How Do You Normalize a Function to Unity?

[ehrenfest]

Homework Statement

I understand how you normalize vectors to unity, but how would you normalize a function to unity?

For example, how would you show that the function (2/L)^(1/2) sin(n*pi*x/L) is normalized to unity? You cannot just just take its modulus and set it to one since because x is variable...right?

Homework Equations

The Attempt at a Solution

 

[Dick]

Define the norm. Looks like QM, so you want L^2. The integral over the range of x of psi(x)*conjugate(psi(x))=1 defines a normalized function.

 

[nrqed]

Homework Statement

I understand how you normalize vectors to unity, but how would you normalize a function to unity?

For example, how would you show that the function (2/L)^(1/2) sin(n*pi*x/L) is normalized to unity? You cannot just just take its modulus and set it to one since because x is variable...right?

Homework Equations

The Attempt at a Solution

It's important to realize that the modulus squared (not just the modulus), $| \psi(x)|^2$ of a wavefunction is a probability density , not a probability. This means that the quantity $| \psi(x)|^2 dx$ is aprobability, which represents the probability of finding the particle in the infinitesimal interval (x,x+dx). What must be normalized to one is therefore

$$\int_{- \infty}^{\infty} | \psi(x)|^2 dx =1$$

It seems as if you are considering an infinite squqre well located from x=0 to x=L. In that case, the wavefunction is zero outside of the well and the above condition reduces to


$$\int_{0}^{L} | \psi(x)|^2 dx =1$$


hope this helps

Patrick

 

[ehrenfest]

I see. Thanks.