Okay, hello.

We are just discussing Bohm's theory, or the pilot wave theory as it's called.

And so that's surprisingly simple ansatz to obtain particle trajectories, which then,

if you average over an ensemble, completely reproduce all statistical predictions of quantum

mechanics.

And so just let me remind you, the idea is you have a definite trajectory R of t, but

at the same time you also time evolve the Schrödinger evolution for the wave function.

And you need this because you write the wave function as the magnitude times e to the i

the phase, and this phase is then needed to provide the equation of motion for the trajectory.

So the equation of motion for the trajectory will be the velocity at any given time is

given by the gradient of the phase of the wave function evaluated at the particle position.

With some suitable pre-factor which has to be chosen h bar over m.

Because if it's chosen like this, then the following holds.

If we imagine an ensemble of such particles starting out at time zero, and it just so

happens that the density of the ensemble matches the quantum mechanical probability density,

this will hold for all times.

Psi squared will be equal to rho for all times.

And so in so far as measurements are just position measurements, you are guaranteed

that all the outcomes, all the statistical outcomes just match quantum mechanical predictions.

Now we went a little bit further last time.

We said, okay, let's say this is my velocity field v, and now I think of a hydrodynamic

situation.

So either I can say, aha, my particle moves along the gradient of the phase of the wave

function, which means perpendicular to the lines of constant phase.

Or else I can think of an ensemble of particles moving along such trajectories.

And in order to understand these trajectories, I can imagine that at any given moment in

time, I plot the velocity field, like I would do for the motion of water.

So a particle will drift along this velocity field.

And then we ask ourselves, so what is the equation of motion for the velocity field?

And we found the following, the time derivative of the velocity field plus v times gradient

applied to the velocity field equals some acceleration.

An acceleration that is given by a suitable force, I would call that F-bomb, because it's

different from the Newtonian force, divided by the mass.

Now the left-hand side is very well known to anyone who does hydrodynamics.

This just tells you if the velocity field, even if the velocity field is stationary and

the first term vanishes, then still particles following these streamlines which bend must

be accelerated, obviously.

So this term is the so-called effective term.

So the left-hand side really is the acceleration of the particle.

What is interesting is the right-hand side.

So the Bromian force consists of two parts.

The first part is just the standard Newtonian force.

And so if I do have a potential, it's minus gradient of the potential.

But then there is another contribution.

Let me just write everything in terms of the potential.

Because for this there is added a term that depends on how the modulus of the wave function

depends on position.

So let me write it down.

This would be minus h bar squared over 2m Laplacian of the modulus of psi divided by

the modulus of psi.