Okay, now let me just repeat what we have been doing. We have been discussing the Einstein-Podolski-Rosen

Gedanken experiment and you can remember what was the inspiration for this because if we like to have

a microscopic theory that sort of explains quantum mechanics just like Newtonian mechanics explains

statistical physics, then it's obviously a challenge if we are not allowed to talk about the position

and the momentum of a particle at the same time, if we are not allowed to talk about trajectories,

if we are not allowed to talk about what goes on before the measurement. And all the attempts of

Einstein to show that you could indeed talk about position and momentum at the same time had been refuted

by Bohr and various Gedanken experiments. So here this was a new attempt and the new trick was to take

two particles and to put them in such a quantum state such that if you measure the position of one of the

particles you can deduce what the position of the other particle should be, but then you can still choose

to say measure the momentum and since your deduction of the position cannot possibly have perturbed in any way

the second particle, you are sure that somehow, at least according to this argument, you are able to talk

about both position and momentum before the measurement. That doesn't change anything about the usual

situation that once you measure the position you very much perturb the momentum or once you measure the

momentum you very much perturb the position, but it is intended to show that you are allowed at least

to think conceptually of position and momentum being there even before the measurement.

And so the state they looked at was very simple. You have two particles saying one dimension and the state

they took was just like this. So whenever you measure particle position x1 you are sure that in a subsequent

measurement of particle position x2 you will get the same value. On the other hand though you can also

write this as an integral over our momentum and then an exponential and I can write this exponential as a

product of two exponentials, one of them relating to each of the particles and so that would be e to the plus

e over h bar x1 and e to the minus i p over h bar x2. So you know this is elementary mathematics that this

integral over our momentum will give you the delta function and the difference of the positions.

On the other hand this is just a plane wave for particle number one with momentum p and this is just a plane wave

with momentum minus p. So you will see that the wave function is just a superposition of many different

combinations of plane waves but each of those combinations has opposite and equal momentum for the two particles.

So then in summary there is x1 equals x2 if you choose to measure the positions and there is p1 equals minus p2

if you choose to measure the momentum and then the reasoning goes well for example I might choose to measure

the position of particle number one and then I can as I just said immediately deduce the position of particle

number two but since I already can deduce it with certainty and I can always check that this is really right

I can as well measure momentum of particle number two and then I know both momentum and position or also I can

deduce things the other way once I have measured the momentum of particle number two I can deduce the momentum

of particle number one and I also have measured the position of particle number one so in the end I really know

everything either indirectly by deduction or directly by measurement. And since this is the case the only reasonable

interpretation of this seems to be that well these properties had been there all along even before the measurement

and the only reason that in quantum mechanics I cannot measure both of them with infinite precision at the same time

is somehow that I always invariably perturb them in a measurement that is still admitted but you are allowed to talk

about them before the measurement. Okay and if you are allowed to do that but quantum mechanics itself never gives you

a complete description of both position and momentum at the same time that should mean that quantum mechanics

is somehow incomplete and the challenge is then to find a more microscopic complete theory. That was the reasoning

of Einstein-Gutertzky and Bose. Now when this was published in 1935 it sent Bohr into thinking very hard

about this and it turned out that he didn't really come up with a very convincing answer. So if you read his paper

most of the time he spends in repeating those simple gedankene experiments that show that for a single particle

you can never possibly know both position and momentum at the same time and then he says well you see for a single particle

this is because I have to make a choice. Either I measure position then I use a certain kind of setup or I measure momentum

then I use a completely different kind of setup and I have to make this choice and so I can only get position or momentum

it's only either or and so I contend that since this is always the case we are not allowed to think of position

and momentum existing even before the measurement they become real only via the measurement. And then he just goes on

and says well you see basically the situation here is not really different so I should consider the whole combination

of settings that I choose. So for example if I choose to measure position here and position there that's one kind of