Phase factor in quantum mechanics

[Shreya]

Homework Statement

I watched a lecture in which the professor called the circle expression as a pure phase and took its absolute value to be 1. I don't understand how it's absolute value is 1.
Please refer the image for the circled expression.

Relevant Equations

Wave function at x,t

Please be kind to help.I would be grateful

 

[PeroK]

Any complex number of the form $e^{i\theta}$ has absolute value (modulus) $1$.

 

[Shreya]

Can you please explain why is it so?

 

[Gaussian97]

Can you please explain why is it so?

The norm of a complex number can be calculated using the number $z$ and $z^*=\bar{z}$ the complex conjugate.
So, given $z=e^{i\theta}, \quad \theta \in \mathbb{R}$, you should be able to compute $z^*$ and its norm.
You will see that the norm is 1 independent on $\theta$.

 

[PeroK]

Can you please explain why is it so?

Any complex number can be written $re^{i\theta}$, where $r$ is the modulus and $\theta$ is the angle in the complex plane. That's fairly elementary, I'm sorry to say!

 

[Shreya]

Thanks a lot Pero K and Gaussian97! It helped a lot.