Can You Split the Exponential in This Quantum Mechanics Integral?

[iScience]

i'm trying to do the following integral:

$$\int{e^{\frac{-2amx^2}{ħ}}dx}$$ (in case this is hard to see, the exponent is $$\frac{-2amx^2}{ħ}$$)

where a, m are real constants

but inside the integral can't i split this up into two exponentials?

$$\int{e^{\frac{-2am}{ħ}}e^{x^2}dx} = e^{\frac{-2am}{ħ}}\int{e^{x^2}dx}$$


if not, then why not?..

 

[strangerep]

No, you can't split the exponential in that way. This is due to basic properties of the exponential function.

Alternatively, you could perform a change of variable $$x \to x' = x \sqrt{2am/\hbar} ~.$$
Maybe you first try to do the "easier" integral $$\int e^{-x^2} dx$$ (though perhaps this will still be quite difficult since you're apparently unfamiliar with the properties of the exponential function).

 

[SteamKing]

Even QM can't get around the math.

 

[Chopin]

Since that integral is non-trivial unless you know the trick, you may want to read up on it at http://en.wikipedia.org/wiki/Gaussian_integral. It's well worth getting very comfortable with this type of integral, too, as it comes up again and again in QM. There's a reason for the old saying that the only integral a theoretical physicist knows how to do is a Gaussian.