Week of 9/4

Try the following notebook

Download manipulating complex numbers in Mathematica.

Also, take a look at Mathematica Tips and Tricks.

A real amplitude and phase measurement. Top figure shows the amplitude of the transmitted and reflected electric fields. Bottom are the corresponding amplitudes. Click to download the annotated pdf.

Download amplitude/phase of measured E field

when is it ok to assume physical expressions are complex?

Consider the forced, damped, simple harmonic oscillator.

$\ddot{x} + \gamma \dot{x} + \omega_0 ^2 x_r = F/m$

Let's pretend that both x and F are actually complex variables. So we would write

$x = x_r + i x_i \ \ \ \ F = F_r + i F_i \,$

Plugging these into the equation above we get

$\ddot{x_r} + \gamma \dot{x_r} + \omega_0 ^2 x_r + i\left[\ddot{x_i} + \gamma \dot{x_i} + \omega_0 ^2 x_i\right] = F_r/m + i F_i /m$

Now, whenever you have a complex equation this is equivalent to two real equations, among the real parts and the imaginary parts. Here we have

$\ddot{x_r} + \gamma \dot{x_r} + \omega_0 ^2 x_r = F_r/m$

and

$\ddot{x_i} + \gamma \dot{x_i} + \omega_0 ^2 x_i = F_i/m$

Now the trick is to solving real physical problems (at least linear ones) is to throw away the imaginary part.

Brief Digression on operators.

Download operators

The trick of letting a variable be complex, performing operations to get a solution, and then taking the real part of that solution, only works if all the operations are linear. Think about what would happen if you did something like take $E \bar{E}$ where $\, E$ is a field assumed to have the form $E_0 e^{i \omega t} \,$?

Now back to the damped, forced Simple Harmonic oscillator

Download 9/8/06 lecture notes