Question

3, Explain a single quantum measurement process. (10%)

Answer 1

We consider an experiment in which the position is accurately known at the beginning and the momentum is measured. We shall see that the measurement gives an inaccurate momentum but also introduces an uncertainty into the position.

We assume that the particle is an atom in an excited state, which will give off a photon that has the frequency $\nu _{0}$ if the atom is at rest.Because of the doppler effect, motion of the atom toward the observer with speed v means that the observed frequency is given approximately by

$\nu =\nu _{0}(1+v/c)$ so that $v\approx c(\nu /\nu _{0}-1)$ .

Accurate measurement of the momentum mv by measurement of the frequency $\nu$ requires a relatively long time $\tau$; the minimum error in the frequency measurement can be shown to be $\Delta \nu \sim 1/\tau$ .

The instant at which the photon is emitted is uncertain by $\tau$; at this instant the momentum of the atom decreases by $h\nu /c$, and its velocity decreases by $h\nu /mc$. This makes the subsequent position of the atom uncertain by the amount

$\Delta x=h\nu \tau /mc$ eq 1).since the later the photon is emitted, the longer the atom has the higher velocity and the farther it would have travelled.

This position uncertainty is because of the finiteness of $\tau$.

The momentum uncertainty is $\Delta p_{x}=m\Delta v\approx mc\Delta \nu /\nu _{0}\sim mc/\nu _{0}\tau$ eq 2).

The combination of eq 1) and eq 2) leads to Heisenberg's uncertainty relation.