Okay, hello, welcome back. Let me first start off by reminding you of what we did. We said

that based on a few principles you can derive the Schrödinger equation as the wave equation

for meta waves because it gives you just the right dispersion relation. And so that would

be Schrödinger to pi h bar time derivative of psi equals and then minus h bar squared

so like 2m that plus and applied to psi plus the potential might be pi to the psi. So that

would be the Schrödinger equation. Then you can start to try and solve it for different

potentials for the particle in a box, for the harmonic oscillator, for the hydrogen

atom in particular and you realize that for the hydrogen atom yes you get the correct

energy levels that you would also have according to Bohr's POV and that fit experiment. Okay

and then the question comes up what is the meaning of this wave field because for other

wave fields we know what is the meaning of the wave field if we have a sound wave then

we have pressure and density perturbations or if we have water waves then we just look

at the height of the water surface so there is direct meaning of the wave. Even for electromagnetic

waves in principle we can measure electric and magnetic fields by having small charged

test particles so in all of these cases at least it's clear in principle how to measure

the wave even if it's not always clear what is the microscopic mechanism of this wave

propagation. It's clear if you look into elastic waves there's the forces between the atoms

that make up the crystal but it's completely unclear for the electromagnetic field and

if you start to ask what is the medium in which electromagnetic waves propagate then

you already run into trouble. Okay but at least you can measure the electric and the

magnetic fields. Now there is no way that we know of how to measure psi itself directly

and in the beginning of quantum mechanics it was therefore very natural to us what is

the meaning of this strange wave field which by the way is complex although of course if

you insist that all wave fields that have physical meaning should be real valued you

could just take the real and imaginary parts of psi and then write down complete equations

so that is not the central problem. Now we started the last time by saying that typically

if we have a wave equation like this which is a linear wave equation if you add two solutions

you get another solution if you multiply the solution by a number you get another correct

solution so for such linear wave equations typically we expect that we can build a quadratic

construct out of the wave field which gives rise to a conservation law which gives rise

to energy conservation or momentum conservation and so on so more generally speaking there

are conserved densities and often this gives some insight into the meaning of the wave

field because for example for a sound wave propagating in a crystal lattice you would

have the elastic energy stored in the sound wave and so what is the general concept for

such a local conservation law well you find a density rule which might be the energy density

or the particle density and you find a current density which is a vector field and then you

have an equation like this the time derivative of the rule plus the divergence of this current

density is zero and both of the density and the current density you would be able to express

by the wave field at some point and its derivatives. So this would be usually then called the equation

of continuity and the picture to have in mind is really like in hydrodynamics so for example

in hydrodynamics the current density is very simply connected to the density and the flow

field because you just have J equals rho times the velocity field and then you can try to

draw a situation a typical situation so you could have this flow field like here and then

obviously what happens is it just transports along the particles they stream along from

left to right and because they do that you don't expect the density to change if the

density is constant everywhere then the density will all repulsively remain constant and that's

correct and is compatible with this equation because if you look at it the divergence of

the current density in this situation would just be zero there is no source and no sink

and so everything is fine but then if you have a situation where suddenly the current

density lines converge to a point obviously here the divergence would be negative and