 '24 at 17:11
  • Or this: https://math.stackexchange.com/questions/1297096/gaussian-integral-with-a-shift-in-the-complex-plane – hft Mar 24 '24 at 17:12

1 Answers1

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Given your integral:

$$ \langle p'|\alpha\rangle = \int dx' \frac{1}{\sqrt{2 \pi \hbar}} \exp\left(-\frac{i p'x'}{\hbar}\right) N \exp\left[-(a+ib)x'^2 +(c+id)x'\right] $$

You can complete the square within the exponent term involving $x'$: $$ -(a+ib)x'^2 +(c+id)x' = -a(x' - \frac{c+id}{2a})^2 + \left(\frac{(c+id)^2}{4a} - ib\right) $$

Then, you have: $$ \langle p'|\alpha\rangle = N\frac{1}{\sqrt{2 \pi \hbar}} \exp\left(\frac{(c+id)^2}{4a} - ib\right) \int dx' \exp\left[-a\left(x' - \frac{c+id}{2a}\right)^2\right] \exp\left(-\frac{i p'x'}{\hbar}\right) $$

Now the integral looks more tractable as it's a Gaussian integral and you can use the standard results for Gaussian integrals to evaluate it.

Adversing
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