How do I solve this integration problem in quantum mechanics?

[Hyperreality]

I've recently posted a thread asking for help in the Homework forum in solving this problem, but till now I still can't figure out how do find the solution to this equation.

Here is the problem.

Show that $$A=(\frac{m\omega}{\hbar\pi})^{1/4}.$$

From the normalisation condition

$$|A|^2\int_{-\infty}^{\infty} e^{-2ax^{2}}=1$$

Where $$a =\frac{\sqrt{km}}{2\hbar}$$

I'm really have no idea on how do solve this problem. Does this solving this problem require the use of Fourier Transform?

 

[dextercioby]

I've recently posted a thread asking for help in the Homework forum in solving this problem, but till now I still can't figure out how do find the solution to this equation.

Here is the problem.

Show that $$A=(\frac{m\omega}{\hbar\pi})^{1/4}.$$

From the normalisation condition

$$|A|^2\int_{-\infty}^{\infty} e^{-2ax^{2}}=1$$

Where $$a =\frac{\sqrt{km}}{2\hbar}$$

I'm really have no idea on how do solve this problem. Does this solving this problem require the use of Fourier Transform?

No,just the Poisson integral:
$$I_{1}(a)=:\int_{-\infty}^{+\infty} e^{-ax^{2}} dx =\sqrt{\frac{\pi}{a}}$$

Daniel.

PS.In physics,this integral is widely used...

 

[Hyperreality]

Ohhhhh...

Never heard of it :grumpy: That made things whole a lot simpler.

 

[dextercioby]

Ohhhhh...

Never heard of it :grumpy: That made things whole a lot simpler.

Then how the heck were u supposed to do that integral??I'm sure it's about HLO in QM...You should know a lotta calculus for QM...

Daniel.