Linear algebra application
A basis is one of the most fun-
damental concepts in linear al-
gebra (p. 965). We don’t al-
ways need to choose a basis,
and if we do, the choice is ours
to make.b/A beam of oxygen molecules,
with`= 1, is filtered through two
Stern-Gerlach spectrometers.Completeness
For any system of interest, there exists a set of compatible observ-
ables, called a complete set, such that any state of the system can be
expressed as a sum of wavefunctions having definite values of these
observables.The completeness postulate was discussed at a more elementary level
in section 13.4.3, p. 923.
The set of wavefunctions referred to above is called a basis. (The
terminology comes from linear algebra.) If we require normalization
and ignore the undetectable phase, then choosing a complete set
of observables is equivalent to choosing a basis. Therefore “choice
of basis” and “choice of a complete set of observables” are nearly
synonyms, so we will usually use the shorter phrase. Normally there
is more than one possible choice of basis. The choice from among
these possibilities is arbitrary, and nature doesn’t care which one
we pick. That is, there isno preferred basis. An example of this
principle is the fact that we habitually talk about “up” and “down”
for the spin of an electron, which we are free to do, although it would
be equally permissible to talk about left and right. Another good
example is the discussion of the double degeneracy of the quantum
moat on p. 920, where we were free to talk about a basis consisting
of either two standing waves or two traveling waves.
As an example of the completeness principle, we have seen in
the example in fig. d, p. 923, that for a rotor, the state with= 1 andx= 0 can be written as a sum of the states withz =− 1 andz= 1. In the language of the completeness postulate, we can
express this as follows. Let our system be the set of possible states
of a rotor. The observablesLandLzare compatible, and they turn
out (although we will not prove it here) to be a complete set of
observables for this system. The completeness postulate is satisfied
in this example because the state withx= 0 can be expressed as |z=− 1 〉/
√
2 +|`z= 1〉/√
2.
Translating this scenario into a hypothetical real-world exper-
iment, suppose that, as in figure b, we pass a beam of randomly
oriented oxygen molecules (referred to as an unpolarized beam)
through a Stern-Gerlach spectrometer that disperses them into beams
withx=−1, 0, and +1. All three states are present, and in fact the beam is split into three beams of equal intensity, 1/3 that of the orig- inal beam.^7 Then we throw away all but the molecules havingx=
0, and pass these through a second spectrometer, this one select-
equally likely to be anywhere in the box, i.e., its wavefunction is supposed to be
constant throughout the box. Ignoring normalization, this constant wavefunc-
tion can be expressed as an infinite series in terms of the states of definite energy
as +^13 +^15 +...This kind of representation of a function as an infinite
sum of sine waves is called a Fourier series.
(^7) The equality of these three intensities is not obvious geometrically, but be-
Section 14.6 The underlying structure of quantum mechanics, part 2 987