This is going to get very involved and very maths abstract gan :-)
You will need to give me some background do you know about matrix mathematics and hermite curves in you want to go deeper than what I am about to do for simplicity?
So you have your hamiltonian operator which is how your systems is evolving over time, which you understand.
Since by definition your hamiltonian has the dimension of energy then if we are describing that system from a waveform input as in QM (schrodinger wave equation) then we can describe your hamiltonian via a matrix calculation and that matrix eigenvalues are real.
Essentially what we are saying is we can tie the movement of our input waveforms through to your hamiltonian system.
Many matrix solutions provide non real solutions so your hamiltonian will work when we look at a static value but it will not play out in the correct order if we looked at it over time. So given a wave input your hamiltonian values would plot to some correct value but when we slowly move the input wave the output would jump all over the place to a value on our output but the time sequence would be all wrong.
A hermitian operator is a solution from your input waveform to your hamiltonian system so it is time consistant or smooth as opposed to one that jumps around all over the place.
I guess if you imagine a joystick as your waveform input connected thru your computer to some machine. If you imagine the joystick has 256 positions what you want is a fluent movement not the 256 positions being represented all weirdly on the machine so it jumped around all over the place. Such a nice mapping I would call hermitian in operation.
In QM all observable properties must be represented by hermitian operators as we don't see balls jumping from one location to another then back they move fluently.
The reason for that requirement is the mathematics will give you many solutions but many will be of that jumpy, jerky type which are not hermitian.
The background of this sort of movement is hermite curves and hence the expression.
So hermitian operators are what we would call possible real world solutions there will be many other jerky, disjointed solutions.
I have taken a few liberties again because this is very difficult to bring down to this sort of level.
Last edited by Orac; 11/13/11 05:13 PM.
