How can we know initial state of a particle

[bob900]

In textbooks (such as Griffiths, for example), the general method for a solution to the Shrodinger's equation for a single particle, for some V(x), is given as : 1. Get the stationary state solutions 2. Combine them into a linear combination and figure out the coefficients from the known initial state at time 0, ψ(x,0).

But it never states how exactly do we know this initial ψ(x,0) in the first place? If we are conducting an experiment with the particle, how do we prepare it to be in exactly the right initial state?

 

[capandbells]

If you can get ahold of Ballentine's QM book, Ch. 8 explains this well.

 

[bhobba]

If you can get ahold of Ballentine's QM book, Ch. 8 explains this well.

Yes. As well it develops QM in a much more logical way. The only issue is it is a graduate level book. It would be quite reasonable to just accept at the level of Griffiths there are a few issues that get resolved in more advanced treatments and wait until you study them.

I was concerned about such things when I first studied QM from books like Griffiths many moons ago. I took long detours sorting them out and while I learned a lot and managed to resolve the issues, I now think it was probably better simply to wait until I was ready for the more advanced texts.

Thanks
Bil

 

[kith]

Preparations are measurements. A typical QM experiment goes like this:

1) you have an unknown initial state
2) you prepare it by a measurement (the postulates of QM guarantee you that you know the state afterwards)
3) you let your system evolve in time
4) you do your measurement of interest

 

[bob900]

Preparations are measurements. A typical QM experiment goes like this:

1) you have an unknown initial state
2) you prepare it by a measurement (the postulates of QM guarantee you that you know the state afterwards)
3) you let your system evolve in time
4) you do your measurement of interest

But do you really know the state fully after step 2? Let's say you have a detector that measures the position of a particle (in one dimension). Any detector has some finite accuracy, so the result of the measurement only tells you that the particle was found in some (very small) range [x1,x2]. In other words, you know that

ψ(x,0) = 0 | x outside of [x1,x2]

but ψ(x,0) is still completely unspecified inside [x1,x2]. There are infinitely many ways to specify it inside there. Which one do you use for your calculations thereafter?

 

[bhobba]

But do you really know the state fully after step 2?

No practical measurement will allow you to know the state after the measurement exactly - but to good approximation for most if not all theoretical calculations assuming it is in an exact eigenstate is good enough. It's the principle that counts - namely conceptually it is possible.

There are other ways of determining the initial state as well such as what is called filtering - see the Chapter 8 mentioned previously. But my advice is not to worry about it for now - getting sidetracked on this sort of stuff can be time consuming and counterproductive to the aims of a book at the level of Griffiths.

Thanks
Bill