🚀 go-pugleaf

This is a common misconception in quantum mechanics. It's taught in most
undergraduate courses and it's only in specialist courses on theoretical
physics this seems to be covered. The fact is that ANY two non-commuting
variables CANNOT be measured simultaneously - with reprdoducible results.

This is easily explained, (I'll miss out the maths for those who havent
studied mch linear maths/ QM maths formalism).

When we make a measurement of a quantum system in state A (which we assume
to be a linear superposition of states) the system "collapses" after the
measurement into an eigenvector of variable A. If we now look at a variable
B, incompatible with A, we find that the list of possible eigenstates for
this variable does not coincide with those of A; i.e they have no common
eigenvectors. Whenever we make a measurement of A we drive the system into
an eigenvector of A; when we make a measurement of B we similarly drive the
state into an eigenvector of B.

If the eigenvectors do no co-incide (non-commuting or Incompatible
variables) then simultaneous measurement is impossible as we cannot drive
the state into two states at once.



In reality we can, however, force the measurement. Say we make push a system
into stationary eigenstate X of position variable. When we measure position
we always get the same measured value x1. If we now make a simultaneous
measurement of position and momentum we will likel find position = x1,
momentum = p1. If we repeat this experiment many times we find that x1 will
stay constant but that the value p1 will be completely random. The
statistical spread of p1 will likely be infinite. It seems therefore that
mathematically, and physically this type of expreiment is impossible. (If
you can try to get hold of a paper by Prof. Ulf Leonhardt [my quantum optics
lecturer - in the news a lot due to his work on sonic black holes] you'll
see that such measurements might be possible if one is very "fuzzy" as to
what simultaneous measurements means).

As for the bubble chamber, it is clear that the idea of "trajectory" in
quantum mechanics is meaningless. The properties from a bubble chamber are
inferred but simultaneous measurement cannot take place.

[and as for your question "how do you make a position measurement?"

- well I don't - I'm afraid I wouldn't know how :o) I'm a theoretician and
leave that up to people who can carry out experiments without breaking the
equipment :o) ]

Have fun

Bob




"Charles Francis" <charles@clef.demon.co.uk> wrote in message
news:<nxRwN7AhJDn8EwnG@clef.demon.co.uk>...

> In message <a77sai$4uf$1@sparkie.st-andrews.ac.uk>, bp6

> <bp6@st-andrews.ac.uk> writes

> >This was incorrect; clearly (as shown by Jon Von Neumann)

> >simultaneous measurements of momentum and position is impossible in

> >the first place. Therefore there is no limitng accuracy for such

> >impossible measurements.

>

> And how can you measure the momentum of a particle if you do not know

> where it is? Any such measurement, e.g. in a cloud chamber, is a

> simultaneous measurement of momentum and position.

>

>

> Regards

>