# Quantum mechanics Dirac delta representation with integral

So I’m doing QM and found bunch of problems for beginners and I’m struggling with this one: $$\lim_{a\rightarrow 0}\int^{\infty}_{-\infty}e^{\frac{ip x}{\hbar}-a x^2}dx=2\pi\hbar\delta(p).$$ If I swap the limit and the integral I get the correct answer, but can I do that?

• Compete the square in the exponent and do the Gaussian integral. Don't swap. Commented Oct 6, 2020 at 0:00
• related question on math stack exchange: math.stackexchange.com/q/253696 Commented Oct 6, 2020 at 0:05
• Would Mathematics be a better home for this question? Commented Oct 6, 2020 at 5:15
• I believe the OP could provide more context to this question. This is a more general problem of Fourier Transform of a Gaussian state. The physics is interesting, may be add more context to be useful to a larger audience? Commented Oct 6, 2020 at 5:45

Using $$\int_{-\infty}^{\infty}e^{-\alpha x^2+\beta x}=\sqrt{\frac{\pi}{\alpha}}\exp{\frac{\beta^2}{4\alpha}}$$
put $$n=1/a$$ we get

$$\lim_{n \to \infty}\sqrt{\pi n \cdot\frac {\pi 4 \hbar^2}{\pi 4 \hbar^2}}\exp{\frac{-p^2n}{4\hbar^2}}$$ Now define $$n=>n/4\hbar^2$$

Now,$$e^{-nx^2}\sqrt{\frac{n}{\pi}}$$ defines a generalised function $$\delta(x)$$ such that

$$\int_{-\infty}^{\infty}\delta(x)F(x)=F(o)$$ in limit $$n\rightarrow\infty$$ can be easily proved (ref.M.J.Lighthill Fourier and Generalised Functions for more)

We get your answer $$2\pi\hbar\delta(p)$$

You can just rewrite the integral first(make substitutions $$x = -x$$, $$\frac{p}{\hbar}=k$$), say $$I=\int_{-\infty}^{\infty}dx \ e^{-i k.x} e^{-ax^2}$$ Now, this is a like a Fourier transform of a Gaussian function.

Where $$I = \sqrt{\frac{\pi}{a}}e^{-\frac{k^2}{4a}}$$

Therefore, you can look at the definitions of dirac delta as limit, and write $$2\pi \ \lim_{a\rightarrow 0} \frac{1}{2\sqrt{{\pi a}}}e^{\frac{-k^2}{4a}} = 2\pi \delta(k)$$ Now, using the property of dirac delta $$\delta(ax) = \frac{1}{|a|}\delta(x)$$, you can get to your final form.

PS: But, it would be interesting to think of the physical motivation of this manipulation for the context in which OP might be looking at this question.