Okay, hello. Last time we discussed the probability interpretation of the wave function and we

found that while in some situations you can have quite a naive picture that would naturally

give rise to this probability interpretation. If you want to apply it to more complicated

settings such as interference setups, then you run into real trouble as to what really

goes on in such interference setups. Can I ascribe a trajectory to a particle going through

only one slit but not the other? And the fact that this gives rise to all kinds of trouble

then in the standard Copenhagen interpretation led to the dictum that you are not allowed

to have trajectories in quantum mechanics. You are not allowed to think of the position

and the momentum of a particle to be real before the measurement. And in the measurement

you have to choose either one or the other. Now before I want to end this chapter, let

me still recall how the probability interpretation then is also carried over to many particle

situations. So what do we have in terms of a many particle wave function? I already briefly

mentioned this when we discussed the Schrödinger equation itself. So the many body wave function

is just a wave function that depends on all the positions simultaneously. And this might

be natural if you come from classical statistical physics. It might be unnatural if you come

from another point of view but that's the way it is. So you could have thought that

well if one particle is described by one meta wave then maybe if I have ten particles I

have just ten different meta waves. But this is not the correct approach. If this were

the correct approach then it would be relatively easy to cook up a simple underlying microscopic

theory for quantum mechanics and make sense of it at all. But rather what we have is one

big wave function describing all the ten million particles at the same time and this is a complex

function of all the coordinates simultaneously. And so you would call this space in which

all the coordinates live simultaneously configuration space. Configuration space is also where you

would formulate classical mechanics but still in classical mechanics it's different. It's

just an artificial mathematical description but you can still think pictorially of each

particle individually moving along its individual trajectory whereas apparently in quantum mechanics

you are not allowed to do this. Now if you come to the probability interpretation this

is fairly simple. So initially we said that psi of x squared is the probability density

of finding the particle of position x in a measurement. Now here apparently since the

wave function depends on all coordinates simultaneously the real question to ask is if I try to measure

all the particle positions at the same instant then what is the configuration that I will

do? So you would then square this wave function that depends on all the particle positions

and by that you would get the probability of finding particle number one at this spot,

particle number two at that spot, particle number three maybe close by and so on. Now

more precisely of course just as for the single particle case this is the probability density

so really the probability to find them at exactly these locations would be zero but

we can give them a little bit of freedom by prescribing these intervals and then I have

to multiply this density by the size of the intervals in order really to get the probability

to find this configuration within this freedom that I have allowed. And so I have drawn it

here like in one dimension where all the particles would lie on a line but of course we can have

the same in multiple dimensions so if I take any system of many electrons or atoms what

the multi-particle wave function gives me is the amplitude or the probability density

for finding this configuration or the other exponentially many configurations that I can

have. And then you can extend it to fields you can also ask what is the probability density

to find a field in exactly this configuration and again of course you have to allow for

a little bit of freedom in the field amplitudes and then you can extend the concept to fields.

So this is nice and this is not so surprising if you come from classical statistical physics

because there you would also say my Boltzmann distribution or my canonical distribution

tells me what is the probability to find my n particles in just this configuration if

I were to measure them. And so again there the probability density is a probability density