Normalizing Functions: Basics & Examples

[laser123]

How does you normalize a function? Could someone explain it very basically and give an example.

 

[Chopin]

In Quantum Mechanics, when we talk about normalizing a function, we mean that we want its integral over all space to be equal to 1. So a function is normalized if $\int^\infty_{-\infty}f(x)\:dx = 1$

If a function is not normalized, then we can normalize it by dividing the function by whatever its total integral is. So if we define $A = \int^\infty_{-\infty}f(x)\:dx$, and we define a new function $f'(x) = \frac{f(x)}{A}$, then by definition, $\int^\infty_{-\infty}f'(x)\:dx = 1$, so $f'(x)$ is normalized.

 

[laser123]

Thank you! :D

 

[Chopin]

You're welcome, glad to help!

Also, I should point out that what I gave above was a very generic definition of normalization. In the specific case of quantum mechanics, what we're usually normalizing is the probability density, which is the magnitude squared of the wavefunction. So in practice, the way that you will usually see normalization conditions written is $\int^\infty_{-\infty}|\Psi(x)|^2\:dx = 1$.