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1.8 Physical interpretation of the wave function 33
orientation of the magnetic ®eld gradient, vertical (up or down), horizontal (left
or right), or any angle in between, the wave function of the atom is always
given by equation (1.52) with á parallel and â antiparallel to the magnetic ®eld
gradient. Since the atomic magnetic moments are initially randomly oriented,
half of the wave functions collapse to á and half to â.
In the Stern±Gerlach experiment with two magnets having parallel magnetic
®eld gradients±the `®rst arrangement' described in Section 1.7±all the atoms
entering the second magnet are in state á and therefore are all de¯ected in the
same direction by the second magnetic ®eld gradient. Thus, it is clear that the
wave function Ø before any interaction is permanently changed by the inter-
action with the ®rst magnet.
In the `second arrangement' of the Stern±Gerlach experiment, the atoms
emerging from the ®rst magnet and entering the second magnet are all in the
same state, say á. (Recall that the other beam of atoms in state â is blocked.)
The wave function á may be regarded as the weighted sum of two states á9
and â9
á c9 á á9 c9 â â9
where á9 and â9 refer to states with atomic magnetic moments parallel and
antiparallel, respectively, to the second magnetic ®eld gradient and where c9 á
and c9 â are constants related by
2
2
jc9 á j jc9 â j 1
In the `second arrangement', the second magnetic ®eld gradient is perpendicu-
lar to the ®rst, so that
2 2 1
jc9 á j jc9 â j
2
and
1
á p (á9 â9)
2
The interaction of the atoms in state á with the second magnet collapses the
wave function á to either á9 or â9 with equal probabilities.
In the `third arrangement', the right beam of atoms emerging from the
second magnet (all atoms being in state á9), passes through a third magnetic
®eld gradient parallel to the ®rst. In this case, the wave function á9 may be
expressed as the sum of states á and â
1
á9 p (á â)
2
The interaction between the third magnetic ®eld gradient and each atom
collapses the wave function á9 to either á or â with equal probabilities.
The interpretation of the various arrangements in the Stern±Gerlach experi-
