Linear algebra application
Observables are represented a
linear operators (p. 965). We
also require that this operator
have real eigenvalues.
When we carried over the classical kinetic energy observable to
quantum mechanics, we weren’t going blind. For example, the factor
of−h^2 / 2 min front is tightly constrained by requirements like units
and the need for a traveling sine wave to have positive energy. But
for the superposition of two states, classical mechanics will never
give us any guidance. For example, what is the body temperature
of Schr ̈odinger’s cat? For the energy operators appearing in the
Schr ̈odinger equation, we used linear operators. The result was that
our law of physics was perfectly linear, and this is a hard require-
ment, for the reasons described on p. 915. It therefore seems natural
to require thatallobservables be represented by linear operators,O(Ψ 1 + Ψ 2 ) =OΨ 1 +OΨ 2.Indeed, if they were not linear, then quantum mechanics would lack
self-consistency, for the act of measurement can be described by
applying the Schr ̈odinger equation to a big system consisting of the
system being observed interacting with the measuring device.
Finally, we have one more requirement, which is that the linear
operator representing an observable should have eigenvalues that
are real. This isn’t because the results of a measurement must log-
ically be real — e.g., we can measure complex impedances. But in
any real-world application of the complex number system, we must
always choose some arbitrary phase conventions, such as that an
inductor has a positive imaginary impedance to represent the fact
that the voltage leads the current by 90 degrees. (Such phase con-
ventions are always arbitrary because we defineias√
−1, but this
doesn’t distinguishifrom−i.) These phase conventions are all in-
dependent of one another, and the classical ones are independent of
the convention used for wavefunctions in quantum mechanics, which
is that a state with positive energy twirls clockwise in the complex
plane. (See also example 14, p. 989.)Observables
In quantum mechanics, any observable is represented by a linear
operator that takes a wavefunction as an input and has real eigen-
values.Some important examples of observables are momentum (example
5 below), position (example 7), energy, and angular momentum.
These are represented by linear operatorsOx,Op, OE, and OL,
respectively.
The momentum operator example 5
Quantum mechanics represents motion as a dependence of the
wavefunction on position, so that a constant wavefunction has
no motion. This suggests defining the momentum operator as
the derivative with respect to position. This almost works, but
needs to be tweaked a little. We expect that a state of definite976 Chapter 14 Additional Topics in Quantum Physics