Postulates of quantum mechanics 375the quantum mechanical position and momentum operators, respectively, then these
will satisfy eigenvalue equations of the form
xˆ|x〉=x|x〉, pˆ|p〉=p|p〉, (9.2.37)wherexandpare the continuous eigenvalues. In place of the discrete orthonormality
and completeness relations, we have continuous analogs, which take the form
〈x|x′〉=δ(x−x′), 〈p|p′〉=δ(p−p′)∫
dx|x〉〈x|=I,ˆ∫
dp|p〉〈p|=Iˆ|φ〉=∫
dx|x〉〈x|φ〉, |φ〉=∫
dp|p〉〈p|φ〉. (9.2.38)The last line shows how to expand an arbitrary vector|φ〉in terms of the position or
momentum eigenvectors.
Quantum mechanics postulates that the position and momentum of aparticle
are not compatible observables. That is, no experiment can measure both properties
simultaneously. This postulate is known as theHeisenberg uncertainty principleafter
the German physicist Werner Heisenberg (1901–1976) and is expressed as a relation
between the statistical uncertainties ∆x≡
√
〈xˆ^2 〉−〈xˆ〉^2 and ∆p≡√
〈pˆ〉^2 −〈pˆ〉^2 ,
namely
∆x∆p≥̄h
2. (9.2.39)
Since ∆xand ∆pare inversely proportional, the more certainty we have about a
particle’s position, the less certain we are about its momentum, and vice versa. Thus,
any experiment designed to measure a particle’s position with a small uncertainty
must cause a large uncertainty in the particle’s momentum. The uncertainty principle
also tells us that the concepts of classical microstates and phase spaces are fictions,
as these require a specification of a particle’s position and momentumsimultaneously.
Thus, a point in phase space cannot correspond to anything physical. The uncertainty
principle, therefore, supports the idea of a “coarse-graining” ofphase space, which was
considered in Problem 2.6 and in Section 3.2. A two-dimensional phase space should
be represented as a tiling with squares of minimum area ̄h/2. These squares would
represent the smallest area into which the particle’s position and momentum can be
localized. Similarly, the phase space of anN-particle system should be coarse-grained
into hypervolumes of size ( ̄h/2)^3 N. In the classical limit, which involves letting ̄h→0,
we recover the notion of a continuous phase space as an approximation.
The action of the operators ˆxand ˆpon an arbitrary Hilbert-space vector|φ〉can
be expressed in terms of a projection of the resulting vector ontothe basis of either
position or momentum eigenvectors. Consider the vector ˆx|φ〉and multiply on the left
by〈x|, which yields〈x|xˆ|φ〉. Since〈x|xˆ=〈x|x, this becomesx〈x|φ〉. Remembering
that the eigenvaluexis continuous, the vectors|x〉form a continuous set of vectors,
and hence, the inner product〈x|φ〉is a continuous function ofx, which we can denote