I want to prove that $|\hat{\psi}(p)|^2= \frac{1}{2\pi} \int W_\psi \mathrm{d}x $ where $W_\psi $ is the Wigner function.
Starting with the definition I get ($z=-y$ and $u=x+z/2$):
$$\frac{1}{2\pi}\iint \psi^*\left(x-\frac{y}{2}\right)\mathrm{e}^{\mathrm{i}py} \psi\left(x+\frac{y}{2}\right) \mathrm{d}y \mathrm{d}x= \frac{1}{2\pi}\iint \psi^*\left(u\right)\psi\left(u-z\right) \mathrm{d}u\: \mathrm{e}^{-\mathrm{i}pz}\mathrm{d}z$$
Next I want to use the Convolution Theorem to get the product of the Fourier Transforms:
$$\frac{1}{2\pi}\iint \psi^*\left(u\right)\phi\left(z-u\right) \mathrm{d}u\: \mathrm{e}^{-\mathrm{i}pz}\mathrm{d}z= \frac{1}{\sqrt{2\pi}}\mathcal{F}(\psi^{*}*\phi)(p)=\hat\psi^*(-p)\hat{\psi}(-p) $$
But with the definition $\phi(x):=\psi(-x)$ I get $|\hat{\psi}(-p)|^2$. Does somebody know where the problem is?
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$\begingroup$ Your exponent on the left hand side has a very wrong sign ! This amounts to a p flipped around.... $\endgroup$– Cosmas ZachosMay 18, 2022 at 15:29
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$\begingroup$ @CosmasZachos in the first line? $\endgroup$– SilasMay 18, 2022 at 15:35
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$\begingroup$ On the left hand side of the first line. You are mis-defining p. $\endgroup$– Cosmas ZachosMay 18, 2022 at 15:41
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$\begingroup$ @CosmasZachos but is $W_\psi(x,p)=\int\psi^*\left(x-\frac{y}{2}\right)\mathrm{e}^{\mathrm{i}py} \psi\left(x+\frac{y}{2}\right) \mathrm{d}y $ not the definition of the Wigner function? $\endgroup$– SilasMay 18, 2022 at 15:45
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1$\begingroup$ @CosmasZachos ahh yes thanks. I didn‘t see that the WP definition has the plus sign in the complex conjugated wavefunction. $\endgroup$– SilasMay 18, 2022 at 16:50
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1 Answer
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Use $$ \psi\left(x\pm\frac{y}{2}\right)=\int dx e^{ip\left(x\pm\frac{y}{2}\right)}\bar{\psi}(p) $$ and $$ \int dxe^{i(p-p')}=2\pi\delta(p-p') $$
answered May 18, 2022 at 15:39