c/A solution to the Schrodinger ̈
equation found by separability.
Positive values are shown as light
colors, negative ones as dark
colors.
But ifXandY are solutions of the one-dimensional equation, then
both terms on the left are constants, so we have a valid solution to
the two-dimensional equation.
As an example, we know that sinkxis a solution to the one-
dimensional Schr ̈odinger equation, so the function sinkxsinky is
also a solution. The result, shown in figure c, can be chopped off
and made into a solution of the two-dimensional particle in a box.
Solutions similar to this one are found in real-life examples such
as microwave photons in a microwave oven. For more about sep-
arability, and how it compares with entanglement, see sec. 14.11,
p. 1004.14.6 The underlying structure of quantum me-
chanics, part 2
14.6.1 Observables
By the time my first-year mechanics students have been in class
for a week, they know how to answer when I ask them the velocity
of the tape dispenser at the front of the classroom: “We don’t know,
it depends on your frame of reference.” Theabsolutevelocity of an
object is a meaningless concept, part of the mythical dungeons-and-
dragons cosmology of Aristotelian physics. Quantum mechanics is
as great a break from Newton as Newton was from Aristotle, and
similar care is required in redefining what concepts areobservables
— meaningful things to talk about measuring.
Classically, we describe the state of the system as a point in phase
space (sec. 5.4.2, p. 328) — which is just a fancy way of saying that
we specify all the positions and momenta of its particles — and an
observable is defined as a function that takes that point as an input
and produces a real number as an output. (By the way, the word
“phase” in “phase space” doesn’t refer directly to the phase of a
wave, which we’ll also be discussing below.) For example, kinetic
energy is a classical observable, andK( ) = 0, where the picture
represents a tennis ball at rest. For a moving tennis ball with one
unit of energy,K( ) = 1. For a vibrating violin string, we could
haveU( ) = 1, andU( ) = 4 (where doubling the amplitude
gives four times the energy).
Quantum-mechanically, the Heisenberg uncertainty principle tells
us that we can’t independently dial in the desired values of a par-
ticle’s position and momentum. They aren’t two variables that are
independent of one another. Therefore we don’t have a phase space,
so an observable has to be represented by a function whose input is
a wavefunction. Furthermore, we expect that:- The output shouldn’t depend on the phase^1 of the wavefunc-
(^1) “Phase” as in the phase of a wave, not as in “phase space.”
974 Chapter 14 Additional Topics in Quantum Physics