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I want to calculate the Fourier transform of the Feynman propagator for a free particle $$G(q_f,q_i;E) = \left ( -\frac{i}{\hbar} \right ) \int_{0}^{\infty} K(q_f t\mid q_i 0)e^{\frac{iEt}{\hbar}} dt \tag{1} $$

to do that I`m recall that the propagator of a free particle looks like:

$$K(q_f t\mid q_i 0) = \left ( \frac{m}{2\pi i\hbar t} \right )^{1/2}\exp \left ( \frac{im(q_f- q_i)^2}{2\hbar t} \right ) .\tag{2} $$

This expression confused me a lot, because here I have $1/t$ in the exponent. Any tips on how I could do this?

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  • $\begingroup$ HI StanLinch. Welcome to Phys.SE. So the question is not about formula (2) but how to evaluate the $t$-integral (1)? Are you following a reference? Which page? In which context? The free particle has continuous spectrum. $\endgroup$
    – Qmechanic
    Oct 27, 2022 at 8:10
  • $\begingroup$ @Qmechanic Yes, shure. This is not a task from some textbook. I just want to calculate the density of states and for this I want to use such an integral. $\endgroup$
    – StanLinch
    Nov 4, 2022 at 9:57

1 Answer 1

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If we insert OP's formula (2) into OP's integral (1), and Wick-rotate $\tau=it$, then OP's integral (1) becomes (up to a sign)

$$\begin{align} &\frac{1}{\hbar}\int_{\mathbb{R}_+}\!d\tau \sqrt{\frac{m}{2\pi\hbar\tau}} \exp\left(-\frac{m(\Delta q)^2}{2\hbar\tau}+\frac{E\tau}{\hbar} \right) \cr ~=~& \frac{1}{\hbar}\sqrt{-\frac{m}{2E}} \exp\left(-\sqrt{-2mE}\frac{|\Delta q|}{\hbar} \right), \qquad E~<~0.\end{align}\tag{A}$$

Eq. (A) matches the Greens function

$$G(\Delta q;E)~=~\frac{m}{\hbar^2\kappa} \exp\left(- \kappa|\Delta q|\right), \qquad \kappa~:=~\frac{\sqrt{-2mE}}{\hbar},\tag{B}$$

for

$$ \left(-\frac{\hbar^2}{2m}\frac{\partial}{\partial q_f^2}-E\right)G(\Delta q;E)~=~\delta(\Delta q), \qquad \Delta q~:=~q_f-q_i.\tag{C}$$

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  • $\begingroup$ thank you so much $\endgroup$
    – StanLinch
    Dec 5, 2022 at 14:17

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