366 Quantum mechanics
the detector in some small region. This observation suggests an object that exhibits
“wave-like” behavior over one that follows a precise particle-like trajectory predictable
from a deterministic equation of motion.
The notions of energy quantization, unpredictability of individual experimental
outcomes, and particle–wave duality are aspects of the modern theory of the micro-
scopic world known asquantum mechanics. Yet even this particle/wave description is
incomplete. For what exactly does it mean for a particle to behave likea wave and
a wave to behave like a particle? To answer this, we need to specify more precisely
what we mean by “wave” and “particle.” In general, a wave is a type offield describing
something that can vary over an extended region of space as a function of time. Exam-
ples are the displacement of a plucked string over its length or the airpressure inside
of an organ pipe. Mathematically, a wave is described by anamplitude,A(x,t) (in
one dimension) that depends on both space and time. In classical wave mechanics, the
form ofA(x,t) is determined by solving the (classical) wave equation. Quantum theory
posits that the probability of an experimental outcome is determined from a partic-
ular “wave” that assigns to each possible outcome a (generally complex)probability
amplitudeΨ. If, for example, we are interested in the probability that a particle will
strike a detector at a locationxat timet, then there is an amplitude Ψ(x,t) for this
outcome. From the amplitude, the probability that the particle will strike the detector
in a small region dxabout the pointxat timetis given byP(x,t)dx=|Ψ(x,t)|^2 dx.
Here,
P(x,t) =|Ψ(x,t)|^2 (9.1.1)is known as theprobability densityorprobability distribution. Such probability am-
plitudes are fundamental in quantum mechanics because they directly relate to the
possible outcomes of experiments and lead to predictions of average quantities ob-
tained over many trials of an experiment. These averages are known asexpectation
values. The spatial probability amplitude, Ψ(x,t), is determined by a particular type
of wave equation known as the Schr ̈odinger equation. As we will seeshortly, the frame-
work of quantum mechanics describes how to compute the probabilities and associated
expectation values of any type of physical observable beyond thespatial probability
distribution.
We now seek to understand what is meant by “particle” in quantum mechanics. A
particularly elegant description was provided by Richard Feynman in the context of his
path integralformalism (to be discussed in detail in Chapter 12). As we noted above,
the classical notion that particles follow precise, deterministic trajectories breaks down
in the microscopic realm. Indeed, if an experiment can have many possible outcomes
with different associated probabilities, then it should follow that a particle can follow
many different possible paths between the initiation and detection points of an ex-
perimental setup. Moreover, it must trace all of these pathssimultaneously! In order
to build up a probability distributionP(x,t), the different paths that a particle can
follow will have different associated weights or amplitudes. Since the particle evolves
unobserved between initiation and detection, it is impossible to conclude that a parti-
cle follows a particular path from one point to the other, and according to Feynman’s
concept, physical predictions can only be made by summing overallpossible paths