If we measure E once and we find Ei as outcome we know that the system is in the ith eigenstate of the Hamiltonian. That certainty means that if we measure the energy again we must find Ei again. This is called the ``collapse of the wave function'': before the first measurement we couldn't predict the outcome of the experiment, but the first measurements prepares the wave function of the system in one particuliar state, and there is only one component left!
Now what happens if we measure two different observables? Say, at
12 o'clock we measure the position of a particle, and a little later
its momentum. How do these measurements relate?
Measuring
to be x0 makes the wavefunction collapse to
(x - x0), whatever it was before. Now mathematically it can
be shown that
| (8.31) |
|
incompatible operators
The reason is that
The way to show this is to calculate
for arbitrary f (x). A little algebra shows that
In operatorial notation,
where the operator
The reason these are now called ``incompatible operators'' is that an
eigenfunction of one operator is not one of the other: if
If |
Now what happens if we initially measure
= x0 with finite
acuracy
x? This means that the wave function collapses to a
Gaussian form,
| (8.37) |
| exp(- (x - x0)2/ |
(8.38) |
| (8.39) |