I hope this is the correct place for my question. I posted it here, because it`s from Peskin & Schroeder:(adsbygoogle = window.adsbygoogle || []).push({});

"Consider the amplitude for a free particle to propagate from [tex] \mathbf{x}_{0} [/tex] to [tex] \mathbf{x} [/tex] :

[tex]U(t)=\left\langle \mathbf{x}\right|e^{-iHt}\left|\mathbf{x_{0}}\right\rangle[/tex]

In nonrelativistic quantum mechanics we have E=p^2/2m, so

[tex] U(t)&=&\left\langle \mathbf{x}\right|e^{-i(\mathbf{p}^{2}/2m)t}\left|\mathbf{x}_{0}\right\rangle [/tex]

[tex]

=\int\frac{d^{3}p}{(2\pi)^{3}}\left\langle \mathbf{x}\right|e^{-i(\mathbf{p}^{2}/2m)t}\left|\mathbf{p}\right\rangle \left\langle \mathbf{p}\right.\left|\mathbf{x_{0}}\right\rangle [/tex]

[tex]

=\frac{1}{(2\pi)^{3}}\int d^{3}p\, e^{-i(\mathbf{p}^{2}/2m)t}\cdot e^{i\mathbf{p}\cdot(\mathbf{x}-\mathbf{x}_{0})}

[/tex]

[tex]

=\left(\frac{m}{2\pi it}\right)^{3/2}\, e^{im(\mathbf{x}-\mathbf{x}_{0})/2t}}

[/tex]

."

I don't understand the last equation.

Why I can't use the fourier-transformation:

[tex]

=\frac{1}{(2\pi)^{3}}\int d^{3}p\, e^{-i\mathbf{p}\cdot(\mathbf{x}_{0}-\mathbf{x})}\widetilde{f}(\mathbf{p})

[/tex]

[tex]

=f(\mathbf{p})

[/tex]

[tex]

=e^{-i(\mathbf{x}-\mathbf{x}_{0})^{2}/2m)t}

[/tex]

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# Amplitude for a free particle

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