This audio is presented by the University of Erlangen-Nürnberg.

This audio is presented by the University of Erlangen-Nürnberg.

This audio is presented by the University of Erlangen-Nürnberg.

This audio is presented by the University of Erlangen-Nürnberg.

This audio is presented by the University of Erlangen-Nürnberg.

This audio is presented by the University of Erlangen-Nürnberg.

This audio is presented by the University of Erlangen-Nürnberg.

This audio is presented by the University of Erlangen-Nürnberg.

... needs to account for several things.

I'll mention three.

The first one is that...

measurements of observables range over range unlike in classical mechanics not

only over some interval I in R. So I underline this whole statement and I'll

give an example for that. So first of all in classical mechanics I remind you an

observable was a map, a map F that takes an element of the phase space which in

most cases is given as the cotangent bundle of some configuration space, phase

space, the Q's and the P's. It takes an element from phase space which are the

states of the system. If you know an element in here you know what the state

in which the system is which is exactly the information you know in order to

evaluate each observable of the system and takes this to the real numbers and

because it's classical mechanics these maps at least are continuous. Alright I

mean usually you want them to be differentiable and so on but they're at

least continuous. What does that mean? Well if you have the entire phase space

here and you equip this with the standard topology and you have a topology

on here it means you can only the image of this F will always be an interval on

R. It cannot jump. So this continuity means that if you look at all the values

this F could take for all the states so I plug the whole set in then what you get

is an interval because of the continuity. Okay it's not possible for instance for

the energy to get energies from 0 to 20 and then from 40 to 60 it's just not

possible because of this continuity. So consider for instance the two-body

problem. The two-body problem for a potential or classical mechanics right

for a potential V of R goes like minus 1 over R so the typical Kepler potential

or as you met it in classical electrodynamics to electrostatic

spherically symmetric potential to Coulomb potential. So you know that if

you have a two-body problem that you can reduce it to a one-body problem with an

effective potential V effective which then if the angular momentum of the

system is not zero will look roughly like this for angular momentum L non

zero. You have an effective potential like this right and so this is the

relative coordinate between the two, it's no color, between the two now between the

two bodies and now you know of course if you're here in this two-body problem

it's reduced to a one-body problem then the system doesn't move and that's the

circular motion. So this is the exact circular motion. Now if you have an energy

so this is if you have a total energy that is like here then the system will

have the particles will have relative distance between this R minimal

and this R maximal so if you have an energy like this so this was the energy

for the circular motion if you have an energy like this then you get elliptic

motion and that it's elliptic motion that discloses has to do with this being

a 1 over R potential. Okay now if you go to this energy here which here happens

to be zero then you actually get the parabolic motion so it's already

scattering of one particle on the other and if you go to even higher energies