Unravelling real and complex parts of material parameters

[swooshfactory]

Hi,

I'm writing a term paper and am having a lot of trouble understanding what is meant by the real and imaginary parts of various quantities. It seems a lot of textbooks have their own conventions and it's hard to understand whether they've redefined some symbol when they "complexify" them.

I do understand certain things, like how in the Drude model a complex electric field implies a complex magnitude of velocity for the oscillating electrons, which in turn means that they pick up some offset phase from the electric field.

An example of my discomfort is captured by the following excerpt from Wooten's Optical Properties of Solids, available (with his consent) online.

"Although D-hat can be written in complex notation, the values for the physical quantities that D-hat represents are not ovtained by taking the real part of D-hat. The quantity D-hat is truly a complex quantity and represents two real quantities D and J. The true values from D-hat must be obtained from the right-hand side of D-hat = D + i 4 pi J/ omega by taking the real parts of D and J, i.e., D-hat(true) = Re(D-hat) + i (4 pi / omega) Re (J). Having recognized that there is a truly complex D-hat, we shall from here on generally follow convention and simply write D."

He says "two real quantities D and J" then takes Re(J).

It's difficult to capture my confusion, because I now mistrust all of this complex stuff and never know what's supposed to be real and what's complex and what either of those mean.

Help, please!

 

[eljose79]

Hi there,

I completely understand your confusion and frustration with understanding the real and imaginary parts of complex quantities. It can be a tricky concept to grasp, especially when different textbooks use different conventions. But don't worry, I'm here to help clarify things for you!

First, let's define what we mean by "real" and "imaginary" parts of a complex quantity. A complex quantity is made up of two parts - a real part and an imaginary part. The real part represents the magnitude or size of the quantity, while the imaginary part represents the phase or direction of the quantity. In other words, the real part is the "normal" or "ordinary" part of the quantity, while the imaginary part is the "extra" or "special" part that gives it a unique direction or angle.

Now, when we talk about complex quantities in physics or science, we often use the notation of a complex number, with a real part and an imaginary part separated by the letter "i" (which represents the square root of -1). For example, a complex quantity could be written as a + bi, where a is the real part and bi is the imaginary part. This notation is commonly used to represent complex quantities in equations and calculations.

In the excerpt from Wooten's book that you provided, he is discussing a complex quantity D-hat, which represents two real quantities D and J. However, in order to obtain the true values of D and J from D-hat, we must take the real parts of both D-hat and J, as he explains in the equation D-hat(true) = Re(D-hat) + i (4 pi / omega) Re (J). This means that the real part of D-hat is D and the real part of J is J (since the imaginary part is multiplied by i). So, in this case, the real parts of D-hat and J are the true values that we are looking for.

In summary, the real and imaginary parts of a complex quantity represent different aspects of the quantity - the real part is the magnitude or size, while the imaginary part is the phase or direction. When dealing with complex quantities in physics or science, it is important to understand how to obtain the true values from the complex notation, as explained in the excerpt from Wooten's book. I hope this helps clarify things for you and makes the concept of complex quantities a little less confusing. Let