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.

 

[paranormal]

I understand your thought process in wanting to split the integral into two exponentials. However, in quantum mechanics, we must consider the wave function, which is described by a complex-valued function. When splitting the integral into two exponentials, you are assuming that the wave function can be separated into two independent parts, which is not always the case. The wave function is a fundamental concept in quantum mechanics and must be treated with care. In this case, it is not appropriate to split the integral into two exponentials. Instead, you can use techniques like substitution or integration by parts to solve the integral. I suggest consulting a textbook or seeking guidance from a professor or colleague for further assistance.