Why does intensity mean anything if there's a complex number

[yosimba2000]

So say a wave is described by Acos(Φ), completely real.

Then the to use Euler's Eq, we we say the wave is Ae^iΦ, which is expanded to Acos(Φ) + iAsin(Φ). We tell ourselves that we just ignore the imaginary part and only keep the real part.

And if intensity is |Ae^iΦ|², which is (Acos(Φ) + iAsin(Φ)) * (Acos(Φ) - iAsin(Φ)), we get A²cos²(Φ) + A²sin²(Φ).

So why do we take the sin(Φ) part in the intensity result, instead of just taking the cos(Φ) part?

 

[PeterDonis]

say a wave is described by Acos(Φ), completely real.

Then the to use Euler's Eq, we we say the wave is ^AeiΦ,

You just contradicted yourself. Either the wave is described by $A \cos \Phi$, or it's described by $A e^{i \Phi}$. You have to pick one; it can't be both.

It might help if you would give an actual concrete problem, preferably from a reference such as a textbook or peer-reviewed paper, where you encountered this.

 

[yosimba2000]

You just contradicted yourself. Either the wave is described by $A \cos \Phi$, or it's described by $A e^{i \Phi}$. You have to pick one; it can't be both.

It might help if you would give an actual concrete problem, preferably from a reference such as a textbook or peer-reviewed paper, where you encountered this.

Can you explain? I thought Euler's equation was only used as convenience because it's easier to write down or something. For example when dealing with sinusoidal voltages, they are expressed in Euler's equation, but only the real part of it has any meaning.

 

[PeterDonis]

I thought Euler's equation was only used as convenience

In which case your question makes no sense, because it assumes that there is some actual difference between writing $A \cos \Phi$ and $A e^{i \Phi}$.

You really need to give a specific reference for where you are getting all this from.

 

[yosimba2000]

In which case your question makes no sense, because it assumes that there is some actual difference between writing $A \cos \Phi$ and $A e^{i \Phi}$.

You really need to give a specific reference for where you are getting all this from.

I'm getting this from the first page of Chapter 1 at the bottom paragraph.

 

[PeterDonis]

I'm getting this from the first page of Chapter 1 at the bottom paragraph.

So which case do you want to discuss, a classical wave or a quantum wave function? The notes you reference are about the quantum case; they only mention the classical case for comparison.

 

[yosimba2000]

I want to discuss the quantum case. I was assuming the reason why we used Euler's formula for the classical case (convenience) was also the same reason why we used it in the quantum case.

 

[PeterDonis]

I was assuming the reason why we used Euler's formula for the classical case (convenience) was also the same reason why we used it in the quantum case.

You assumed incorrectly, as the passage you referenced from the notes you referenced should make obvious.

 

[PeterDonis]

I want to discuss the quantum case.

If you want more discussion of that case, in case the response I gave in post #8 just now isn't enough, please start a new thread in the Quantum Physics forum, and please formulate your question to make it clear that you're asking about the quantum case, and what exactly you are asking about.

This thread is closed.