Why Use Absolute Value for Squaring in Wave Function Normalization?

[Niles]

Homework Statement

Hi all.

I have a wave function given by

$$\Psi \left( {x,0} \right) = A\frac{x}{a}$$

I have to normalize it, which is OK. But in the solution to this problem, the teacher uses |A|² when squaring A. Is there any particular reason for this? I mean, if you square the constant, then why bother with the signs?

I thought that it maybe because A is a complex constant, but still - I cannot see what difference it would make taking the absolute value of A before squaring.

 

[Redbelly98]

If A is not real, then |A|² and A² are different. Allowing for A to be complex is the only reason I can think of for including the absolute value signs.

 

[nicksauce]

For complex numbers, |z|^2 is not the same as z^2. Suppose z = 1 + i. Then |z| = sqrt(2), so |z|^2 = 2. But z^2 = 1 + 2i + i^2 = 2i. |z|^2 always gives a nonnegative real number, which is required to interpret the wave function as a probability density.

 

[Niles]

Great, thanks to both of you.

 

[Niles]

I have another question related to this.

When I find the constant A, then am I finding the complex number A or the modulus of the complex number A, |A|?

 

[Redbelly98]

It's |A|.