# A complex integral from the text

Homework Helper
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In my book QM book, Gasiorowicz says that

$$\int_{-\infty}^{\infty} e^{i(p-p')x/\hbar}dx = 2\pi \hbar \delta(p-p')$$

Where does that come from? I mean, set i(p-p')/h = K. Then the solution is

$$\frac{\hbar e^{i(p-p')x/\hbar}}{i(p-p')}$$

and evaluate at infinity, it doesn't exists as the limit of cos and sin at infinity do not exist.

There must be something in the fact that the integral is complex but I haven't studies complex analysis yet so go easy on me plz.

Homework Helper
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What, no one?

George Jones
Staff Emeritus
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This takes a bit of work. Are you familiar with Fourier transforms?

Regards,
George

Homework Helper
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$$\int_{-\infty}^{\infty}e^{2ikx}dx = 0$$

George Jones
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quasar987 said:
$$\int_{-\infty}^{\infty}e^{2ikx}dx = 0$$

This isn't consistent with your first post in this thread. To see this, set $p'=0$, $\hbar =1$, and $x = 2u$ in the integral in your first post.

Regards,
George

Homework Helper
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Well they are respectively equation 3-40 and 3-54 in Gasiorowizc.

George Jones
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It's okay if $k \neq 0$.

$$\int_{-\infty}^{\infty}e^{2ikx}dx = \frac{1}{2} \int_{-\infty}^{\infty}e^{iku}du = \pi \delta (k)$$

This equals zero if $k \neq 0$.

Regards,
George