Week of 9/4
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and then taking the real part of that solution, only works if all the operations | 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 | are linear. Think about what would happen if you did something like take | ||
| − | <math>E \bar{E}</math> where E is a field assumed to have the form <math>E_0 e^{i \omega t}</math>? | + | <math>E \bar{E}</math> where <math>\, E</math> is a field assumed to have the form <math>E_0 e^{i \omega t} \,</math>? |
==Now back to the damped, forced Simple Harmonic oscillator == | ==Now back to the damped, forced Simple Harmonic oscillator == | ||
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[[Image:Resonance.gif]] | [[Image:Resonance.gif]] | ||
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| + | {{PDF|filename=9 8 06.pdf|title=9/8/06 lecture notes}} | ||
Latest revision as of 17:11, 8 September 2006
Try the following notebook
 Download manipulating complex numbers in Mathematica.
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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
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when is it ok to assume physical expressions are complex?
Consider the forced, damped, simple harmonic oscillator.
Let's pretend that both x and F are actually complex variables. So we would write
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](/csm/wiki/images/math/2/0/0/200ca3de1b5126a171fd0ec143de9e02.png)
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
and
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
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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
where 
is a field assumed to have the form 
?
Now back to the damped, forced Simple Harmonic oscillator
| Download 9/8/06 lecture notes
|
