Okay, so welcome again. I want to start by having a small correction. When we calculated

the, when I gave you the result for the overlap of the wave packets in the Stern-Gerlach apparatus,

there was a little mistake in my formula and so I'll write down just the correct result.

So the magnitude of this overlap between the wave packets going up or down in the Stern-Gerlach apparatus

is just given by the following expression, P to the minus P bar squared over P delta P squared.

I will define it in a moment. Minus X bar squared over 2 delta X squared.

And so P bar is actually the momentum that depends on time and depends on the force that is exerted by this

in homogeneous magnetic fields, so it would be F times T. X bar would be the position of my wave packet

according to the acceleration by this force, so it's one half acceleration times T squared.

And that would be F over 2m T squared. And finally delta P is the width in momentum space,

that's where we did the calculation, and delta X is the corresponding width in real space

and because this is a Gaussian wave packet it's actually related to the width in momentum space by Heisenberg.

So it's H bar over 2 delta P. And the first term is the one I wrote down, not in this form,

but it's the first term that I wrote down. The second term I omitted because it turns out that in those phase factors

that enter the wave functions, you also have a term that depends both on the momentum and on the force.

And so when you calculate the skater product you have to keep that phase factor because it depends on the momentum

when you carry out the integration. And so if you keep that you get the second term.

Now it turns out that if the dispersion of the wave packets doesn't matter until the overlap has gone to zero,

the second term will not be important. And so that would be the situation that you typically would like to have.

Now I want to go on and discuss this question of reversibility. Remember we had said that if I have one Stern-Gerland apparatus

I can turn the information that was initially in the spin direction into a spatial separation of the two wave packets.

And so it seems like this is already a measurement and the orbital degree of freedom is a little bit like the measurement apparatus.

So it's another degree of freedom which couples to the spin variable that is to be measured.

But we realize that this cannot be the full story because actually I can imagine a setup where after producing these deflected beams

I can undo the deflection. And so by having a suitable arrangement of Stern-Gerland magnets

I can just recombine these two beams and I will end up with exactly the input state and no measurement has been done.

So that is what is known as the quantum eraser. And so then we ask ourselves what else do we need in order to really do a measurement?

And so we briefly came up with the idea that apparently it's still possible, it's still necessary to have this information

that is already in the position converted into a real event, whatever that means. And the idea is that reversibility helps.

And so in order to understand this, let us imagine what happens when such a particle strikes a crystal lattice, for example.

So here's the incoming atom, it will strike the crystal lattice and it will emit sound waves into the crystal lattice.

And the basic idea that I want to tell you about is simply that it's very hard to bring these sound waves back, as you might imagine.

And in order to appreciate why this is important for our task, imagine a slightly different situation.

Again, I have my atom coming along here and I want to arrange it such that this atom reveals its presence by giving a kick to some other atom.

And this other atom I imagine to be harmonically bound to some substrate.

And then I can imagine this atom comes along, it gives a kick to this harmonic oscillator, and afterwards the energy of the harmonic oscillator has increased,

so somehow I have detected the presence of this atom going along because I can now measure that the harmonic oscillator has increased energy.

So that would be the idea. And in principle this is correct.

But you can easily imagine a situation where again this can be undone, just like in our quantum eraser, it's only just a little bit harder.

And what we would need to do is the following.

First, we would have to reverse the velocity of the atom so that it has a chance to come back a second time to my harmonic oscillator,

because it will have to interact a second time in order to undo the measurement, in order to decrease the overlap of the two different states,

chi up and chi down to zero, so to speak.

But that's not everything. What I also need is the following.

If I look at the harmonic oscillator, what happens after this first kick is it just acquires some momentum and it starts to oscillate.

And what I want to arrange is a situation where it comes back exactly to the initial point,

because only then do I have a chance in the subsequent interaction to completely undo this measurement.

And now here for a single oscillator this is still sort of conceivable, although it would be very unlikely to happen by chance.

And what I want to argue is that if I replace the single harmonic oscillator by a full crystal lattice that you can think of as being composed of infinitely many normal modes,

then it becomes not only a little bit unlikely but practically impossible to recover all these sound waves that have been emitted and to refocus them on the spot of input.